@@ -1,6755 +1,6755 @@ -REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... - - -% -% Testing file for REDUCE Special Functions Package -% -% Chris Cannam, ZIB Berlin -% October 1992 -> Feb 1993 -% (only some of the time, of course) -% -% Corrections and comments to neun@sc.zib-berlin.de -% - - -on savesfs; - - % just in case it's off for some reason - -off bfspace; - - % to provide more similarity between runs - % with old & new bigfloats - -let {sinh (~x) => (exp(x) - exp (-x))/2 }; - - - % this will improve some results - - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - -% 1. Bernoulli numbers - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - - -off rounded; - - - -procedure do!*one!*bern(x); - write ("Bernoulli ", x, " is ", bernoulli x); - - -do*one*bern - - -do!*one!*bern(1); - - - - 1 -Bernoulli 1 is ------ - 2 - -do!*one!*bern(2); - - - 1 -Bernoulli 2 is --- - 6 - -do!*one!*bern(3); - - -Bernoulli 3 is 0 - -do!*one!*bern(13); - - -Bernoulli 13 is 0 - -do!*one!*bern(14); - - - 7 -Bernoulli 14 is --- - 6 - -do!*one!*bern(300); - - -Bernoulli 300 is ( - -1863878995204859011995045341848156066182191846635905937518715320655775958174360523134990756922303410810482600528769479642021001218415879006164302955370460829146434807964717737195356935144151583424833154250047747433575584999029126775186293388721514970183351129809976971603227633930434923843984829580311593372565398574762880028289167635570012415606941367995702212211519561707046505473575241 - )/ - - 866054419230 - -do!*one!*bern(-2); - - -Bernoulli -2 is bernoulli(-2) - -do!*one!*bern(0); - - -Bernoulli 0 is 1 - - -for n := 2 step 2 until 100 do do!*one!*bern n; - - - 1 -Bernoulli 2 is --- - 6 - - - 1 -Bernoulli 4 is ------ - 30 - - 1 -Bernoulli 6 is ---- - 42 - - - 1 -Bernoulli 8 is ------ - 30 - - 5 -Bernoulli 10 is ---- - 66 - - - 691 -Bernoulli 12 is -------- - 2730 - - 7 -Bernoulli 14 is --- - 6 - - - 3617 -Bernoulli 16 is --------- - 510 - - 43867 -Bernoulli 18 is ------- - 798 - - - 174611 -Bernoulli 20 is ----------- - 330 - - 854513 -Bernoulli 22 is -------- - 138 - - - 236364091 -Bernoulli 24 is -------------- - 2730 - - 8553103 -Bernoulli 26 is --------- - 6 - - - 23749461029 -Bernoulli 28 is ---------------- - 870 - - 8615841276005 -Bernoulli 30 is --------------- - 14322 - - - 7709321041217 -Bernoulli 32 is ------------------ - 510 - - 2577687858367 -Bernoulli 34 is --------------- - 6 - - - 26315271553053477373 -Bernoulli 36 is ------------------------- - 1919190 - - 2929993913841559 -Bernoulli 38 is ------------------ - 6 - - - 261082718496449122051 -Bernoulli 40 is -------------------------- - 13530 - - 1520097643918070802691 -Bernoulli 42 is ------------------------ - 1806 - - - 27833269579301024235023 -Bernoulli 44 is ---------------------------- - 690 - - 596451111593912163277961 -Bernoulli 46 is -------------------------- - 282 - - - 5609403368997817686249127547 -Bernoulli 48 is --------------------------------- - 46410 - - 495057205241079648212477525 -Bernoulli 50 is ----------------------------- - 66 - - - 801165718135489957347924991853 -Bernoulli 52 is ----------------------------------- - 1590 - - 29149963634884862421418123812691 -Bernoulli 54 is ---------------------------------- - 798 - - - 2479392929313226753685415739663229 -Bernoulli 56 is --------------------------------------- - 870 - - 84483613348880041862046775994036021 -Bernoulli 58 is ------------------------------------- - 354 - - - 1215233140483755572040304994079820246041491 -Bernoulli 60 is ------------------------------------------------ - 56786730 - - 12300585434086858541953039857403386151 -Bernoulli 62 is ---------------------------------------- - 6 - - - 106783830147866529886385444979142647942017 -Bernoulli 64 is ----------------------------------------------- - 510 - - 1472600022126335654051619428551932342241899101 -Bernoulli 66 is ------------------------------------------------ - 64722 - - - 78773130858718728141909149208474606244347001 -Bernoulli 68 is ------------------------------------------------- - 30 - - 1505381347333367003803076567377857208511438160235 -Bernoulli 70 is --------------------------------------------------- - 4686 - - - 5827954961669944110438277244641067365282488301844260429 -Bernoulli 72 is ------------------------------------------------------------ - 140100870 - - 34152417289221168014330073731472635186688307783087 -Bernoulli 74 is ---------------------------------------------------- - 6 - - - 24655088825935372707687196040585199904365267828865801 -Bernoulli 76 is ---------------------------------------------------------- - 30 - - 414846365575400828295179035549542073492199375372400483487 -Bernoulli 78 is ----------------------------------------------------------- - 3318 - -Bernoulli 80 is ( - - - 4603784299479457646935574969019046849794257872751288919656867)/ - - 230010 - - 1677014149185145836823154509786269900207736027570253414881613 -Bernoulli 82 is --------------------------------------------------------------- - 498 - -Bernoulli 84 is ( - - -2024576195935290360231131160111731009989917391198090877281083932477 - )/3404310 - -Bernoulli 86 is 660714619417678653573847847426261496277830686653388931761996983/ - - 6 - -Bernoulli 88 is ( - - -1311426488674017507995511424019311843345750275572028644296919890574047 - )/61410 - -Bernoulli 90 is - -1179057279021082799884123351249215083775254949669647116231545215727922535 - /272118 - -Bernoulli 92 is ( - - -1295585948207537527989427828538576749659341483719435143023316326829946247 - )/1410 - -Bernoulli 94 is - -1220813806579744469607301679413201203958508415202696621436215105284649447 - /6 - -Bernoulli 96 is ( - -211600449597266513097597728109824233673043954389060234150638733420050668349987259 - )/4501770 - -Bernoulli 98 is - -67908260672905495624051117546403605607342195728504487509073961249992947058239 - /6 - -Bernoulli 100 is ( - -94598037819122125295227433069493721872702841533066936133385696204311395415197247711 - )/33330 - - -on rounded; - - -precision 100; - - -12 - - -do!*one!*bern(1); - - -Bernoulli 1 is - 0.5 - -do!*one!*bern(2); - - -Bernoulli 2 is 0.166666666666666666666666666666666666666666666666666666666666666 - - 6666666666666666666666666666666666667 - -do!*one!*bern(3); - - -Bernoulli 3 is 0 - -do!*one!*bern(13); - - -Bernoulli 13 is 0 - -do!*one!*bern(14); - - -Bernoulli 14 is 1.16666666666666666666666666666666666666666666666666666666666666 - - 6666666666666666666666666666666666667 - -do!*one!*bern(300); - - -Bernoulli 300 is - 2.1521499732799868297198173767560881985734565727838067018493 - - 43925846796995824343474766557085976828531e+375 - -do!*one!*bern(-2); - - -Bernoulli -2 is bernoulli(-2) - -do!*one!*bern(0); - - -Bernoulli 0 is 1 - -do!*one!*bern(38); - - -Bernoulli 38 is 4.88332318973593166666666666666666666666666666666666666666666666 - - 6666666666666666666666666666666666667e+14 - -do!*one!*bern(400); - - -Bernoulli 400 is - 6.8469448558064533606162585823108835976782300971806257424147 - - 81150311357197834589950328990573681886442e+549 - - - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - -% 2. Gamma function - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - - -on rounded; - - -off complex; - - -precision 40; - - -100 - - -algebraic procedure wg(x); - write ("gamma (", x, ") ==> ", gamma x); - - -wg - - -algebraic procedure wp(x); - write ("-- precision ", x, ", from ", precision(x)); - - -wp - - -wg (1/2); - - -gamma (0.5) ==> 1.772453850905516027298167483341145182798 - -wg (3/2); - - -gamma (1.5) ==> 0.8862269254527580136490837416705725913988 - - -write ("sqrt(pi)/2 ==> ", sqrt(pi)/2); - - -sqrt(pi)/2 ==> 0.8862269254527580136490837416705725913988 - - -wp(10); - - --- precision 10, from 40 - - -for x := 0 step 5 until 100 do - << wg (1 + x/1000); - wg (-1 - x/13); - wp (8+floor(x/4)) >>; - - -gamma (1) ==> 1 - -gamma (-1) ==> gamma(-1) - --- precision 8, from 10 - -gamma (1.005) ==> 0.99713854 - -gamma ( - 1.3846154) ==> 2.7320314 - --- precision 9, from 8 - -gamma (1.01) ==> 0.994325851 - -gamma ( - 1.76923077) ==> 2.89933597 - --- precision 10, from 9 - -gamma (1.015) ==> 0.9915612888 - -gamma ( - 2.153846154) ==> - 2.919307224 - --- precision 11, from 10 - -gamma (1.02) ==> 0.98884420326 - -gamma ( - 2.5384615385) ==> - 0.91247160689 - --- precision 13, from 11 - -gamma (1.025) ==> 0.9861739631483 - -gamma ( - 2.923076923077) ==> - 2.407817725014 - --- precision 14, from 13 - -gamma (1.03) ==> 0.98354995055382 - -gamma ( - 3.3076923076923) ==> 0.42665848359037 - --- precision 15, from 14 - -gamma (1.035) ==> 0.980971560550586 - -gamma ( - 3.69230769230769) ==> 0.250121998146955 - --- precision 16, from 15 - -gamma (1.04) ==> 0.9784382009142447 - -gamma ( - 4.076923076923077) ==> - 0.4868211588210416 - --- precision 18, from 16 - -gamma (1.045) ==> 0.975949291822951489 - -gamma ( - 4.46153846153846154) ==> - 0.064315864992688343 - --- precision 19, from 18 - -gamma (1.05) ==> 0.9735042655627756432 - -gamma ( - 4.846153846153846154) ==> - 0.07308480130893001268 - --- precision 20, from 19 - -gamma (1.055) ==> 0.9711025662416699039 - -gamma ( - 5.2307692307692307692) ==> 0.026504298643014994546 - --- precision 21, from 20 - -gamma (1.06) ==> 0.968743649511638364209 - -gamma ( - 5.61538461538461538462) ==> 0.00947958151841097406813 - --- precision 23, from 21 - -gamma (1.065) ==> 0.96642698229883993296188 - -gamma (-6) ==> gamma(-6) - --- precision 24, from 23 - -gamma (1.07) ==> 0.964152042541366448869499 - -gamma ( - 6.38461538461538461538462) ==> - 0.00224562428754660955672785 - --- precision 25, from 24 - -gamma (1.075) ==> 0.9619183189344448686422338 - -gamma ( - 6.769230769230769230769231) ==> - 0.0014913552210721080976799 - --- precision 26, from 25 - -gamma (1.08) ==> 0.95972531068282223532653464 - -gamma ( - 7.1538461538461538461538462) ==> 0.0009821350464726696000373901 - --- precision 28, from 26 - -gamma (1.085) ==> 0.9575725272601010524841387298 - -gamma ( - 7.538461538461538461538461538) ==> 0.0002081367552774423831456972591 - --- precision 29, from 28 - -gamma (1.09) ==> 0.95545948817480124076971838245 - -gamma ( - 7.9230769230769230769230769231) ==> 0.000383724703597219454150804519 - -77 - --- precision 30, from 29 - -gamma (1.095) ==> 0.95338572274293305256571929565 - -gamma ( - 8.30769230769230769230769230769) ==> - - - 0.0000487302698231202992921248405302 - --- precision 31, from 30 - -gamma (1.1) ==> 0.9513507698668731836292487177265 - -gamma ( - 8.692307692307692307692307692308) ==> - - - 0.00002092711689267207910068115762837 - --- precision 33, from 31 - - -wg(1/1000000003); - - -gamma (0.000000000999999997000000008999999973) ==> 1.0000000024227843360875231 - -3084681e+9 - - -off rounded; - - - -gamma(17/2); - - - 2027025*sqrt(pi) ------------------- - 256 - -gamma(-17/2); - - - - 512*pi -------------------- - 34459425*sqrt(pi) - -gamma(4); - - -6 - -gamma(0); - - -gamma(0) - -gamma(-4); - - -gamma(-4) - -gamma(-17/3); - - - 6*pi ------------------------- - 17 - 17*sqrt(3)*gamma(----) - 3 - - -p := gamma(x**2) * gamma(x-y**gamma(y)) - (1/(gamma(4*(x-y)))); - - - 2 gamma(y) - gamma(x )*gamma( - y + x)*gamma(4*x - 4*y) - 1 -p := -------------------------------------------------------- - gamma(4*x - 4*y) - -y := 1/4; - - - 1 -y := --- - 4 - -p; - - - 2 1 gamma(1/4) - gamma(x )*gamma( - (---) + x)*gamma(4*x - 1) - 1 - 4 ------------------------------------------------------------- - gamma(4*x - 1) - -x := 3; - - -x := 3 - -p; - - - 1 gamma(1/4) - 146313216000*gamma( - (---) + 3) - 1 - 4 ------------------------------------------------- - 3628800 - -y := -3/8; - - - - 3 -y := ------ - 8 - -p; - - -(128*(2490343877896875*sqrt(pi) - - - 3 (8*pi)/(3*gamma(3/8)*sin((3*pi)/8)) - 3*(------) - 1 - 8 - *gamma(---------------------------------------------------) - 64))/( - - 3 (8*pi)/(3*gamma(3/8)*sin((3*pi)/8)) - (------) - 8 - - 7905853580625*sqrt(pi)) - - -on rounded, complex; - - -*** Domain mode rounded changed to complex-rounded - -precision 50; - - -33 - - -p; - - - - 0.00000000058461000084165968732153392127582134179078414159599 - - + 3.7721251013859508830301986850709684723938237902095e-60*i - - -off rounded, complex; - - -*** Domain mode complex-rounded changed to complex - -clear y; - - - -p; - - - gamma(y) - 40320*gamma( - y + 3)*gamma( - 4*y + 12) - 1 ------------------------------------------------------- - gamma( - 4*y + 12) - - - - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - -% 3. Beta function. Not very interesting - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - - -algebraic procedure do!*one!*beta(x,y); - write ("Beta (", x, ",", y, ") = ", beta(x,y)); - - -do*one*beta - - -do!*one!*beta(0,1); - - -Beta (0,1) = beta(0,1) - -do!*one!*beta(2,-3); - - -Beta (2,-3) = beta(2,-3) - -do!*one!*beta(3,2); - - - 1 -Beta (3,2) = ---- - 12 - -do!*one!*beta(a+b,(c+d)**(b-a)); - - - b - (c + d) - gamma(a + b)*gamma(----------) - b a - (c + d) (c + d) -Beta (a + b,----------) = --------------------------------------------- - a a a b - (c + d) (c + d) *a + (c + d) *b + (c + d) - gamma(------------------------------------) - a - (c + d) - -do!*one!*beta(-3,4); - - -Beta (-3,4) = beta(-3,4) - -do!*one!*beta(-3,2); - - -Beta (-3,2) = beta(-3,2) - -do!*one!*beta(-3,-7.5); - - - - 15 - 15 -Beta (-3,-------) = beta(-3,-------) - 2 2 - -do!*one!*beta((pi * 10), exp(5)); - - - 5 - 5 gamma(e )*gamma(10*pi) -Beta (10*pi,e ) = ------------------------ - 5 - gamma(e + 10*pi) - - -on rounded; - - -precision 30; - - -50 - - -do!*one!*beta(0,1); - - -Beta (0,1) = beta(0,1) - -do!*one!*beta(2,-3); - - -Beta (2,-3) = beta(2,-3) - -do!*one!*beta(3,2); - - -Beta (3,2) = 0.0833333333333333333333333333333 - -do!*one!*beta(a+b,(c+d)**(b-a)); - - - b - (c + d) - gamma(a + b)*gamma(----------) - b a - (c + d) (c + d) -Beta (a + b,----------) = --------------------------------------------- - a a a b - (c + d) (c + d) *a + (c + d) *b + (c + d) - gamma(------------------------------------) - a - (c + d) - -do!*one!*beta(-3,4); - - -Beta (-3,4) = beta(-3,4) - -do!*one!*beta(-3,2); - - -Beta (-3,2) = beta(-3,2) - -do!*one!*beta(-3,-7.5); - - -Beta (-3, - 7.5) = beta(-3, - 7.5) - -do!*one!*beta((pi * 10), exp(5)); - - -Beta (31.4159265358979323846264338328,148.413159102576603421115580041) = 3.26162 - -024071771351768890966259e-37 - - - - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - -% 4. Pochhammer notation - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - - -off rounded; - - - -pochhammer(4,5); - - -6720 - -pochhammer(-4,5); - - - - 24 ---------------- - factorial(-1) - -pochhammer(4,-5); - - - factorial(-2) ---------------- - 6 - -pochhammer(-4,-5); - - - - 1 -------- - 15120 - -pochhammer(17/2,12); - - - 157783444591397625 --------------------- - 4096 - -pochhammer(-17/2,12); - - - - 516891375 --------------- - 4096 - -pochhammer(1/3,14)*pochhammer(2/3,15); - - - 148260333813553014031851192320000000 --------------------------------------- - 68630377364883 - - -q := pochhammer(1/5,11)*pochhammer(2/5,11)*pochhammer(3/5,11)* - pochhammer(1-1/5,11)*pochhammer(1,11)*pochhammer(6/5,11)* - pochhammer(70/50,11)*pochhammer(8/5,11)*pochhammer(9/5,11); - - -q := -30835943859725167092317151562947659325673016884166249458427053145347526260421323040989302952930905614937458388095527696384131072 - / - - 1292469707114105741986576081359316958696581423282623291015625 - - -on complex; - - - -pochhammer(a+b*i,c)*pochhammer(a-b*i,c); - - -pochhammer(a - i*b,c)*pochhammer(a + i*b,c) - - -a := 2; - - -a := 2 - -b := 3; - - -b := 3 - -c := 5; - - -c := 5 - - -pochhammer(a+b*i,c)*pochhammer(a-b*i,c); - - -8950500 - - -off complex; - - -on rounded; - - - -pochhammer(1/5,11)*pochhammer(2/5,11)*pochhammer(3/5,11)* - pochhammer(1-1/5,11)*pochhammer(1,11)*pochhammer(6/5,11)* - pochhammer(70/50,11)*pochhammer(8/5,11)*pochhammer(9/5,11); - - -2.38581559706937212593793381562e+67 - - -q; - - -2.38581559706937212593793381562e+67 - - -pochhammer(pi,floor (pi**8)); - - -2.47253079057195612235973919163e+33625 - -pochhammer(-pi,floor (pi**7)); - - -5.91750008101140123889628058136e+9185 - -pochhammer(1.5,floor (pi**8)); - - -1.88808885937650373473836451368e+33619 - - - - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - -% 5. Digamma function - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - - -procedure do!*one!*psi(x); - << precision (precision(0) + 4)$ - write("Psi of ", x, " is ", psi(x) ) >> ; - - -do*one*psi - - -clear x, y; - - - -z := x * ((x+y)**2 + (x**y)); - - - y 2 2 -z := x*(x + x + 2*x*y + y ) - - -off rounded; - - - -do!*one!*psi(3); - - - 1 - 2*log(2) + psi(---) + psi(1) + 3 - 2 -Psi of 3 is ---------------------------------- - 2 - -do!*one!*psi(pi); - - -Psi of pi is psi(pi) - -do!*one!*psi(1.005); - - - 201 1 -Psi of ----- is psi(-----) + 200 - 200 200 - -do!*one!*psi(1.995); - - - pi 1 - 199*cot(-----)*pi + 199*psi(-----) + 200 - 399 200 200 -Psi of ----- is ------------------------------------------ - 200 199 - -do!*one!*psi(74); - - -Psi of 74 is (269426837164032294756360054754050*log(2) - - 1 - + 134713418582016147378180027377025*psi(---) - 2 - - + 2138308231460573767907619482175*psi(1) - - + 667084944417653637854891458725877)/ - - 136851726813476721146087646859200 - -do!*one!*psi(-1/2); - - - - 1 1 -Psi of ------ is psi(---) + 2 - 2 2 - -do!*one!*psi(-3); - - -Psi of -3 is infinity - -do!*one!*psi(z); - - - y 2 2 y 3 2 2 -Psi of x*(x + x + 2*x*y + y ) is psi(x *x + x + 2*x *y + x*y ) - - -on rounded; - - -precision 100; - - -62 - - -do!*one!*psi(3); - - -Psi of 3 is 0.922784335098467139393487909917597568957840664060076401194232765115 - - 13227322233532906305293670825325048537 - -do!*one!*psi(pi); - - -Psi of 3.14159265358979323846264338327950288419716939937510582097494459230781640 - - 628620899862803482534211706798214809 is 0.9772133079420067332920694864061 - -82343640834609994325638009523286531810592477714131730207565436292873435576949 - -do!*one!*psi(1.005); - - -Psi of 1.005 is - 0.56902091134438283168829569095002880926796887026346909469924 - - 37771372094405771031990077045450115105840960011306543 - -do!*one!*psi(1.995); - - -Psi of 1.995 is 0.41955460302810862865162027228724395469212997650340596127576800 - - 237211077024989654159358528162438486780461247996589393 - -do!*one!*psi(74); - - -Psi of 74 is 4.29729311880466763910635036256036713791964393500098361596464857079 - - 142072638532815275329153012799334519059885201047316254 - -do!*one!*psi(-1/2); - - -Psi of - 0.5 is 0.0364899739785765205590236670012444328068403953395658929528727 - - 461283450292829458978513262827154158754013655907090515460516684 - - 6 - -do!*one!*psi(-3); - - -Psi of -3 is infinity - -do!*one!*psi(z); - - - y 2 2 y 3 2 2 -Psi of x*(x + x + 2*x*y + y ) is psi(x *x + x + 2*x *y + x*y ) - - -precision 15; - - -132 - - -x := 8/3; - - -x := 2.66666666666667 - -y := 7/1000; - - -y := 0.007 - -do!*one!*psi(z); - - -Psi of 21.74768766103287773 is 3.056340330052438423 - - -off rounded; - - - -clear x, y; - - - -df(psi(z), x); - - - y 3 2 2 y y 2 2 -polygamma(1,x *x + x + 2*x *y + x*y )*(x *y + x + 3*x + 4*x*y + y ) - -df(df(psi(z), y),x); - - - 2*y y 3 2 2 -x *log(x)*polygamma(2,x *x + x + 2*x *y + x*y )*x*y - - 2*y y 3 2 2 - + x *log(x)*polygamma(2,x *x + x + 2*x *y + x*y )*x - - y y 3 2 2 3 - + 3*x *log(x)*polygamma(2,x *x + x + 2*x *y + x*y )*x - - y y 3 2 2 2 - + 4*x *log(x)*polygamma(2,x *x + x + 2*x *y + x*y )*x *y - - y y 3 2 2 2 - + x *log(x)*polygamma(2,x *x + x + 2*x *y + x*y )*x*y - - y y 3 2 2 - + x *log(x)*polygamma(1,x *x + x + 2*x *y + x*y )*y - - y y 3 2 2 - + x *log(x)*polygamma(1,x *x + x + 2*x *y + x*y ) - - y y 3 2 2 2 - + 2*x *polygamma(2,x *x + x + 2*x *y + x*y )*x *y - - y y 3 2 2 2 - + 2*x *polygamma(2,x *x + x + 2*x *y + x*y )*x - - y y 3 2 2 2 - + 2*x *polygamma(2,x *x + x + 2*x *y + x*y )*x*y - - y y 3 2 2 - + 2*x *polygamma(2,x *x + x + 2*x *y + x*y )*x*y - - y y 3 2 2 - + x *polygamma(1,x *x + x + 2*x *y + x*y ) - - y 3 2 2 4 - + 6*polygamma(2,x *x + x + 2*x *y + x*y )*x - - y 3 2 2 3 - + 14*polygamma(2,x *x + x + 2*x *y + x*y )*x *y - - y 3 2 2 2 2 - + 10*polygamma(2,x *x + x + 2*x *y + x*y )*x *y - - y 3 2 2 3 - + 2*polygamma(2,x *x + x + 2*x *y + x*y )*x*y - - y 3 2 2 - + 4*polygamma(1,x *x + x + 2*x *y + x*y )*x - - y 3 2 2 - + 2*polygamma(1,x *x + x + 2*x *y + x*y )*y - -int(psi(z), z); - - - y 3 2 2 -log(gamma(x *x + x + 2*x *y + x*y )) - - -on rounded; - - - -for k := 1 step 0.1 until 2 do do!*one!*psi(k); - - -Psi of 1 is - 0.57721566490153286060651 - -Psi of 1.09999999999999999999999999 is - 0.423754940411076795168216226 - -Psi of 1.19999999999999999999999999483 is - 0.2890398965921882955472079690017 - -Psi of 1.2999999999999999999999999948304368 is - - - 0.16919088886679965563116117990224309 - -Psi of 1.39999999999999999999999999483043679294 is - - - 0.0613845445851161457306754873481741603465 - -Psi of 1.499999999999999999999999994830436792938162 is 0.03648997397857652055902 - -366216872537099062413 - -Psi of 1.5999999999999999999999999948304367929381615219 is 0.1260474527734762519 - -0600271342271679469086378102 - -Psi of 1.69999999999999999999999999483043679293816152191799 is 0.208547874873493 - -95667996417990930337500109911872963 - -Psi of 1.799999999999999999999999994830436792938161521917991479 is 0.28499143329 - -38615406087023630333301945809176983421812009 - -Psi of 1.8999999999999999999999999948304367929381615219179914789724 is 0.3561841 - -6116405971922472708037210055174041765918986424485621 - -Psi of 1.99999999999999999999999999483043679293816152191799147897240477 is 0.422 - -784335098467139393487906583570145998489083710421411058968004 - - -off rounded; - - - -% PSI_SIMP.TST F.J.Wright, 2 July 1993 - -on evallhseqp; - - -factor psi; - - on rat, intstr, div; - - % for neater output -% Do not try using "off mcd"! - -psi(x+m) - psi(x+m-1) = 1/(x+m-1); - - - 1 1 ------------=----------- - m + x - 1 m + x - 1 - -psi(x+2) - psi(x+1) + 2*psi(x) = 1/(x+1) + 2*psi(x); - - - 1 1 -2*psi(x) + -------=2*psi(x) + ------- - x + 1 x + 1 - -psi(x+2) + 3*psi(x) = 4*psi(x) + 1/x + 1/(x+1); - - - -1 -1 - x + 2 x + 2 -4*psi(x) + ---------=4*psi(x) + --------- - x + 1 x + 1 - - -psi(x + 1) = psi(x) + 1/x; - - - -1 -1 -psi(x) + x =psi(x) + x - -psi(x + 3/2) = psi(x + 1/2) + 1/(x + 1/2); - - - 1 2 1 2 -psi(x + ---) + ---------=psi(x + ---) + --------- - 2 2*x + 1 2 2*x + 1 - -psi(x - 1/2) = psi(x + 1/2) - 1/(x - 1/2); - - - 1 2 1 2 -psi(x + ---) - ---------=psi(x + ---) - --------- - 2 2*x - 1 2 2*x - 1 - -psi((x + 3a)/a); - - - -1 - 1 2*(3*x + 8*x + 12) -psi(---*x) + ---------------------- - 2 2 - x + 6*x + 8 - psi(x/y + 3); - - - -1 2 - -1 y*(3*x + 2*x *y + 6*y) -psi(x*y ) + -------------------------- - 2 2 - x + 3*x*y + 2*y - - -off rat, intstr, div; - - on rational; - - - -psi(x+m) - psi(x+m-1) = 1/(x+m-1); - - - 1 1 ------------=----------- - m + x - 1 m + x - 1 - -psi(x+2) - psi(x+1) + 2*psi(x) = 1/(x+1) + 2*psi(x); - - - 2*psi(x)*x + 2*psi(x) + 1 2*psi(x)*x + 2*psi(x) + 1 ----------------------------=--------------------------- - x + 1 x + 1 - -psi(x+2) + 3*psi(x) = 4*psi(x) + 1/x + 1/(x+1); - - - 2 2 - 4*psi(x)*x + 4*psi(x)*x + 2*x + 1 4*psi(x)*x + 4*psi(x)*x + 2*x + 1 -------------------------------------=------------------------------------ - 2 2 - x + x x + x - - -psi(x + 1) = psi(x) + 1/x; - - - psi(x)*x + 1 psi(x)*x + 1 ---------------=-------------- - x x - -psi(x + 3/2) = psi(x + 1/2) + 1/(x + 1/2); - - - 1 1 1 1 1 1 - psi(x + ---)*x + ---*psi(x + ---) + 1 psi(x + ---)*x + ---*psi(x + ---) + 1 - 2 2 2 2 2 2 ----------------------------------------=--------------------------------------- - 1 1 - x + --- x + --- - 2 2 - -psi(x - 1/2) = psi(x + 1/2) - 1/(x - 1/2); - - - 1 1 1 1 1 1 - psi(x + ---)*x - ---*psi(x + ---) - 1 psi(x + ---)*x - ---*psi(x + ---) - 1 - 2 2 2 2 2 2 ----------------------------------------=--------------------------------------- - 1 1 - x - --- x - --- - 2 2 - -psi((x + 3a)/a); - - - 1 2 2 8 - psi(---*x)*x*(x + 6*x + 8) + 6*(x + 4*x + ---) - 2 3 --------------------------------------------------- - 2 - x*(x + 6*x + 8) - psi(x/y + 3); - - - x 2 2 2 2 2 - psi(---)*x*(x + 3*x*y + 2*y ) + 3*y*(x + 2*x*y + ---*y ) - y 3 ------------------------------------------------------------- - 2 2 - x*(x + 3*x*y + 2*y ) - - -off rational; - - - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - -% 6. Polygamma functions - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - - -procedure do!*one!*pg(n,x); - write ("Polygamma (", n, ") of ", x, " is ", polygamma(n,x)); - - -do*one*pg - - -off rounded; - - - -do!*one!*pg(1,1/2); - - - 1 1 -Polygamma (1) of --- is polygamma(1,---) - 2 2 - -do!*one!*pg(1,1); - - - 2 - pi -Polygamma (1) of 1 is ----- - 6 - -do!*one!*pg(1,3/2); - - - 2 - 3 pi - 8 -Polygamma (1) of --- is --------- - 2 2 - -do!*one!*pg(1,1.005); - - - 201 201 -Polygamma (1) of ----- is polygamma(1,-----) - 200 200 - -do!*one!*pg(1,1.995); - - - 399 399 -Polygamma (1) of ----- is polygamma(1,-----) - 200 200 - -do!*one!*pg(1,1e-10); - - - 1 1 -Polygamma (1) of ------------- is polygamma(1,-------------) - 10000000000 10000000000 - -do!*one!*pg(2,1.45); - - - 29 29 -Polygamma (2) of ---- is polygamma(2,----) - 20 20 - -do!*one!*pg(3,1.99); - - - 199 199 -Polygamma (3) of ----- is polygamma(3,-----) - 100 100 - -do!*one!*pg(4,-8.2); - - - - 41 - 41 -Polygamma (4) of ------- is polygamma(4,-------) - 5 5 - -do!*one!*pg(5,0); - - -Polygamma (5) of 0 is infinity - -do!*one!*pg(6,-5); - - -Polygamma (6) of -5 is infinity - -do!*one!*pg(7,200); - - -Polygamma (7) of 200 is ( -17726903568079021516229662669344581999513453462651624712682433450930396569413638060869523209196416456149981841411350319061723276968944831274405326841106087088051862580482604049179034792880997036306450715491134920388121743532178874504889199026667348488846246287054278592284199023548572837863642592371308428279823642858774260678182876022899303888558378920013392172441409952343098219076353830605697314645020537255667718925997023946826863459069360054494011515187931095594830347819766125471751832158677864778297413490571816668099535649328289203181141151677544880676008798702466775519536091686976761627732998715124407667063278175501392084867643186474774933465396962161891279302009565617687762042880000000000000000000000 - 8 - *pi - -168202274324609919911886603179553832485831631970776943913214724306758011473978941132413094748296521617074167117661053498983406945306976674698602419479427329202365001144489134708765120356232374484239386020849061429951991522474455058548094378661234695692419462486402225369012574020278308089090741397551205530205566242796380021997272735088025054393694714504643986352458446128585149055841912367918690170304028682697422603414064799513570602284100682485151815951427876744662169487167539572896985974962832514966524381217035170993953423670212502913136819535576467936946089479260579835796147751358821073694464012485544356514018523618843496912692835966265011413341612149624871304965950289830955139972163546512706990894178649281 - )/ -33237944190148165342930617505021091249087725242471796336279562720494493567650571364130356017243280855281215952646281848240731144316771558639509987827073913290097242338404882592210690236651869443074595091545877975727728269122835389696667248175001278416586711788226772360532873169153574070994329860696203303024669330360201738771592892542936194791046960475025110323327643660643309160768163432385682464959413507354376972986244419900300368985755050102176271590977370804240306902162061485259534685297520996459307650294822156252686629342490542255964639659395396651267516497567125204099130171913081428051999372590858264375743646579065110159126830974640203000247619304053546148691267935533164553830400000000000000000000000 - - -on rounded; - - -precision 100; - - -63 - - -do!*one!*pg(1,1/2); - - -Polygamma (1) of 0.5 is 4.934802200544679309417245499938075567656849703620395313 - -206674688110022411209602621500886701859276116 - -do!*one!*pg(1,1); - - -Polygamma (1) of 1 is 1.64493406684822643647241516664602518921894990120679843773 - -5558229370007470403200873833628900619758705 - -do!*one!*pg(1,3/2); - - -Polygamma (1) of 1.5 is 0.934802200544679309417245499938075567656849703620395313 - -2066746881100224112096026215008867018592761159 - -do!*one!*pg(1,1.005); - - -Polygamma (1) of 1.005 is 1.6329941567556809752535869786110269899606654266292810 - -17423768306472667920740136690588658095268582677 - -do!*one!*pg(1,1.995); - - -Polygamma (1) of 1.995 is 0.6469608286405823512476399271078276952436864784225077 - -824256922477813603546774633500406650324443323374 - -do!*one!*pg(1,1e-10); - - -Polygamma (1) of 0.0000000001 is 1.000000000000000000016449340666078150558729660 - -065732639916420752057756424780180942170336642993019363e+20 - -do!*one!*pg(2,1.45); - - -Polygamma (2) of 1.45 is - 0.90383740307625768823230953852986719286009030811144 - -6658245367522559171482627166928803699348863802546 - -do!*one!*pg(3,1.99); - - -Polygamma (3) of 1.99 is 0.50290713241686532018051099086257019718615168956130804 - -44480934082942919777908933506993776280887706183 - -do!*one!*pg(4,-8.2); - - -Polygamma (4) of - 8.2 is 74935.51259577427052712029230781313552295905493466277 - -917617629718610711391593937121099755096024434196 - -do!*one!*pg(5,0); - - -Polygamma (5) of 0 is infinity - -do!*one!*pg(6,-5); - - -Polygamma (6) of -5 is infinity - -do!*one!*pg(7,200); - - -Polygamma (7) of 200 is 5.724093725392558373837029681544727245346807368195956154 - -743567847385926644129753317064645661270233713e-14 - - -off rounded; - - -clear x; - - - - -% Polygamma differentiation has already -% been tried a bit in the psi section - -df(int(int(int(polygamma(3,x),x),x),x),x); - - -polygamma(1,x) - - -clear w, y, z; - - - - - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - -% 7. Zeta function - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - - -procedure do!*one!*zeta(n); - write ("Zeta of ", n, " is ", zeta n); - - -do*one*zeta - - -off rounded; - - -clear x, y, z; - - - -z := x * ((x+y)**5 + (x**y)); - - - y 5 4 3 2 2 3 4 5 -z := x*(x + x + 5*x *y + 10*x *y + 10*x *y + 5*x*y + y ) - - -do!*one!*zeta(0); - - - - 1 -Zeta of 0 is ------ - 2 - - -for k := 4 step 2 until 35 do - do!*one!*zeta(k); - - - 4 - pi -Zeta of 4 is ----- - 90 - - 6 - pi -Zeta of 6 is ----- - 945 - - 8 - pi -Zeta of 8 is ------ - 9450 - - 10 - pi -Zeta of 10 is ------- - 93555 - - 12 - 691*pi -Zeta of 12 is ----------- - 638512875 - - 14 - 2*pi -Zeta of 14 is ---------- - 18243225 - - 16 - 3617*pi -Zeta of 16 is -------------- - 325641566250 - - 18 - 43867*pi -Zeta of 18 is ---------------- - 38979295480125 - - 20 - 174611*pi -Zeta of 20 is ------------------ - 1531329465290625 - - 22 - 155366*pi -Zeta of 22 is ------------------- - 13447856940643125 - - 24 - 236364091*pi -Zeta of 24 is ----------------------- - 201919571963756521875 - - 26 - 1315862*pi -Zeta of 26 is ---------------------- - 11094481976030578125 - - 28 - 6785560294*pi -Zeta of 28 is -------------------------- - 564653660170076273671875 - - 30 - 6892673020804*pi -Zeta of 30 is ------------------------------ - 5660878804669082674070015625 - -Zeta of 32 is zeta(32) - -Zeta of 34 is zeta(34) - - -do!*one!*zeta(-17/3); - - - - 17 - 17 -Zeta of ------- is zeta(-------) - 3 3 - -do!*one!*zeta(190); - - -Zeta of 190 is zeta(190) - -do!*one!*zeta(300); - - -Zeta of 300 is zeta(300) - -do!*one!*zeta(0); - - - - 1 -Zeta of 0 is ------ - 2 - -do!*one!*zeta(-44); - - -Zeta of -44 is 0 - - -on rounded; - - -clear x, y; - - - -for k := 3 step 3 until 36 do << - precision (31+k*3); - do!*one!*zeta(k) >>; - - -Zeta of 3 is 1.202056903159594285399738161511449990765 - -Zeta of 6 is 1.017343061984449139714517929790920527901817490033 - -Zeta of 9 is 1.002008392826082214417852769232412060485605851394888756549 - -Zeta of 12 is 1.0002460865533080482986379980477396709604160884580034045330409521 - - 33 - -Zeta of 15 is 1.0000305882363070204935517285106450625876279487068581775065699328 - - 93332267156 - -Zeta of 18 is 1.0000038172932649998398564616446219397304546972189533311431744299 - - 87630039542650045638 - -Zeta of 21 is 1.0000004769329867878064631167196043730459664466947849376002074873 - - 76596839087898159833876638564 - -Zeta of 24 is 1.0000000596081890512594796124402079358012275039188373027958642469 - - 72321724495355468544848206832825003614 - -Zeta of 27 is 1.0000000074507117898354294919810041706041194547190318825658299932 - - 39578352147606271570867900837100313523764933952 - -Zeta of 30 is 1.0000000009313274324196681828717647350212198135679551368161850086 - - 13360441960672940496363503624604027929086312123388047291 - -Zeta of 33 is 1.0000000001164155017270051977592973835456309516522471727635932565 - - 17739947029124624567548673934974376008810870912845774213829513369 - -Zeta of 36 is 1.0000000000145519218910419842359296322453184209838088941240380691 - - 395422185717458650302201529989423295781853630847913399997790928914 - - 91916899 - - -precision 20; - - -139 - - -do!*one!*zeta(-17/3); - - -Zeta of - 5.6666666666666666667 is - 0.0018766468228592287697 - -do!*one!*zeta(z); - - - y 5 4 3 2 2 3 4 5 -Zeta of x*(x + x + 5*x *y + 10*x *y + 10*x *y + 5*x*y + y ) is zeta( - - y 6 5 4 2 3 3 2 4 5 - x *x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + x*y ) - - -y := 3; - - -y := 3 - -x := pi; - - -x := 3.1415926535897932385 - - -do!*one!*zeta(z); - - -Zeta of 27548.203250393209469 is 1 - -do!*one!*zeta(190); - - -Zeta of 190 is 1.0 - -do!*one!*zeta(300); - - -Zeta of 300 is 1 - -do!*one!*zeta(0); - - -Zeta of 0 is - 0.5 - -do!*one!*zeta(-44); - - -Zeta of -44 is 0 - - -off rounded; - - - - - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - -% 8. Kummer functions - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - - -off rounded; - - - -t!*kummer!*a := { 1, 2.4, -1397/10 }$ - - -t!*kummer!*b := { 0, 1, pi, -pi, 26 }$ - - - -for each a in t!*kummer!*a do - for each b in t!*kummer!*a do - for each z in t!*kummer!*a do - << write "KummerM(", a, ",", b, ",", z, ") = ", - kummerm(a,b,z); - write "KummerU(", a, ",", b, ",", z, ") = ", - kummeru(a,b,z) >>; - - -KummerM(1,1,1) = e - -KummerU(1,1,1) = kummeru(1,1,1) - - 12 2/5 2 -KummerM(1,1,----) = e *e - 5 - - 12 12 -KummerU(1,1,----) = kummeru(1,1,----) - 5 5 - - - 1397 1 -KummerM(1,1,---------) = ------------ - 10 7/10 139 - e *e - - - 1397 - 1397 -KummerU(1,1,---------) = kummeru(1,1,---------) - 10 10 - - 12 12 -KummerM(1,----,1) = kummerm(1,----,1) - 5 5 - - 2 12 12 - 2*gamma(---)*(gamma(----)*e - kummerm(1,----,1)) - 12 5 5 5 -KummerU(1,----,1) = -------------------------------------------------- - 5 12 - 5*gamma(----) - 5 - - 12 12 12 12 -KummerM(1,----,----) = kummerm(1,----,----) - 5 5 5 5 - - 12 12 -KummerU(1,----,----) = - 5 5 - - 2 2/5 2/5 12 2 4/5 2/5 12 12 - gamma(---)*(5*e *5 *gamma(----)*e - 12*2 *3 *kummerm(1,----,----)) - 5 5 5 5 ------------------------------------------------------------------------------ - 4/5 2/5 12 - 30*2 *3 *gamma(----) - 5 - - 12 - 1397 12 - 1397 -KummerM(1,----,---------) = kummerm(1,----,---------) - 5 10 5 10 - - 12 - 1397 2 -KummerU(1,----,---------) = (2*gamma(---)*( - 5 10 5 - - 7/10 2/5 12 - 1397 139 - - 1397*e *1397 *kummerm(1,----,---------)*e - 5 10 - - 2/5 12 7/10 2/5 12 139 - - 10*10 *gamma(----)))/(6985*e *1397 *gamma(----)*e ) - 5 5 - - - 1397 - 1397 -KummerM(1,---------,1) = kummerm(1,---------,1) - 10 10 - - - 1397 -KummerU(1,---------,1) = - 10 - - 1397 - 1397 3*pi - 1397*gamma(------)*kummerm(1,---------,1)*sin(------) - 10*e*pi - 10 10 10 ------------------------------------------------------------------ - 1417 3*pi - 10*gamma(------)*sin(------) - 10 10 - - - 1397 12 - 1397 12 -KummerM(1,---------,----) = kummerm(1,---------,----) - 10 5 10 5 - - - 1397 12 -KummerU(1,---------,----) = ( - - 10 5 -48689401707698071401637346296148709517635305466315546784369062686038955387470672028214421172492426257970894859785608060010798787187515790195560655355904 - 2/5 - *e - - 2/5 7/10 2 - *2 *3 *e *pi + -20045966895736519148343810333245571434480905551887283539400953780162950579324387945234775543212890625 - 7/10 1397 - 1397 12 3*pi - *5 *gamma(------)*kummerm(1,---------,----)*sin(------)) - 10 10 5 10 - -/( -143492962746861268062589909328887411843098822848155215027923792270314606867032125592231750488281250 - 7/10 1417 3*pi - *5 *gamma(------)*sin(------)) - 10 10 - - - 1397 - 1397 - 1397 - 1397 -KummerM(1,---------,---------) = kummerm(1,---------,---------) - 10 10 10 10 - - - 1397 - 1397 -KummerU(1,---------,---------) = (1397*( - - 10 10 -152159982579315173600331019038628152238885433889572221061155552992026662790488679161680024820756008730757778143037042693923932846444113056473446432154278794079710763645820340853851602689287025084815217783532480956517472235723731181252401977856952544597761994384769996636245679210281354768553595796003820698711669598622602752713703151447293557592795912091076094345980353443096817710446904761297306235868715700448137787799653833690839289133 - 7/10 - *( - 1397) *pi + -10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 - 7/10 7/10 - *e *10 - - 1397 - 1397 - 1397 3*pi 139 - *gamma(------)*kummerm(1,---------,---------)*sin(------)*e ))/( - 10 10 10 10 -100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 - 7/10 7/10 - *e *10 - - 1417 3*pi 139 - *gamma(------)*sin(------)*e ) - 10 10 - - 12 12 -KummerM(----,1,1) = kummerm(----,1,1) - 5 5 - - 12 12 -KummerU(----,1,1) = kummeru(----,1,1) - 5 5 - - 12 12 12 12 -KummerM(----,1,----) = kummerm(----,1,----) - 5 5 5 5 - - 12 12 12 12 -KummerU(----,1,----) = kummeru(----,1,----) - 5 5 5 5 - - 12 - 1397 12 - 1397 -KummerM(----,1,---------) = kummerm(----,1,---------) - 5 10 5 10 - - 12 - 1397 12 - 1397 -KummerU(----,1,---------) = kummeru(----,1,---------) - 5 10 5 10 - - 12 12 -KummerM(----,----,1) = e - 5 5 - - 2 - 2 2*pi - 2*gamma(---)*kummerm(1,------,1)*sin(------) + 5*e*pi - 12 12 5 5 5 -KummerU(----,----,1) = ------------------------------------------------------- - 5 5 12 2*pi - 5*gamma(----)*sin(------) - 5 5 - - 12 12 12 2/5 2 -KummerM(----,----,----) = e *e - 5 5 5 - - 12 12 12 -KummerU(----,----,----) = - 5 5 5 - - 2/5 4/5 2/5 2 2/5 2 - 2 12 2*pi - 6*e *2 *3 *e *pi + 5 *gamma(---)*kummerm(1,------,----)*sin(------) - 5 5 5 5 ------------------------------------------------------------------------------ - 4/5 2/5 12 2*pi - 6*2 *3 *gamma(----)*sin(------) - 5 5 - - 12 12 - 1397 1 -KummerM(----,----,---------) = ------------ - 5 5 10 7/10 139 - e *e - - 12 12 - 1397 -KummerU(----,----,---------) = ( - 5 5 10 - - 7/10 2/5 2 - 2 - 1397 2*pi 139 - - 4*e *10 *gamma(---)*kummerm(1,------,---------)*sin(------)*e - 5 5 10 5 - - 2/5 7/10 2/5 12 2*pi 139 - + 1397*1397 *pi)/(1397*e *1397 *gamma(----)*sin(------)*e ) - 5 5 - - 12 - 1397 12 - 1397 -KummerM(----,---------,1) = kummerm(----,---------,1) - 5 10 5 10 - - 12 - 1397 1431 1431 1417 -KummerU(----,---------,1) = ( - 10*gamma(------)*kummerm(------,------,1)*pi + - 5 10 10 10 10 - - 1417 1397 12 12 - 1397 - 1397*gamma(------)*gamma(------)*gamma(----)*kummerm(----,---------,1) - 10 10 5 5 10 - - 3*pi 1431 1417 12 3*pi - *sin(------))/(10*gamma(------)*gamma(------)*gamma(----)*sin(------)) - 10 10 10 5 10 - - 12 - 1397 12 12 - 1397 12 -KummerM(----,---------,----) = kummerm(----,---------,----) - 5 10 5 5 10 5 - - 12 - 1397 12 -KummerU(----,---------,----) = ( - 5 10 5 -20045966895736519148343810333245571434480905551887283539400953780162950579324387945234775543212890625 - 7/10 1417 1397 12 - *5 *gamma(------)*gamma(------)*gamma(----) - 10 10 5 - - 12 - 1397 12 3*pi - *kummerm(----,---------,----)*sin(------) - - 5 10 5 10 -48689401707698071401637346296148709517635305466315546784369062686038955387470672028214421172492426257970894859785608060010798787187515790195560655355904 - 2/5 - *2 - - 7/10 1431 1431 1417 12 - *3 *gamma(------)*kummerm(------,------,----)*pi)/( - 10 10 10 5 -143492962746861268062589909328887411843098822848155215027923792270314606867032125592231750488281250 - 7/10 1431 1417 12 3*pi - *5 *gamma(------)*gamma(------)*gamma(----)*sin(------)) - 10 10 5 10 - - 12 - 1397 - 1397 12 - 1397 - 1397 -KummerM(----,---------,---------) = kummerm(----,---------,---------) - 5 10 10 5 10 10 - - 12 - 1397 - 1397 -KummerU(----,---------,---------) = (1397*( - - 5 10 10 -152159982579315173600331019038628152238885433889572221061155552992026662790488679161680024820756008730757778143037042693923932846444113056473446432154278794079710763645820340853851602689287025084815217783532480956517472235723731181252401977856952544597761994384769996636245679210281354768553595796003820698711669598622602752713703151447293557592795912091076094345980353443096817710446904761297306235868715700448137787799653833690839289133 - 7/10 1431 - *( - 1397) *gamma(------) - 10 - - 1431 1417 - 1397 - *kummerm(------,------,---------)*pi + - 10 10 10 -10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 - 7/10 - *10 - - 1417 1397 12 12 - 1397 - 1397 - *gamma(------)*gamma(------)*gamma(----)*kummerm(----,---------,---------) - 10 10 5 5 10 10 - - 3*pi - *sin(------)))/( - 10 -100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 - 7/10 - *10 - - 1431 1417 12 3*pi - *gamma(------)*gamma(------)*gamma(----)*sin(------)) - 10 10 5 10 - - - 1397 - 1397 -KummerM(---------,1,1) = kummerm(---------,1,1) - 10 10 - - - 1397 - 1397 -KummerU(---------,1,1) = kummeru(---------,1,1) - 10 10 - - - 1397 12 - 1397 12 -KummerM(---------,1,----) = kummerm(---------,1,----) - 10 5 10 5 - - - 1397 12 - 1397 12 -KummerU(---------,1,----) = kummeru(---------,1,----) - 10 5 10 5 - - - 1397 - 1397 - 1397 - 1397 -KummerM(---------,1,---------) = kummerm(---------,1,---------) - 10 10 10 10 - - - 1397 - 1397 - 1397 - 1397 -KummerU(---------,1,---------) = kummeru(---------,1,---------) - 10 10 10 10 - - - 1397 12 - 1397 12 -KummerM(---------,----,1) = kummerm(---------,----,1) - 10 5 10 5 - - - 1397 12 -KummerU(---------,----,1) = ( - 10 5 - - 1411 - 1397 12 pi - 7055*gamma(------)*kummerm(---------,----,1)*sin(----)*pi + 2794 - 10 10 5 10 - - 1397 12 2 - 1411 - 2 3*pi - *gamma(------)*gamma(----)*gamma(---)*kummerm(---------,------,1)*sin(------) - 10 5 5 10 5 10 - - 2*pi 12 2*pi - *sin(------))/(50*gamma(----)*sin(------)*pi) - 5 5 5 - - - 1397 12 12 - 1397 12 12 -KummerM(---------,----,----) = kummerm(---------,----,----) - 10 5 5 10 5 5 - - - 1397 12 12 2/5 1397 12 2 -KummerU(---------,----,----) = (1397*5 *gamma(------)*gamma(----)*gamma(---) - 10 5 5 10 5 5 - - - 1411 - 2 12 3*pi 2*pi - *kummerm(---------,------,----)*sin(------)*sin(------) - 10 5 5 10 5 - - 4/5 2/5 1411 - 1397 12 12 pi - + 8466*2 *3 *gamma(------)*kummerm(---------,----,----)*sin(----)*pi)/( - 10 10 5 5 10 - - 4/5 2/5 12 2*pi - 60*2 *3 *gamma(----)*sin(------)*pi) - 5 5 - - - 1397 12 - 1397 - 1397 12 - 1397 -KummerM(---------,----,---------) = kummerm(---------,----,---------) - 10 5 10 10 5 10 - - - 1397 12 - 1397 -KummerU(---------,----,---------) = ( - 10 5 10 - - 2/5 1411 - 1397 12 - 1397 pi - 1411*1397 *gamma(------)*kummerm(---------,----,---------)*sin(----)*pi - 4 - 10 10 5 10 10 - - 2/5 1397 12 2 - *10 *gamma(------)*gamma(----)*gamma(---) - 10 5 5 - - - 1411 - 2 - 1397 3*pi 2*pi 2/5 - *kummerm(---------,------,---------)*sin(------)*sin(------))/(10*1397 - 10 5 10 10 5 - - 12 2*pi - *gamma(----)*sin(------)*pi) - 5 5 - - - 1397 - 1397 -KummerM(---------,---------,1) = e - 10 10 - - - 1397 - 1397 -KummerU(---------,---------,1) = - 10 10 - - 1397 1417 1417 - 1397*gamma(------)*(gamma(------)*e - kummerm(1,------,1)) - 10 10 10 ------------------------------------------------------------- - 1417 - 10*gamma(------) - 10 - - - 1397 - 1397 12 2/5 2 -KummerM(---------,---------,----) = e *e - 10 10 5 - - - 1397 - 1397 12 1397 -KummerU(---------,---------,----) = (1397*gamma(------)*( - 10 10 5 10 -71746481373430634031294954664443705921549411424077607513961896135157303433516062796115875244140625 - 2/5 7/10 1417 2 - *e *5 *gamma(------)*e - - 10 -24344700853849035700818673148074354758817652733157773392184531343019477693735336014107210586246213128985447429892804030005399393593757895097780327677952 - 2/5 - *2 - - 7/10 1417 12 - *3 *kummerm(1,------,----)))/( - 10 5 -717464813734306340312949546644437059215494114240776075139618961351573034335160627961158752441406250 - 7/10 1417 - *5 *gamma(------)) - 10 - - - 1397 - 1397 - 1397 1 -KummerM(---------,---------,---------) = ------------ - 10 10 10 7/10 139 - e *e - - - 1397 - 1397 - 1397 1397 -KummerU(---------,---------,---------) = (1397*gamma(------)*( - - 10 10 10 10 -212567495663303297519662433596963528677722951143732392822434307529861247918312684788866994674596144196868616065822748643411734186482425939893404665719527475329355936813211016172830688956933974043486859243594875896254908713306052460209605563066162704803073506155523685300835213856763052611669373327017337516100202429275776045541043302571869099957135889191233303801334553760006254341494325951532336811508595833526048489556116405666102486918801 - 7/10 7/10 - *e *( - 1397) - - 1417 - 1397 139 - *kummerm(1,------,---------)*e + - 10 10 -100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 - 7/10 - *10 - - 1417 - *gamma(------)))/( - 10 -1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 - 7/10 7/10 - *e *10 - - 1417 139 - *gamma(------)*e ) - 10 - - -on rounded; - - -precision 30; - - -20 - -t!*k!*c := 7; - - -t*k*c := 7 - - -% To test each and every possible combination of -% three arguments from t!*kummer!*b would take too -% long, but we want the possibility of trying most -% special cases. Compromise: test every seventh -% possibility. - -for each a in t!*kummer!*b do - for each b in t!*kummer!*b do - for each z in t!*kummer!*b do - << if t!*k!*c = 7 - then << write "KummerM(", a, ",", b, ",", z, ") = ", - kummerm(a,b,z); - write "KummerU(", a, ",", b, ",", z, ") = ", - kummeru(a,b,z); - t!*k!*c := 0 >>; - t!*k!*c := t!*k!*c + 1 >>; - - -KummerM(0,0,0) = 1 - -KummerU(0,0,0) = kummeru(0,0,0) - -KummerM(0,1,3.14159265358979323846264338328) = 1 - -KummerU(0,1,3.14159265358979323846264338328) = kummeru(0,1, - - 3.14159265358979323846264338328) - -KummerM(0,3.14159265358979323846264338328,26) = 1 - -KummerU(0,3.14159265358979323846264338328,26) = kummeru(0, - - 3.14159265358979323846264338328,26) - -KummerM(0,26,1) = 1 - -KummerU(0,26,1) = kummeru(0,26,1) - -KummerM(1,0, - 3.14159265358979323846264338328) = kummerm(1,0, - - - 3.14159265358979323846264338328) - -KummerU(1,0, - 3.14159265358979323846264338328) = kummeru(1,0, - - - 3.14159265358979323846264338328) - -KummerM(1,3.14159265358979323846264338328,0) = 1 - -KummerU(1,3.14159265358979323846264338328,0) = kummeru(1, - - 3.14159265358979323846264338328,0) - -KummerM(1, - 3.14159265358979323846264338328,3.14159265358979323846264338328) = - -2692.89480079631357528203659153 - -KummerU(1, - 3.14159265358979323846264338328,3.14159265358979323846264338328) = - -0.129554194296952806409644905659 - -KummerM(1,26,26) = 7.74565667206271943920803547594 - -KummerU(1,26,26) = kummeru(1,26,26) - -KummerM(3.14159265358979323846264338328,1,1) = 10.2259987795328570162092950355 - -KummerU(3.14159265358979323846264338328,1,1) = kummeru( - - 3.14159265358979323846264338328,1,1) - -KummerM(3.14159265358979323846264338328,3.14159265358979323846264338328, - - - 3.14159265358979323846264338328) = 0.0432139182637722497744177371717 - -KummerU(3.14159265358979323846264338328,3.14159265358979323846264338328, - - - 3.14159265358979323846264338328) = ( - 0.137891580772667438638357954721 - - 2.14159265358979323846264338328 - *( - 3.14159265358979323846264338328) - - + 1.22938452238615186004844111454)/ - - 2.14159265358979323846264338328 -( - 3.14159265358979323846264338328) - -KummerM(3.14159265358979323846264338328,26,0) = 1 - -KummerU(3.14159265358979323846264338328,26,0) = kummeru( - - 3.14159265358979323846264338328,26,0) - -KummerM( - 3.14159265358979323846264338328,0,3.14159265358979323846264338328) = - -kummerm( - 3.14159265358979323846264338328,0,3.14159265358979323846264338328) - -KummerU( - 3.14159265358979323846264338328,0,3.14159265358979323846264338328) = - -kummeru( - 3.14159265358979323846264338328,0,3.14159265358979323846264338328) - -KummerM( - 3.14159265358979323846264338328,1,26) = 6.185222265647228001735135594 - -62e+5 - -KummerU( - 3.14159265358979323846264338328,1,26) = kummeru( - - - 3.14159265358979323846264338328,1,26) - -KummerM( - 3.14159265358979323846264338328, - 3.14159265358979323846264338328,1 - -) = 2.71828182845904523536028747135 - -KummerU( - 3.14159265358979323846264338328, - 3.14159265358979323846264338328,1 - -) = 19.2419564406028465661346237348 - -KummerM( - 3.14159265358979323846264338328,26, - 3.14159265358979323846264338328 - -) = 1.42892253084220157246185820464 - -KummerU( - 3.14159265358979323846264338328,26, - 3.14159265358979323846264338328 - -) = kummeru( - 3.14159265358979323846264338328,26, - - - 3.14159265358979323846264338328) - -KummerM(26,1,0) = 1 - -KummerU(26,1,0) = kummeru(26,1,0) - -KummerM(26,3.14159265358979323846264338328,3.14159265358979323846264338328) = 3. - -91638029828702661403357541917e+5 - -KummerU(26,3.14159265358979323846264338328,3.14159265358979323846264338328) = 4. - -18995913372628050180833640475e-32 - -KummerM(26, - 3.14159265358979323846264338328,26) = 2.19471945265322419268333614 - -674e+34 - -KummerU(26, - 3.14159265358979323846264338328,26) = 0 - - -off rounded; - - -clear x, y, z, t!*k!*c; - - - -df(df(kummerM(a,b,z),z),z); - - - 2 -(kummerm(2,3,z)*z - 2*kummerm(2,3,z)*z + 2*kummerm(2,3,z) + 2*kummerm(1,3,z)*z - - 2 - - 4*kummerm(1,3,z) + 2*kummerm(0,3,z))/z - -df(kummerU(a,b,z),z); - - - kummeru(2,3,z)*z - kummeru(2,3,z) - kummeru(1,3,z) ----------------------------------------------------- - z - - -z := ((x^2 + y)^5) + (x^(x+y)); - - - x + y 10 8 6 2 4 3 2 4 5 -z := x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y - - -df(df(kummerM(a,b,z),y),x); - - - 4*x + 4*y -(x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 4*x + 4*y - *log(x) *x + x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 4*x + 4*y - *x + x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 3*x + 3*y - *y + 2*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 11 3*x + 3*y - *log(x) *x + 10*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 9 3*x + 3*y - *log(x) *x *y + 20*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 7 2 3*x + 3*y - *log(x) *x *y + 20*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 5 3 3*x + 3*y - *log(x) *x *y + 10*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 3 4 3*x + 3*y - *log(x) *x *y + 2*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 5 3*x + 3*y - *log(x) *x*y - x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 3*x + 3*y - *log(x) *x + 2*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 11 3*x + 3*y - *x + 2*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 10 3*x + 3*y - *x *y + 10*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 10 3*x + 3*y - *x + 10*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 9 3*x + 3*y - *x *y + 5*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 9 3*x + 3*y - *x + 10*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 8 2 3*x + 3*y - *x *y + 40*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 8 3*x + 3*y - *x *y + 20*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 7 2 3*x + 3*y - *x *y + 20*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 7 3*x + 3*y - *x *y + 20*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 6 3 3*x + 3*y - *x *y + 60*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 6 2 3*x + 3*y - *x *y + 20*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 5 3 3*x + 3*y - *x *y + 30*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 5 2 3*x + 3*y - *x *y + 20*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 4 4 3*x + 3*y - *x *y + 40*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 4 3 3*x + 3*y - *x *y + 10*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 3 4 3*x + 3*y - *x *y + 20*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 3 3 3*x + 3*y - *x *y + 10*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 2 5 3*x + 3*y - *x *y + 10*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 2 4 3*x + 3*y - *x *y + 2*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 5 3*x + 3*y - *x*y + 5*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 4 3*x + 3*y - *x*y - x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 3*x + 3*y - *x + 2*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 6 3*x + 3*y - *y - x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 3*x + 3*y - *y + 5*x - - x + y 10 8 6 2 4 3 2 4 5 9 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + 5 - - 3*x + 3*y - *x - - x + y 10 8 6 2 4 3 2 4 5 8 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + - - 3*x + 3*y - 20*x - - x + y 10 8 6 2 4 3 2 4 5 7 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + - - 3*x + 3*y - 20*x - - x + y 10 8 6 2 4 3 2 4 5 6 2 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - 3*x + 3*y - + 30*x - - x + y 10 8 6 2 4 3 2 4 5 5 2 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - 3*x + 3*y - + 30*x - - x + y 10 8 6 2 4 3 2 4 5 4 3 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - 3*x + 3*y - + 20*x - - x + y 10 8 6 2 4 3 2 4 5 3 3 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - 3*x + 3*y - + 20*x - - x + y 10 8 6 2 4 3 2 4 5 2 4 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - 3*x + 3*y - + 5*x - - x + y 10 8 6 2 4 3 2 4 5 4 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x*y + - - 3*x + 3*y - 5*x - - x + y 10 8 6 2 4 3 2 4 5 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*y + - - 3*x + 3*y - x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + 2 - - 3*x + 3*y - *x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 3*x + 3*y - *log(x) *x + 2*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 3*x + 3*y - *x + 2*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 2*x + 2*y - *y + x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 21 2*x + 2*y - *log(x) *x + 10*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 19 2*x + 2*y - *log(x) *x *y + 45*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 17 2 2*x + 2*y - *log(x) *x *y + 120*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 15 3 2*x + 2*y - *log(x) *x *y + 210*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 13 4 2*x + 2*y - *log(x) *x *y + 252*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 11 5 2*x + 2*y - *log(x) *x *y + 210*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 9 6 2*x + 2*y - *log(x) *x *y + 120*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 7 7 2*x + 2*y - *log(x) *x *y + 45*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 5 8 2*x + 2*y - *log(x) *x *y + 10*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 3 9 2*x + 2*y - *log(x) *x *y + x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 10 2*x + 2*y - *log(x) *x*y + x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 2 2*x + 2*y - *log(x) *x + x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 21 2*x + 2*y - *x + x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 20 2*x + 2*y - *x *y + 20*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 20 2*x + 2*y - *x + 10*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 19 2*x + 2*y - *x *y + 10*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 19 2*x + 2*y - *x + 10*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 18 2 2*x + 2*y - *x *y + 180*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 18 2*x + 2*y - *x *y + 45*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 17 2 2*x + 2*y - *x *y + 90*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 17 2*x + 2*y - *x *y + 45*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 16 3 2*x + 2*y - *x *y + 720*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 16 2 2*x + 2*y - *x *y + 120*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 15 3 2*x + 2*y - *x *y + 360*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 15 2 2*x + 2*y - *x *y + 120*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 14 4 2*x + 2*y - *x *y + 1680*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 14 3 2*x + 2*y - *x *y + 210*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 13 4 2*x + 2*y - *x *y + 840*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 13 3 2*x + 2*y - *x *y + 210*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 12 5 2*x + 2*y - *x *y + 2520*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 12 4 2*x + 2*y - *x *y + 252*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 11 5 2*x + 2*y - *x *y + 1260*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 11 4 2*x + 2*y - *x *y + 252*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 10 6 2*x + 2*y - *x *y + 2520*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - 10 5 2*x + 2*y - *x *y - 20*x - - x + y 10 8 6 2 4 3 2 4 5 - *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - 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- x + y - + 10*x - - x + y 10 8 6 2 4 3 2 4 5 4 4 - *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - x + y - + 40*x - - x + y 10 8 6 2 4 3 2 4 5 4 3 - *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - x + y - + 5*x - - x + y 10 8 6 2 4 3 2 4 5 3 4 - *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - x + y - + 5*x - - x + y 10 8 6 2 4 3 2 4 5 2 5 - *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - x + y - + 10*x - - x + y 10 8 6 2 4 3 2 4 5 2 4 - *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - x + y - + x - - x + y 10 8 6 2 4 3 2 4 5 5 - *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x*y - - - x + y x + y 10 8 6 2 4 3 2 4 5 - x *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - x + y - *x + x - - x + y 10 8 6 2 4 3 2 4 5 6 - *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*y - - - x + y x + y 10 8 6 2 4 3 2 4 5 - x *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - x + y - *y - x - - x + y 10 8 6 2 4 3 2 4 5 - *kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) - - x + y - *x - x - - x + y 10 8 6 2 4 3 2 4 5 - *kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x - - - x + y x + y 10 8 6 2 4 3 2 4 5 - x *kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - x + y 10 8 6 2 4 3 2 4 5 - *y + 10*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 20 - *x + 90 - - x + y 10 8 6 2 4 3 2 4 5 18 - *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - x + y 10 8 6 2 4 3 2 4 5 - + 360*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 16 2 - *x *y + 840 - - x + y 10 8 6 2 4 3 2 4 5 14 3 - *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - x + y 10 8 6 2 4 3 2 4 5 - + 1260*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 12 4 - *x *y + 1260 - - x + y 10 8 6 2 4 3 2 4 5 10 5 - *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - x + y 10 8 6 2 4 3 2 4 5 - - 10*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 10 - *x + 840 - - x + y 10 8 6 2 4 3 2 4 5 8 6 - *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - x + y 10 8 6 2 4 3 2 4 5 8 - - 40*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x - - *y + 360 - - x + y 10 8 6 2 4 3 2 4 5 6 7 - *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - x + y 10 8 6 2 4 3 2 4 5 6 - - 60*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x - - 2 - *y + 90 - - x + y 10 8 6 2 4 3 2 4 5 4 8 - *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - x + y 10 8 6 2 4 3 2 4 5 4 - - 40*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x - - 3 - *y + 10 - - x + y 10 8 6 2 4 3 2 4 5 2 9 - *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - x + y 10 8 6 2 4 3 2 4 5 2 - - 10*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x - - 4 - *y - 10 - - x + y 10 8 6 2 4 3 2 4 5 10 - *kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x - - - x + y 10 8 6 2 4 3 2 4 5 8 - 40*kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - x + y 10 8 6 2 4 3 2 4 5 - - 60*kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) - - 6 2 - *x *y - 40 - - x + y 10 8 6 2 4 3 2 4 5 4 3 - *kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y - - x + y 10 8 6 2 4 3 2 4 5 2 - - 10*kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x - - 4 x + y 10 8 6 2 4 3 2 4 5 - *y )/(x*(x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )) - - - - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - -% 9. Bessel functions - -% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - - -% Lengthy test of the Bessel functions. This isn't even -% remotely exhaustive of the special cases -- though a -% real person with lots of time could no doubt come up -% with a better lot of tests than this automated rubbish. -% Again, compromise by only actually doing one in five or -% nine. If you want a really thorough test, you can -% easily change this to get it; but it'll take hours to -% run. - -clear p, q; - - - -hankel1(p,q); - - -hankel1(p,q) - -r := df(ws,q); - - - hankel1(p - 1,q)*q - hankel1(p,q)*p -r := ------------------------------------- - q - - -on complex; - - - -r; - - - besselj(p - 1,q)*q - besselj(p,q)*p + i*bessely(p - 1,q)*q - i*bessely(p,q)*p -------------------------------------------------------------------------------- - q - -p:=3/8; - - - 3 -p := --- - 8 - -r; - - - - 5 3*pi - 5 3*pi -(8*i*besselj(------,q)*cos(------)*q + 8*besselj(------,q)*sin(------)*q - 8 8 8 8 - - - 3 5 - + 3*i*besselj(------,q) + 8*i*besselj(---,q)*q - 8 8 - - 3 3*pi 3 3*pi - - 3*i*besselj(---,q)*cos(------) - 3*besselj(---,q)*sin(------))/(8 - 8 8 8 8 - - 3*pi - *sin(------)*q) - 8 - -q := pi; - - -q := pi - -r; - - - - 5 3*pi - 5 3*pi -(8*i*besselj(------,pi)*cos(------)*pi + 8*besselj(------,pi)*sin(------)*pi - 8 8 8 8 - - - 3 5 - + 3*i*besselj(------,pi) + 8*i*besselj(---,pi)*pi - 8 8 - - 3 3*pi 3 3*pi - - 3*i*besselj(---,pi)*cos(------) - 3*besselj(---,pi)*sin(------))/(8 - 8 8 8 8 - - 3*pi - *sin(------)*pi) - 8 - - -on rounded; - - -*** Domain mode complex changed to complex-rounded - - -r; - - - - 0.119366207318921501826662822529 - -*hankel1(0.375,3.14159265358979323846264338328) - - + hankel1( - 0.625,3.14159265358979323846264338328) - - -off complex, rounded; - - -*** Domain mode complex-rounded changed to rounded - - -df(df(besselj(pp,qq)+rr * hankel1(pp*2,qq) * - bessely(pp-qq,qq),qq),qq); - - - 2 -(besselj(pp - 2,qq)*qq - 2*besselj(pp - 1,qq)*pp*qq + besselj(pp - 1,qq)*qq - - 2 - + besselj(pp,qq)*pp + besselj(pp,qq)*pp - - 2 - + bessely(pp - qq,qq)*hankel1(2*pp - 2,qq)*qq *rr - - - 4*bessely(pp - qq,qq)*hankel1(2*pp - 1,qq)*pp*qq*rr - - + bessely(pp - qq,qq)*hankel1(2*pp - 1,qq)*qq*rr - - 2 - + 4*bessely(pp - qq,qq)*hankel1(2*pp,qq)*pp *rr - - + 2*bessely(pp - qq,qq)*hankel1(2*pp,qq)*pp*rr - - 2 - + df(bessely(pp - qq,qq),qq,2)*hankel1(2*pp,qq)*qq *rr - - 2 - + 2*df(bessely(pp - qq,qq),qq)*hankel1(2*pp - 1,qq)*qq *rr - - 2 - - 4*df(bessely(pp - qq,qq),qq)*hankel1(2*pp,qq)*pp*qq*rr)/qq - - -% Possible values for real args -t!*bes!*vr := { 1, pi, -pi, 26 }$ - - - -% Possible values for real and imaginary parts of complex args -t!*bes!*vc := { 0, 3, -41/2 }$ - - - -array s!*bes(4)$ - - - -s!*bes(1) := "BesselJ"$ - - -s!*bes(2) := "BesselY"$ - - -s!*bes(3) := "BesselI"$ - - -s!*bes(4) := "BesselK"$ - - - -pre := 16; - - -pre := 16 - -precision pre; - - -30 - -preord := 10**pre; - - -preord := 10000000000000000 - -t!*b!*c := 3; - - -t*b*c := 3 - - -algebraic procedure do!*one!*bessel(s,n,z); - (if s = 1 then besselj(n,z) - else if s = 2 then bessely(n,z) - else if s = 3 then besseli(n,z) - else besselk(n,z)); - - -do*one*bessel - - -algebraic procedure pr!*bessel(s,n,z,k); - << if t!*b!*c = k - then - << on rounded; - bes1 := do!*one!*bessel(s,n,z); - precision(pre+5); - bes2 := do!*one!*bessel(s,n,z); - if bes1 neq 0 - then disc := floor abs(100*(bes2-bes1)*preord/bes1) - else disc := 0; - precision pre; - write s!*bes(s), "(", n, ",", z, ") = ", bes1; - if not numberp disc then - << precom := !*complex; - on complex; - disc := disc; - if not precom then off complex >>; - if disc neq 0 - then write " (discrepancy ", disc, "% of one s.f.)"; - if numberp disc and disc > 200 then - << write "***** WARNING Significant Inaccuracy."; - write " Lower precision result:"; - write " ", bes1; - write " Higher precision result:"; - precision(pre+5); write " ", bes2; precision pre >>; - off rounded; - t!*b!*c := 0 >>; - t!*b!*c := t!*b!*c + 1 >>; - - -pr*bessel - - -% About to begin Bessel test. We have a list of possible -% values, and we test every Bessel, with every value on the -% list as both order and argument. Every Bessel is computed -% twice, to different precisions (differing by 3), and any -% discrepancy is reported. The value reported is the diff- -% erence between the two computed values, expressed as a -% percentage of the unit of the least significant displayed -% digit. A discrepancy between 100% and 200% means that the -% last digit of the displayed value was found to differ at -% higher precision; values greater than 200% are cause for -% concern. An ideal discrepancy would be between about 1% -% and 20%; if the value is found to be zero, no discrepancy -% is reported. - -off msg; - - - -for s := 1:4 do - << write(" ... Testing ", s!*bes(s), " for real domains ... "); - for each n in t!*bes!*vr do - for each z in t!*bes!*vr do - pr!*bessel(s, n, z, 5) >>; - - - ... Testing BesselJ for real domains ... - -BesselJ(1, - 3.141592653589793) = - 0.2846153431797528 - -BesselJ(3.141592653589793,26) = - 0.006989220174690161 - - (discrepancy 5% of one s.f.) - -BesselJ(26,1) = 3.660826744416803e-35 - - ... Testing BesselY for real domains ... - -BesselY(1,3.141592653589793) = 0.358872916776719 - -BesselY(3.141592653589793, - 3.141592653589793) = - - 6.283185307179586 - 0.1545613960392598*( - 1.570796326794897) - 4.829362563540275 --------------------------------------------------------------------------------- - 3.141592653589793 - ( - 1.570796326794897) - -BesselY( - 3.141592653589793,26) = - 0.1386083623177915 - - ... Testing BesselI for real domains ... - -BesselI(1,1) = 0.565159103992485 - -BesselI(3.141592653589793,3.141592653589793) = 1.011423335928613 - -BesselI( - 3.141592653589793, - 3.141592653589793) = - 0.8856101155917482 - - + 0.4221616153281286*i - -BesselI(26,26) = 68397.86776155122 - - ... Testing BesselK for real domains ... - -BesselK(3.141592653589793,1) = 9.025908765806763 - -BesselK( - 3.141592653589793,3.141592653589793) = 0.1107526602738113 - - (discrepancy 1% of one s.f.) - -BesselK(26, - 3.141592653589793) = besselk(26, - 3.141592653589793) - - -on complex; - - - -for s := 1:3 do - << write (" ... Testing ", s!*bes(s), " for complex domains ... "); - for each nr in t!*bes!*vc do - for each ni in t!*bes!*vc do - for each zr in t!*bes!*vc do - for each zi in t!*bes!*vc do - pr!*bessel(s, nr+ni*i, zr+zi*i, 9) >>; - - - ... Testing BesselJ for complex domains ... - -BesselJ(0, - 20.5 + 3.0*i) = 1.05389016561334 + 1.410918160335249*i - -BesselJ(3*i, - 20.5 + 3.0*i) = 0.01225787392170983 + 0.01066256817009466*i - -BesselJ( - 20.5*i, - 20.5 + 3.0*i) = - 6.607837931625446e+38 - - + 7.203284455482089e+38*i - -BesselJ(3, - 20.5 + 3.0*i) = 1.568613483726435 - 0.7011991107137573*i - -BesselJ(3 + 3*i, - 20.5 + 3.0*i) = 0.007904103001381543 - 0.006566520928092784*i - -BesselJ(3.0 - 20.5*i, - 20.5 + 3.0*i) = - 7.069920310202644e+37 - - - 1.753271554229047e+37*i - -BesselJ( - 20.5, - 20.5 + 3.0*i) = 0.1758742246345278 - 0.332739860634916*i - -BesselJ( - 20.5 + 3.0*i, - 20.5 + 3.0*i) = 0.08815299110072903 - - - 0.1369698556512304*i - -BesselJ( - 20.5 - 20.5*i, - 20.5 + 3.0*i) = - 5.364748129151297e+46 - - + 2.608178375230083e+47*i - - ... Testing BesselY for complex domains ... - -BesselY(0, - 20.5 + 3.0*i) = - 1.404746667469566 + 1.060048452645186*i - -BesselY(3*i, - 20.5 + 3.0*i) = 0.4973091982659732 + 0.7985114801567726*i - -BesselY( - 20.5*i, - 20.5 + 3.0*i) = - 7.203284455482089e+38 - - - 6.607837931625446e+38*i - - (discrepancy 10% of one s.f.) - -BesselY(3, - 20.5 + 3.0*i) = 0.6963128100601111 + 1.576222640523309*i - -BesselY(3 + 3*i, - 20.5 + 3.0*i) = - 1.117333163968302 + 0.9789575771194796*i - -BesselY(3.0 - 20.5*i, - 20.5 + 3.0*i) = 1.753271554229047e+37 - - - 7.069920310202644e+37*i - -BesselY( - 20.5, - 20.5 + 3.0*i) = 0.2353954565395826 + 0.144691313932682*i - -BesselY( - 20.5 + 3.0*i, - 20.5 + 3.0*i) = - 0.1527215881543493 - - + 0.2371137974094512*i - -BesselY( - 20.5 - 20.5*i, - 20.5 + 3.0*i) = - 2.608178375230083e+47 - - - 5.364748129151297e+46*i - - ... Testing BesselI for complex domains ... - -BesselI(0, - 20.5 + 3.0*i) = - 6.891185608459107e+7 - 1.506065792318474e+7*i - -BesselI(3*i, - 20.5 + 3.0*i) = - 6879.511500081906 - 1745.250262122384*i - -BesselI( - 20.5*i, - 20.5 + 3.0*i) = 4.756052726395977e+40 - - - 1.836844915663626e+40*i - -BesselI(3, - 20.5 + 3.0*i) = 5.56813934269752e+7 + 1.026348768124686e+7*i - -BesselI(3 + 3*i, - 20.5 + 3.0*i) = 5917.619873410601 - 1360.623977225443*i - -BesselI(3.0 - 20.5*i, - 20.5 + 3.0*i) = 1.125266362182384e+40 - - - 3.487618645712341e+39*i - -BesselI( - 20.5, - 20.5 + 3.0*i) = - 0.0005935206748136158 - - - 0.001045512507192035*i - -BesselI( - 20.5 + 3.0*i, - 20.5 + 3.0*i) = 0.0001367212493519011 - - - 0.00008899048591346719*i - -BesselI( - 20.5 - 20.5*i, - 20.5 + 3.0*i) = - 4.705541285798881e+45 - - + 6.968045629188714e+45*i - - -off complex; - - - -on msg; - - - -write (" ..."); - - - ... - -write ("Bessel test complete."); - - -Bessel test complete. - - -end; -(TIME: specfn 121310 136641) +REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... + + +% +% Testing file for REDUCE Special Functions Package +% +% Chris Cannam, ZIB Berlin +% October 1992 -> Feb 1993 +% (only some of the time, of course) +% +% Corrections and comments to neun@sc.zib-berlin.de +% + + +on savesfs; + + % just in case it's off for some reason + +off bfspace; + + % to provide more similarity between runs + % with old & new bigfloats + +let {sinh (~x) => (exp(x) - exp (-x))/2 }; + + + % this will improve some results + + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + +% 1. Bernoulli numbers + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + + +off rounded; + + + +procedure do!*one!*bern(x); + write ("Bernoulli ", x, " is ", bernoulli x); + + +do*one*bern + + +do!*one!*bern(1); + + + - 1 +Bernoulli 1 is ------ + 2 + +do!*one!*bern(2); + + + 1 +Bernoulli 2 is --- + 6 + +do!*one!*bern(3); + + +Bernoulli 3 is 0 + +do!*one!*bern(13); + + +Bernoulli 13 is 0 + +do!*one!*bern(14); + + + 7 +Bernoulli 14 is --- + 6 + +do!*one!*bern(300); + + +Bernoulli 300 is ( - +1863878995204859011995045341848156066182191846635905937518715320655775958174360523134990756922303410810482600528769479642021001218415879006164302955370460829146434807964717737195356935144151583424833154250047747433575584999029126775186293388721514970183351129809976971603227633930434923843984829580311593372565398574762880028289167635570012415606941367995702212211519561707046505473575241 + )/ + + 866054419230 + +do!*one!*bern(-2); + + +Bernoulli -2 is bernoulli(-2) + +do!*one!*bern(0); + + +Bernoulli 0 is 1 + + +for n := 2 step 2 until 100 do do!*one!*bern n; + + + 1 +Bernoulli 2 is --- + 6 + + - 1 +Bernoulli 4 is ------ + 30 + + 1 +Bernoulli 6 is ---- + 42 + + - 1 +Bernoulli 8 is ------ + 30 + + 5 +Bernoulli 10 is ---- + 66 + + - 691 +Bernoulli 12 is -------- + 2730 + + 7 +Bernoulli 14 is --- + 6 + + - 3617 +Bernoulli 16 is --------- + 510 + + 43867 +Bernoulli 18 is ------- + 798 + + - 174611 +Bernoulli 20 is ----------- + 330 + + 854513 +Bernoulli 22 is -------- + 138 + + - 236364091 +Bernoulli 24 is -------------- + 2730 + + 8553103 +Bernoulli 26 is --------- + 6 + + - 23749461029 +Bernoulli 28 is ---------------- + 870 + + 8615841276005 +Bernoulli 30 is --------------- + 14322 + + - 7709321041217 +Bernoulli 32 is ------------------ + 510 + + 2577687858367 +Bernoulli 34 is --------------- + 6 + + - 26315271553053477373 +Bernoulli 36 is ------------------------- + 1919190 + + 2929993913841559 +Bernoulli 38 is ------------------ + 6 + + - 261082718496449122051 +Bernoulli 40 is -------------------------- + 13530 + + 1520097643918070802691 +Bernoulli 42 is ------------------------ + 1806 + + - 27833269579301024235023 +Bernoulli 44 is ---------------------------- + 690 + + 596451111593912163277961 +Bernoulli 46 is -------------------------- + 282 + + - 5609403368997817686249127547 +Bernoulli 48 is --------------------------------- + 46410 + + 495057205241079648212477525 +Bernoulli 50 is ----------------------------- + 66 + + - 801165718135489957347924991853 +Bernoulli 52 is ----------------------------------- + 1590 + + 29149963634884862421418123812691 +Bernoulli 54 is ---------------------------------- + 798 + + - 2479392929313226753685415739663229 +Bernoulli 56 is --------------------------------------- + 870 + + 84483613348880041862046775994036021 +Bernoulli 58 is ------------------------------------- + 354 + + - 1215233140483755572040304994079820246041491 +Bernoulli 60 is ------------------------------------------------ + 56786730 + + 12300585434086858541953039857403386151 +Bernoulli 62 is ---------------------------------------- + 6 + + - 106783830147866529886385444979142647942017 +Bernoulli 64 is ----------------------------------------------- + 510 + + 1472600022126335654051619428551932342241899101 +Bernoulli 66 is ------------------------------------------------ + 64722 + + - 78773130858718728141909149208474606244347001 +Bernoulli 68 is ------------------------------------------------- + 30 + + 1505381347333367003803076567377857208511438160235 +Bernoulli 70 is --------------------------------------------------- + 4686 + + - 5827954961669944110438277244641067365282488301844260429 +Bernoulli 72 is ------------------------------------------------------------ + 140100870 + + 34152417289221168014330073731472635186688307783087 +Bernoulli 74 is ---------------------------------------------------- + 6 + + - 24655088825935372707687196040585199904365267828865801 +Bernoulli 76 is ---------------------------------------------------------- + 30 + + 414846365575400828295179035549542073492199375372400483487 +Bernoulli 78 is ----------------------------------------------------------- + 3318 + +Bernoulli 80 is ( - + + 4603784299479457646935574969019046849794257872751288919656867)/ + + 230010 + + 1677014149185145836823154509786269900207736027570253414881613 +Bernoulli 82 is --------------------------------------------------------------- + 498 + +Bernoulli 84 is ( - + +2024576195935290360231131160111731009989917391198090877281083932477 + )/3404310 + +Bernoulli 86 is 660714619417678653573847847426261496277830686653388931761996983/ + + 6 + +Bernoulli 88 is ( - + +1311426488674017507995511424019311843345750275572028644296919890574047 + )/61410 + +Bernoulli 90 is + +1179057279021082799884123351249215083775254949669647116231545215727922535 + /272118 + +Bernoulli 92 is ( - + +1295585948207537527989427828538576749659341483719435143023316326829946247 + )/1410 + +Bernoulli 94 is + +1220813806579744469607301679413201203958508415202696621436215105284649447 + /6 + +Bernoulli 96 is ( - +211600449597266513097597728109824233673043954389060234150638733420050668349987259 + )/4501770 + +Bernoulli 98 is + +67908260672905495624051117546403605607342195728504487509073961249992947058239 + /6 + +Bernoulli 100 is ( - +94598037819122125295227433069493721872702841533066936133385696204311395415197247711 + )/33330 + + +on rounded; + + +precision 100; + + +12 + + +do!*one!*bern(1); + + +Bernoulli 1 is - 0.5 + +do!*one!*bern(2); + + +Bernoulli 2 is 0.166666666666666666666666666666666666666666666666666666666666666 + + 6666666666666666666666666666666666667 + +do!*one!*bern(3); + + +Bernoulli 3 is 0 + +do!*one!*bern(13); + + +Bernoulli 13 is 0 + +do!*one!*bern(14); + + +Bernoulli 14 is 1.16666666666666666666666666666666666666666666666666666666666666 + + 6666666666666666666666666666666666667 + +do!*one!*bern(300); + + +Bernoulli 300 is - 2.1521499732799868297198173767560881985734565727838067018493 + + 43925846796995824343474766557085976828531e+375 + +do!*one!*bern(-2); + + +Bernoulli -2 is bernoulli(-2) + +do!*one!*bern(0); + + +Bernoulli 0 is 1 + +do!*one!*bern(38); + + +Bernoulli 38 is 4.88332318973593166666666666666666666666666666666666666666666666 + + 6666666666666666666666666666666666667e+14 + +do!*one!*bern(400); + + +Bernoulli 400 is - 6.8469448558064533606162585823108835976782300971806257424147 + + 81150311357197834589950328990573681886442e+549 + + + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + +% 2. Gamma function + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + + +on rounded; + + +off complex; + + +precision 40; + + +100 + + +algebraic procedure wg(x); + write ("gamma (", x, ") ==> ", gamma x); + + +wg + + +algebraic procedure wp(x); + write ("-- precision ", x, ", from ", precision(x)); + + +wp + + +wg (1/2); + + +gamma (0.5) ==> 1.772453850905516027298167483341145182798 + +wg (3/2); + + +gamma (1.5) ==> 0.8862269254527580136490837416705725913988 + + +write ("sqrt(pi)/2 ==> ", sqrt(pi)/2); + + +sqrt(pi)/2 ==> 0.8862269254527580136490837416705725913988 + + +wp(10); + + +-- precision 10, from 40 + + +for x := 0 step 5 until 100 do + << wg (1 + x/1000); + wg (-1 - x/13); + wp (8+floor(x/4)) >>; + + +gamma (1) ==> 1 + +gamma (-1) ==> gamma(-1) + +-- precision 8, from 10 + +gamma (1.005) ==> 0.99713854 + +gamma ( - 1.3846154) ==> 2.7320314 + +-- precision 9, from 8 + +gamma (1.01) ==> 0.994325851 + +gamma ( - 1.76923077) ==> 2.89933597 + +-- precision 10, from 9 + +gamma (1.015) ==> 0.9915612888 + +gamma ( - 2.153846154) ==> - 2.919307224 + +-- precision 11, from 10 + +gamma (1.02) ==> 0.98884420326 + +gamma ( - 2.5384615385) ==> - 0.91247160689 + +-- precision 13, from 11 + +gamma (1.025) ==> 0.9861739631483 + +gamma ( - 2.923076923077) ==> - 2.407817725014 + +-- precision 14, from 13 + +gamma (1.03) ==> 0.98354995055382 + +gamma ( - 3.3076923076923) ==> 0.42665848359037 + +-- precision 15, from 14 + +gamma (1.035) ==> 0.980971560550586 + +gamma ( - 3.69230769230769) ==> 0.250121998146955 + +-- precision 16, from 15 + +gamma (1.04) ==> 0.9784382009142447 + +gamma ( - 4.076923076923077) ==> - 0.4868211588210416 + +-- precision 18, from 16 + +gamma (1.045) ==> 0.975949291822951489 + +gamma ( - 4.46153846153846154) ==> - 0.064315864992688343 + +-- precision 19, from 18 + +gamma (1.05) ==> 0.9735042655627756432 + +gamma ( - 4.846153846153846154) ==> - 0.07308480130893001268 + +-- precision 20, from 19 + +gamma (1.055) ==> 0.9711025662416699039 + +gamma ( - 5.2307692307692307692) ==> 0.026504298643014994546 + +-- precision 21, from 20 + +gamma (1.06) ==> 0.968743649511638364209 + +gamma ( - 5.61538461538461538462) ==> 0.00947958151841097406813 + +-- precision 23, from 21 + +gamma (1.065) ==> 0.96642698229883993296188 + +gamma (-6) ==> gamma(-6) + +-- precision 24, from 23 + +gamma (1.07) ==> 0.964152042541366448869499 + +gamma ( - 6.38461538461538461538462) ==> - 0.00224562428754660955672785 + +-- precision 25, from 24 + +gamma (1.075) ==> 0.9619183189344448686422338 + +gamma ( - 6.769230769230769230769231) ==> - 0.0014913552210721080976799 + +-- precision 26, from 25 + +gamma (1.08) ==> 0.95972531068282223532653464 + +gamma ( - 7.1538461538461538461538462) ==> 0.0009821350464726696000373901 + +-- precision 28, from 26 + +gamma (1.085) ==> 0.9575725272601010524841387298 + +gamma ( - 7.538461538461538461538461538) ==> 0.0002081367552774423831456972591 + +-- precision 29, from 28 + +gamma (1.09) ==> 0.95545948817480124076971838245 + +gamma ( - 7.9230769230769230769230769231) ==> 0.000383724703597219454150804519 + +77 + +-- precision 30, from 29 + +gamma (1.095) ==> 0.95338572274293305256571929565 + +gamma ( - 8.30769230769230769230769230769) ==> + + - 0.0000487302698231202992921248405302 + +-- precision 31, from 30 + +gamma (1.1) ==> 0.9513507698668731836292487177265 + +gamma ( - 8.692307692307692307692307692308) ==> + + - 0.00002092711689267207910068115762837 + +-- precision 33, from 31 + + +wg(1/1000000003); + + +gamma (0.000000000999999997000000008999999973) ==> 1.0000000024227843360875231 + +3084681e+9 + + +off rounded; + + + +gamma(17/2); + + + 2027025*sqrt(pi) +------------------ + 256 + +gamma(-17/2); + + + - 512*pi +------------------- + 34459425*sqrt(pi) + +gamma(4); + + +6 + +gamma(0); + + +gamma(0) + +gamma(-4); + + +gamma(-4) + +gamma(-17/3); + + + 6*pi +------------------------ + 17 + 17*sqrt(3)*gamma(----) + 3 + + +p := gamma(x**2) * gamma(x-y**gamma(y)) - (1/(gamma(4*(x-y)))); + + + 2 gamma(y) + gamma(x )*gamma( - y + x)*gamma(4*x - 4*y) - 1 +p := -------------------------------------------------------- + gamma(4*x - 4*y) + +y := 1/4; + + + 1 +y := --- + 4 + +p; + + + 2 1 gamma(1/4) + gamma(x )*gamma( - (---) + x)*gamma(4*x - 1) - 1 + 4 +------------------------------------------------------------ + gamma(4*x - 1) + +x := 3; + + +x := 3 + +p; + + + 1 gamma(1/4) + 146313216000*gamma( - (---) + 3) - 1 + 4 +------------------------------------------------ + 3628800 + +y := -3/8; + + + - 3 +y := ------ + 8 + +p; + + +(128*(2490343877896875*sqrt(pi) + + - 3 (8*pi)/(3*gamma(3/8)*sin((3*pi)/8)) + 3*(------) - 1 + 8 + *gamma(---------------------------------------------------) - 64))/( + - 3 (8*pi)/(3*gamma(3/8)*sin((3*pi)/8)) + (------) + 8 + + 7905853580625*sqrt(pi)) + + +on rounded, complex; + + +*** Domain mode rounded changed to complex-rounded + +precision 50; + + +33 + + +p; + + + - 0.00000000058461000084165968732153392127582134179078414159599 + + + 3.7721251013859508830301986850709684723938237902095e-60*i + + +off rounded, complex; + + +*** Domain mode complex-rounded changed to complex + +clear y; + + + +p; + + + gamma(y) + 40320*gamma( - y + 3)*gamma( - 4*y + 12) - 1 +------------------------------------------------------ + gamma( - 4*y + 12) + + + + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + +% 3. Beta function. Not very interesting + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + + +algebraic procedure do!*one!*beta(x,y); + write ("Beta (", x, ",", y, ") = ", beta(x,y)); + + +do*one*beta + + +do!*one!*beta(0,1); + + +Beta (0,1) = beta(0,1) + +do!*one!*beta(2,-3); + + +Beta (2,-3) = beta(2,-3) + +do!*one!*beta(3,2); + + + 1 +Beta (3,2) = ---- + 12 + +do!*one!*beta(a+b,(c+d)**(b-a)); + + + b + (c + d) + gamma(a + b)*gamma(----------) + b a + (c + d) (c + d) +Beta (a + b,----------) = --------------------------------------------- + a a a b + (c + d) (c + d) *a + (c + d) *b + (c + d) + gamma(------------------------------------) + a + (c + d) + +do!*one!*beta(-3,4); + + +Beta (-3,4) = beta(-3,4) + +do!*one!*beta(-3,2); + + +Beta (-3,2) = beta(-3,2) + +do!*one!*beta(-3,-7.5); + + + - 15 - 15 +Beta (-3,-------) = beta(-3,-------) + 2 2 + +do!*one!*beta((pi * 10), exp(5)); + + + 5 + 5 gamma(e )*gamma(10*pi) +Beta (10*pi,e ) = ------------------------ + 5 + gamma(e + 10*pi) + + +on rounded; + + +precision 30; + + +50 + + +do!*one!*beta(0,1); + + +Beta (0,1) = beta(0,1) + +do!*one!*beta(2,-3); + + +Beta (2,-3) = beta(2,-3) + +do!*one!*beta(3,2); + + +Beta (3,2) = 0.0833333333333333333333333333333 + +do!*one!*beta(a+b,(c+d)**(b-a)); + + + b + (c + d) + gamma(a + b)*gamma(----------) + b a + (c + d) (c + d) +Beta (a + b,----------) = --------------------------------------------- + a a a b + (c + d) (c + d) *a + (c + d) *b + (c + d) + gamma(------------------------------------) + a + (c + d) + +do!*one!*beta(-3,4); + + +Beta (-3,4) = beta(-3,4) + +do!*one!*beta(-3,2); + + +Beta (-3,2) = beta(-3,2) + +do!*one!*beta(-3,-7.5); + + +Beta (-3, - 7.5) = beta(-3, - 7.5) + +do!*one!*beta((pi * 10), exp(5)); + + +Beta (31.4159265358979323846264338328,148.413159102576603421115580041) = 3.26162 + +024071771351768890966259e-37 + + + + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + +% 4. Pochhammer notation + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + + +off rounded; + + + +pochhammer(4,5); + + +6720 + +pochhammer(-4,5); + + + - 24 +--------------- + factorial(-1) + +pochhammer(4,-5); + + + factorial(-2) +--------------- + 6 + +pochhammer(-4,-5); + + + - 1 +------- + 15120 + +pochhammer(17/2,12); + + + 157783444591397625 +-------------------- + 4096 + +pochhammer(-17/2,12); + + + - 516891375 +-------------- + 4096 + +pochhammer(1/3,14)*pochhammer(2/3,15); + + + 148260333813553014031851192320000000 +-------------------------------------- + 68630377364883 + + +q := pochhammer(1/5,11)*pochhammer(2/5,11)*pochhammer(3/5,11)* + pochhammer(1-1/5,11)*pochhammer(1,11)*pochhammer(6/5,11)* + pochhammer(70/50,11)*pochhammer(8/5,11)*pochhammer(9/5,11); + + +q := +30835943859725167092317151562947659325673016884166249458427053145347526260421323040989302952930905614937458388095527696384131072 + / + + 1292469707114105741986576081359316958696581423282623291015625 + + +on complex; + + + +pochhammer(a+b*i,c)*pochhammer(a-b*i,c); + + +pochhammer(a - i*b,c)*pochhammer(a + i*b,c) + + +a := 2; + + +a := 2 + +b := 3; + + +b := 3 + +c := 5; + + +c := 5 + + +pochhammer(a+b*i,c)*pochhammer(a-b*i,c); + + +8950500 + + +off complex; + + +on rounded; + + + +pochhammer(1/5,11)*pochhammer(2/5,11)*pochhammer(3/5,11)* + pochhammer(1-1/5,11)*pochhammer(1,11)*pochhammer(6/5,11)* + pochhammer(70/50,11)*pochhammer(8/5,11)*pochhammer(9/5,11); + + +2.38581559706937212593793381562e+67 + + +q; + + +2.38581559706937212593793381562e+67 + + +pochhammer(pi,floor (pi**8)); + + +2.47253079057195612235973919163e+33625 + +pochhammer(-pi,floor (pi**7)); + + +5.91750008101140123889628058136e+9185 + +pochhammer(1.5,floor (pi**8)); + + +1.88808885937650373473836451368e+33619 + + + + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + +% 5. Digamma function + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + + +procedure do!*one!*psi(x); + << precision (precision(0) + 4)$ + write("Psi of ", x, " is ", psi(x) ) >> ; + + +do*one*psi + + +clear x, y; + + + +z := x * ((x+y)**2 + (x**y)); + + + y 2 2 +z := x*(x + x + 2*x*y + y ) + + +off rounded; + + + +do!*one!*psi(3); + + + 1 + 2*log(2) + psi(---) + psi(1) + 3 + 2 +Psi of 3 is ---------------------------------- + 2 + +do!*one!*psi(pi); + + +Psi of pi is psi(pi) + +do!*one!*psi(1.005); + + + 201 1 +Psi of ----- is psi(-----) + 200 + 200 200 + +do!*one!*psi(1.995); + + + pi 1 + 199*cot(-----)*pi + 199*psi(-----) + 200 + 399 200 200 +Psi of ----- is ------------------------------------------ + 200 199 + +do!*one!*psi(74); + + +Psi of 74 is (269426837164032294756360054754050*log(2) + + 1 + + 134713418582016147378180027377025*psi(---) + 2 + + + 2138308231460573767907619482175*psi(1) + + + 667084944417653637854891458725877)/ + + 136851726813476721146087646859200 + +do!*one!*psi(-1/2); + + + - 1 1 +Psi of ------ is psi(---) + 2 + 2 2 + +do!*one!*psi(-3); + + +Psi of -3 is infinity + +do!*one!*psi(z); + + + y 2 2 y 3 2 2 +Psi of x*(x + x + 2*x*y + y ) is psi(x *x + x + 2*x *y + x*y ) + + +on rounded; + + +precision 100; + + +62 + + +do!*one!*psi(3); + + +Psi of 3 is 0.922784335098467139393487909917597568957840664060076401194232765115 + + 13227322233532906305293670825325048537 + +do!*one!*psi(pi); + + +Psi of 3.14159265358979323846264338327950288419716939937510582097494459230781640 + + 628620899862803482534211706798214809 is 0.9772133079420067332920694864061 + +82343640834609994325638009523286531810592477714131730207565436292873435576949 + +do!*one!*psi(1.005); + + +Psi of 1.005 is - 0.56902091134438283168829569095002880926796887026346909469924 + + 37771372094405771031990077045450115105840960011306543 + +do!*one!*psi(1.995); + + +Psi of 1.995 is 0.41955460302810862865162027228724395469212997650340596127576800 + + 237211077024989654159358528162438486780461247996589393 + +do!*one!*psi(74); + + +Psi of 74 is 4.29729311880466763910635036256036713791964393500098361596464857079 + + 142072638532815275329153012799334519059885201047316254 + +do!*one!*psi(-1/2); + + +Psi of - 0.5 is 0.0364899739785765205590236670012444328068403953395658929528727 + + 461283450292829458978513262827154158754013655907090515460516684 + + 6 + +do!*one!*psi(-3); + + +Psi of -3 is infinity + +do!*one!*psi(z); + + + y 2 2 y 3 2 2 +Psi of x*(x + x + 2*x*y + y ) is psi(x *x + x + 2*x *y + x*y ) + + +precision 15; + + +132 + + +x := 8/3; + + +x := 2.66666666666667 + +y := 7/1000; + + +y := 0.007 + +do!*one!*psi(z); + + +Psi of 21.74768766103287773 is 3.056340330052438423 + + +off rounded; + + + +clear x, y; + + + +df(psi(z), x); + + + y 3 2 2 y y 2 2 +polygamma(1,x *x + x + 2*x *y + x*y )*(x *y + x + 3*x + 4*x*y + y ) + +df(df(psi(z), y),x); + + + 2*y y 3 2 2 +x *log(x)*polygamma(2,x *x + x + 2*x *y + x*y )*x*y + + 2*y y 3 2 2 + + x *log(x)*polygamma(2,x *x + x + 2*x *y + x*y )*x + + y y 3 2 2 3 + + 3*x *log(x)*polygamma(2,x *x + x + 2*x *y + x*y )*x + + y y 3 2 2 2 + + 4*x *log(x)*polygamma(2,x *x + x + 2*x *y + x*y )*x *y + + y y 3 2 2 2 + + x *log(x)*polygamma(2,x *x + x + 2*x *y + x*y )*x*y + + y y 3 2 2 + + x *log(x)*polygamma(1,x *x + x + 2*x *y + x*y )*y + + y y 3 2 2 + + x *log(x)*polygamma(1,x *x + x + 2*x *y + x*y ) + + y y 3 2 2 2 + + 2*x *polygamma(2,x *x + x + 2*x *y + x*y )*x *y + + y y 3 2 2 2 + + 2*x *polygamma(2,x *x + x + 2*x *y + x*y )*x + + y y 3 2 2 2 + + 2*x *polygamma(2,x *x + x + 2*x *y + x*y )*x*y + + y y 3 2 2 + + 2*x *polygamma(2,x *x + x + 2*x *y + x*y )*x*y + + y y 3 2 2 + + x *polygamma(1,x *x + x + 2*x *y + x*y ) + + y 3 2 2 4 + + 6*polygamma(2,x *x + x + 2*x *y + x*y )*x + + y 3 2 2 3 + + 14*polygamma(2,x *x + x + 2*x *y + x*y )*x *y + + y 3 2 2 2 2 + + 10*polygamma(2,x *x + x + 2*x *y + x*y )*x *y + + y 3 2 2 3 + + 2*polygamma(2,x *x + x + 2*x *y + x*y )*x*y + + y 3 2 2 + + 4*polygamma(1,x *x + x + 2*x *y + x*y )*x + + y 3 2 2 + + 2*polygamma(1,x *x + x + 2*x *y + x*y )*y + +int(psi(z), z); + + + y 3 2 2 +log(gamma(x *x + x + 2*x *y + x*y )) + + +on rounded; + + + +for k := 1 step 0.1 until 2 do do!*one!*psi(k); + + +Psi of 1 is - 0.57721566490153286060651 + +Psi of 1.09999999999999999999999999 is - 0.423754940411076795168216226 + +Psi of 1.19999999999999999999999999483 is - 0.2890398965921882955472079690017 + +Psi of 1.2999999999999999999999999948304368 is + + - 0.16919088886679965563116117990224309 + +Psi of 1.39999999999999999999999999483043679294 is + + - 0.0613845445851161457306754873481741603465 + +Psi of 1.499999999999999999999999994830436792938162 is 0.03648997397857652055902 + +366216872537099062413 + +Psi of 1.5999999999999999999999999948304367929381615219 is 0.1260474527734762519 + +0600271342271679469086378102 + +Psi of 1.69999999999999999999999999483043679293816152191799 is 0.208547874873493 + +95667996417990930337500109911872963 + +Psi of 1.799999999999999999999999994830436792938161521917991479 is 0.28499143329 + +38615406087023630333301945809176983421812009 + +Psi of 1.8999999999999999999999999948304367929381615219179914789724 is 0.3561841 + +6116405971922472708037210055174041765918986424485621 + +Psi of 1.99999999999999999999999999483043679293816152191799147897240477 is 0.422 + +784335098467139393487906583570145998489083710421411058968004 + + +off rounded; + + + +% PSI_SIMP.TST F.J.Wright, 2 July 1993 + +on evallhseqp; + + +factor psi; + + on rat, intstr, div; + + % for neater output +% Do not try using "off mcd"! + +psi(x+m) - psi(x+m-1) = 1/(x+m-1); + + + 1 1 +-----------=----------- + m + x - 1 m + x - 1 + +psi(x+2) - psi(x+1) + 2*psi(x) = 1/(x+1) + 2*psi(x); + + + 1 1 +2*psi(x) + -------=2*psi(x) + ------- + x + 1 x + 1 + +psi(x+2) + 3*psi(x) = 4*psi(x) + 1/x + 1/(x+1); + + + -1 -1 + x + 2 x + 2 +4*psi(x) + ---------=4*psi(x) + --------- + x + 1 x + 1 + + +psi(x + 1) = psi(x) + 1/x; + + + -1 -1 +psi(x) + x =psi(x) + x + +psi(x + 3/2) = psi(x + 1/2) + 1/(x + 1/2); + + + 1 2 1 2 +psi(x + ---) + ---------=psi(x + ---) + --------- + 2 2*x + 1 2 2*x + 1 + +psi(x - 1/2) = psi(x + 1/2) - 1/(x - 1/2); + + + 1 2 1 2 +psi(x + ---) - ---------=psi(x + ---) - --------- + 2 2*x - 1 2 2*x - 1 + +psi((x + 3a)/a); + + + -1 + 1 2*(3*x + 8*x + 12) +psi(---*x) + ---------------------- + 2 2 + x + 6*x + 8 + psi(x/y + 3); + + + -1 2 + -1 y*(3*x + 2*x *y + 6*y) +psi(x*y ) + -------------------------- + 2 2 + x + 3*x*y + 2*y + + +off rat, intstr, div; + + on rational; + + + +psi(x+m) - psi(x+m-1) = 1/(x+m-1); + + + 1 1 +-----------=----------- + m + x - 1 m + x - 1 + +psi(x+2) - psi(x+1) + 2*psi(x) = 1/(x+1) + 2*psi(x); + + + 2*psi(x)*x + 2*psi(x) + 1 2*psi(x)*x + 2*psi(x) + 1 +---------------------------=--------------------------- + x + 1 x + 1 + +psi(x+2) + 3*psi(x) = 4*psi(x) + 1/x + 1/(x+1); + + + 2 2 + 4*psi(x)*x + 4*psi(x)*x + 2*x + 1 4*psi(x)*x + 4*psi(x)*x + 2*x + 1 +------------------------------------=------------------------------------ + 2 2 + x + x x + x + + +psi(x + 1) = psi(x) + 1/x; + + + psi(x)*x + 1 psi(x)*x + 1 +--------------=-------------- + x x + +psi(x + 3/2) = psi(x + 1/2) + 1/(x + 1/2); + + + 1 1 1 1 1 1 + psi(x + ---)*x + ---*psi(x + ---) + 1 psi(x + ---)*x + ---*psi(x + ---) + 1 + 2 2 2 2 2 2 +---------------------------------------=--------------------------------------- + 1 1 + x + --- x + --- + 2 2 + +psi(x - 1/2) = psi(x + 1/2) - 1/(x - 1/2); + + + 1 1 1 1 1 1 + psi(x + ---)*x - ---*psi(x + ---) - 1 psi(x + ---)*x - ---*psi(x + ---) - 1 + 2 2 2 2 2 2 +---------------------------------------=--------------------------------------- + 1 1 + x - --- x - --- + 2 2 + +psi((x + 3a)/a); + + + 1 2 2 8 + psi(---*x)*x*(x + 6*x + 8) + 6*(x + 4*x + ---) + 2 3 +-------------------------------------------------- + 2 + x*(x + 6*x + 8) + psi(x/y + 3); + + + x 2 2 2 2 2 + psi(---)*x*(x + 3*x*y + 2*y ) + 3*y*(x + 2*x*y + ---*y ) + y 3 +------------------------------------------------------------ + 2 2 + x*(x + 3*x*y + 2*y ) + + +off rational; + + + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + +% 6. Polygamma functions + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + + +procedure do!*one!*pg(n,x); + write ("Polygamma (", n, ") of ", x, " is ", polygamma(n,x)); + + +do*one*pg + + +off rounded; + + + +do!*one!*pg(1,1/2); + + + 1 1 +Polygamma (1) of --- is polygamma(1,---) + 2 2 + +do!*one!*pg(1,1); + + + 2 + pi +Polygamma (1) of 1 is ----- + 6 + +do!*one!*pg(1,3/2); + + + 2 + 3 pi - 8 +Polygamma (1) of --- is --------- + 2 2 + +do!*one!*pg(1,1.005); + + + 201 201 +Polygamma (1) of ----- is polygamma(1,-----) + 200 200 + +do!*one!*pg(1,1.995); + + + 399 399 +Polygamma (1) of ----- is polygamma(1,-----) + 200 200 + +do!*one!*pg(1,1e-10); + + + 1 1 +Polygamma (1) of ------------- is polygamma(1,-------------) + 10000000000 10000000000 + +do!*one!*pg(2,1.45); + + + 29 29 +Polygamma (2) of ---- is polygamma(2,----) + 20 20 + +do!*one!*pg(3,1.99); + + + 199 199 +Polygamma (3) of ----- is polygamma(3,-----) + 100 100 + +do!*one!*pg(4,-8.2); + + + - 41 - 41 +Polygamma (4) of ------- is polygamma(4,-------) + 5 5 + +do!*one!*pg(5,0); + + +Polygamma (5) of 0 is infinity + +do!*one!*pg(6,-5); + + +Polygamma (6) of -5 is infinity + +do!*one!*pg(7,200); + + +Polygamma (7) of 200 is ( +17726903568079021516229662669344581999513453462651624712682433450930396569413638060869523209196416456149981841411350319061723276968944831274405326841106087088051862580482604049179034792880997036306450715491134920388121743532178874504889199026667348488846246287054278592284199023548572837863642592371308428279823642858774260678182876022899303888558378920013392172441409952343098219076353830605697314645020537255667718925997023946826863459069360054494011515187931095594830347819766125471751832158677864778297413490571816668099535649328289203181141151677544880676008798702466775519536091686976761627732998715124407667063278175501392084867643186474774933465396962161891279302009565617687762042880000000000000000000000 + 8 + *pi - +168202274324609919911886603179553832485831631970776943913214724306758011473978941132413094748296521617074167117661053498983406945306976674698602419479427329202365001144489134708765120356232374484239386020849061429951991522474455058548094378661234695692419462486402225369012574020278308089090741397551205530205566242796380021997272735088025054393694714504643986352458446128585149055841912367918690170304028682697422603414064799513570602284100682485151815951427876744662169487167539572896985974962832514966524381217035170993953423670212502913136819535576467936946089479260579835796147751358821073694464012485544356514018523618843496912692835966265011413341612149624871304965950289830955139972163546512706990894178649281 + )/ +33237944190148165342930617505021091249087725242471796336279562720494493567650571364130356017243280855281215952646281848240731144316771558639509987827073913290097242338404882592210690236651869443074595091545877975727728269122835389696667248175001278416586711788226772360532873169153574070994329860696203303024669330360201738771592892542936194791046960475025110323327643660643309160768163432385682464959413507354376972986244419900300368985755050102176271590977370804240306902162061485259534685297520996459307650294822156252686629342490542255964639659395396651267516497567125204099130171913081428051999372590858264375743646579065110159126830974640203000247619304053546148691267935533164553830400000000000000000000000 + + +on rounded; + + +precision 100; + + +63 + + +do!*one!*pg(1,1/2); + + +Polygamma (1) of 0.5 is 4.934802200544679309417245499938075567656849703620395313 + +206674688110022411209602621500886701859276116 + +do!*one!*pg(1,1); + + +Polygamma (1) of 1 is 1.64493406684822643647241516664602518921894990120679843773 + +5558229370007470403200873833628900619758705 + +do!*one!*pg(1,3/2); + + +Polygamma (1) of 1.5 is 0.934802200544679309417245499938075567656849703620395313 + +2066746881100224112096026215008867018592761159 + +do!*one!*pg(1,1.005); + + +Polygamma (1) of 1.005 is 1.6329941567556809752535869786110269899606654266292810 + +17423768306472667920740136690588658095268582677 + +do!*one!*pg(1,1.995); + + +Polygamma (1) of 1.995 is 0.6469608286405823512476399271078276952436864784225077 + +824256922477813603546774633500406650324443323374 + +do!*one!*pg(1,1e-10); + + +Polygamma (1) of 0.0000000001 is 1.000000000000000000016449340666078150558729660 + +065732639916420752057756424780180942170336642993019363e+20 + +do!*one!*pg(2,1.45); + + +Polygamma (2) of 1.45 is - 0.90383740307625768823230953852986719286009030811144 + +6658245367522559171482627166928803699348863802546 + +do!*one!*pg(3,1.99); + + +Polygamma (3) of 1.99 is 0.50290713241686532018051099086257019718615168956130804 + +44480934082942919777908933506993776280887706183 + +do!*one!*pg(4,-8.2); + + +Polygamma (4) of - 8.2 is 74935.51259577427052712029230781313552295905493466277 + +917617629718610711391593937121099755096024434196 + +do!*one!*pg(5,0); + + +Polygamma (5) of 0 is infinity + +do!*one!*pg(6,-5); + + +Polygamma (6) of -5 is infinity + +do!*one!*pg(7,200); + + +Polygamma (7) of 200 is 5.724093725392558373837029681544727245346807368195956154 + +743567847385926644129753317064645661270233713e-14 + + +off rounded; + + +clear x; + + + + +% Polygamma differentiation has already +% been tried a bit in the psi section + +df(int(int(int(polygamma(3,x),x),x),x),x); + + +polygamma(1,x) + + +clear w, y, z; + + + + + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + +% 7. Zeta function + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + + +procedure do!*one!*zeta(n); + write ("Zeta of ", n, " is ", zeta n); + + +do*one*zeta + + +off rounded; + + +clear x, y, z; + + + +z := x * ((x+y)**5 + (x**y)); + + + y 5 4 3 2 2 3 4 5 +z := x*(x + x + 5*x *y + 10*x *y + 10*x *y + 5*x*y + y ) + + +do!*one!*zeta(0); + + + - 1 +Zeta of 0 is ------ + 2 + + +for k := 4 step 2 until 35 do + do!*one!*zeta(k); + + + 4 + pi +Zeta of 4 is ----- + 90 + + 6 + pi +Zeta of 6 is ----- + 945 + + 8 + pi +Zeta of 8 is ------ + 9450 + + 10 + pi +Zeta of 10 is ------- + 93555 + + 12 + 691*pi +Zeta of 12 is ----------- + 638512875 + + 14 + 2*pi +Zeta of 14 is ---------- + 18243225 + + 16 + 3617*pi +Zeta of 16 is -------------- + 325641566250 + + 18 + 43867*pi +Zeta of 18 is ---------------- + 38979295480125 + + 20 + 174611*pi +Zeta of 20 is ------------------ + 1531329465290625 + + 22 + 155366*pi +Zeta of 22 is ------------------- + 13447856940643125 + + 24 + 236364091*pi +Zeta of 24 is ----------------------- + 201919571963756521875 + + 26 + 1315862*pi +Zeta of 26 is ---------------------- + 11094481976030578125 + + 28 + 6785560294*pi +Zeta of 28 is -------------------------- + 564653660170076273671875 + + 30 + 6892673020804*pi +Zeta of 30 is ------------------------------ + 5660878804669082674070015625 + +Zeta of 32 is zeta(32) + +Zeta of 34 is zeta(34) + + +do!*one!*zeta(-17/3); + + + - 17 - 17 +Zeta of ------- is zeta(-------) + 3 3 + +do!*one!*zeta(190); + + +Zeta of 190 is zeta(190) + +do!*one!*zeta(300); + + +Zeta of 300 is zeta(300) + +do!*one!*zeta(0); + + + - 1 +Zeta of 0 is ------ + 2 + +do!*one!*zeta(-44); + + +Zeta of -44 is 0 + + +on rounded; + + +clear x, y; + + + +for k := 3 step 3 until 36 do << + precision (31+k*3); + do!*one!*zeta(k) >>; + + +Zeta of 3 is 1.202056903159594285399738161511449990765 + +Zeta of 6 is 1.017343061984449139714517929790920527901817490033 + +Zeta of 9 is 1.002008392826082214417852769232412060485605851394888756549 + +Zeta of 12 is 1.0002460865533080482986379980477396709604160884580034045330409521 + + 33 + +Zeta of 15 is 1.0000305882363070204935517285106450625876279487068581775065699328 + + 93332267156 + +Zeta of 18 is 1.0000038172932649998398564616446219397304546972189533311431744299 + + 87630039542650045638 + +Zeta of 21 is 1.0000004769329867878064631167196043730459664466947849376002074873 + + 76596839087898159833876638564 + +Zeta of 24 is 1.0000000596081890512594796124402079358012275039188373027958642469 + + 72321724495355468544848206832825003614 + +Zeta of 27 is 1.0000000074507117898354294919810041706041194547190318825658299932 + + 39578352147606271570867900837100313523764933952 + +Zeta of 30 is 1.0000000009313274324196681828717647350212198135679551368161850086 + + 13360441960672940496363503624604027929086312123388047291 + +Zeta of 33 is 1.0000000001164155017270051977592973835456309516522471727635932565 + + 17739947029124624567548673934974376008810870912845774213829513369 + +Zeta of 36 is 1.0000000000145519218910419842359296322453184209838088941240380691 + + 395422185717458650302201529989423295781853630847913399997790928914 + + 91916899 + + +precision 20; + + +139 + + +do!*one!*zeta(-17/3); + + +Zeta of - 5.6666666666666666667 is - 0.0018766468228592287697 + +do!*one!*zeta(z); + + + y 5 4 3 2 2 3 4 5 +Zeta of x*(x + x + 5*x *y + 10*x *y + 10*x *y + 5*x*y + y ) is zeta( + + y 6 5 4 2 3 3 2 4 5 + x *x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + x*y ) + + +y := 3; + + +y := 3 + +x := pi; + + +x := 3.1415926535897932385 + + +do!*one!*zeta(z); + + +Zeta of 27548.203250393209469 is 1 + +do!*one!*zeta(190); + + +Zeta of 190 is 1.0 + +do!*one!*zeta(300); + + +Zeta of 300 is 1 + +do!*one!*zeta(0); + + +Zeta of 0 is - 0.5 + +do!*one!*zeta(-44); + + +Zeta of -44 is 0 + + +off rounded; + + + + + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + +% 8. Kummer functions + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + + +off rounded; + + + +t!*kummer!*a := { 1, 2.4, -1397/10 }$ + + +t!*kummer!*b := { 0, 1, pi, -pi, 26 }$ + + + +for each a in t!*kummer!*a do + for each b in t!*kummer!*a do + for each z in t!*kummer!*a do + << write "KummerM(", a, ",", b, ",", z, ") = ", + kummerm(a,b,z); + write "KummerU(", a, ",", b, ",", z, ") = ", + kummeru(a,b,z) >>; + + +KummerM(1,1,1) = e + +KummerU(1,1,1) = kummeru(1,1,1) + + 12 2/5 2 +KummerM(1,1,----) = e *e + 5 + + 12 12 +KummerU(1,1,----) = kummeru(1,1,----) + 5 5 + + - 1397 1 +KummerM(1,1,---------) = ------------ + 10 7/10 139 + e *e + + - 1397 - 1397 +KummerU(1,1,---------) = kummeru(1,1,---------) + 10 10 + + 12 12 +KummerM(1,----,1) = kummerm(1,----,1) + 5 5 + + 2 12 12 + 2*gamma(---)*(gamma(----)*e - kummerm(1,----,1)) + 12 5 5 5 +KummerU(1,----,1) = -------------------------------------------------- + 5 12 + 5*gamma(----) + 5 + + 12 12 12 12 +KummerM(1,----,----) = kummerm(1,----,----) + 5 5 5 5 + + 12 12 +KummerU(1,----,----) = + 5 5 + + 2 2/5 2/5 12 2 4/5 2/5 12 12 + gamma(---)*(5*e *5 *gamma(----)*e - 12*2 *3 *kummerm(1,----,----)) + 5 5 5 5 +----------------------------------------------------------------------------- + 4/5 2/5 12 + 30*2 *3 *gamma(----) + 5 + + 12 - 1397 12 - 1397 +KummerM(1,----,---------) = kummerm(1,----,---------) + 5 10 5 10 + + 12 - 1397 2 +KummerU(1,----,---------) = (2*gamma(---)*( + 5 10 5 + + 7/10 2/5 12 - 1397 139 + - 1397*e *1397 *kummerm(1,----,---------)*e + 5 10 + + 2/5 12 7/10 2/5 12 139 + - 10*10 *gamma(----)))/(6985*e *1397 *gamma(----)*e ) + 5 5 + + - 1397 - 1397 +KummerM(1,---------,1) = kummerm(1,---------,1) + 10 10 + + - 1397 +KummerU(1,---------,1) = + 10 + + 1397 - 1397 3*pi + 1397*gamma(------)*kummerm(1,---------,1)*sin(------) - 10*e*pi + 10 10 10 +----------------------------------------------------------------- + 1417 3*pi + 10*gamma(------)*sin(------) + 10 10 + + - 1397 12 - 1397 12 +KummerM(1,---------,----) = kummerm(1,---------,----) + 10 5 10 5 + + - 1397 12 +KummerU(1,---------,----) = ( - + 10 5 +48689401707698071401637346296148709517635305466315546784369062686038955387470672028214421172492426257970894859785608060010798787187515790195560655355904 + 2/5 + *e + + 2/5 7/10 2 + *2 *3 *e *pi + +20045966895736519148343810333245571434480905551887283539400953780162950579324387945234775543212890625 + 7/10 1397 - 1397 12 3*pi + *5 *gamma(------)*kummerm(1,---------,----)*sin(------)) + 10 10 5 10 + +/( +143492962746861268062589909328887411843098822848155215027923792270314606867032125592231750488281250 + 7/10 1417 3*pi + *5 *gamma(------)*sin(------)) + 10 10 + + - 1397 - 1397 - 1397 - 1397 +KummerM(1,---------,---------) = kummerm(1,---------,---------) + 10 10 10 10 + + - 1397 - 1397 +KummerU(1,---------,---------) = (1397*( - + 10 10 +152159982579315173600331019038628152238885433889572221061155552992026662790488679161680024820756008730757778143037042693923932846444113056473446432154278794079710763645820340853851602689287025084815217783532480956517472235723731181252401977856952544597761994384769996636245679210281354768553595796003820698711669598622602752713703151447293557592795912091076094345980353443096817710446904761297306235868715700448137787799653833690839289133 + 7/10 + *( - 1397) *pi + +10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + 7/10 7/10 + *e *10 + + 1397 - 1397 - 1397 3*pi 139 + *gamma(------)*kummerm(1,---------,---------)*sin(------)*e ))/( + 10 10 10 10 +100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + 7/10 7/10 + *e *10 + + 1417 3*pi 139 + *gamma(------)*sin(------)*e ) + 10 10 + + 12 12 +KummerM(----,1,1) = kummerm(----,1,1) + 5 5 + + 12 12 +KummerU(----,1,1) = kummeru(----,1,1) + 5 5 + + 12 12 12 12 +KummerM(----,1,----) = kummerm(----,1,----) + 5 5 5 5 + + 12 12 12 12 +KummerU(----,1,----) = kummeru(----,1,----) + 5 5 5 5 + + 12 - 1397 12 - 1397 +KummerM(----,1,---------) = kummerm(----,1,---------) + 5 10 5 10 + + 12 - 1397 12 - 1397 +KummerU(----,1,---------) = kummeru(----,1,---------) + 5 10 5 10 + + 12 12 +KummerM(----,----,1) = e + 5 5 + + 2 - 2 2*pi + 2*gamma(---)*kummerm(1,------,1)*sin(------) + 5*e*pi + 12 12 5 5 5 +KummerU(----,----,1) = ------------------------------------------------------- + 5 5 12 2*pi + 5*gamma(----)*sin(------) + 5 5 + + 12 12 12 2/5 2 +KummerM(----,----,----) = e *e + 5 5 5 + + 12 12 12 +KummerU(----,----,----) = + 5 5 5 + + 2/5 4/5 2/5 2 2/5 2 - 2 12 2*pi + 6*e *2 *3 *e *pi + 5 *gamma(---)*kummerm(1,------,----)*sin(------) + 5 5 5 5 +----------------------------------------------------------------------------- + 4/5 2/5 12 2*pi + 6*2 *3 *gamma(----)*sin(------) + 5 5 + + 12 12 - 1397 1 +KummerM(----,----,---------) = ------------ + 5 5 10 7/10 139 + e *e + + 12 12 - 1397 +KummerU(----,----,---------) = ( + 5 5 10 + + 7/10 2/5 2 - 2 - 1397 2*pi 139 + - 4*e *10 *gamma(---)*kummerm(1,------,---------)*sin(------)*e + 5 5 10 5 + + 2/5 7/10 2/5 12 2*pi 139 + + 1397*1397 *pi)/(1397*e *1397 *gamma(----)*sin(------)*e ) + 5 5 + + 12 - 1397 12 - 1397 +KummerM(----,---------,1) = kummerm(----,---------,1) + 5 10 5 10 + + 12 - 1397 1431 1431 1417 +KummerU(----,---------,1) = ( - 10*gamma(------)*kummerm(------,------,1)*pi + + 5 10 10 10 10 + + 1417 1397 12 12 - 1397 + 1397*gamma(------)*gamma(------)*gamma(----)*kummerm(----,---------,1) + 10 10 5 5 10 + + 3*pi 1431 1417 12 3*pi + *sin(------))/(10*gamma(------)*gamma(------)*gamma(----)*sin(------)) + 10 10 10 5 10 + + 12 - 1397 12 12 - 1397 12 +KummerM(----,---------,----) = kummerm(----,---------,----) + 5 10 5 5 10 5 + + 12 - 1397 12 +KummerU(----,---------,----) = ( + 5 10 5 +20045966895736519148343810333245571434480905551887283539400953780162950579324387945234775543212890625 + 7/10 1417 1397 12 + *5 *gamma(------)*gamma(------)*gamma(----) + 10 10 5 + + 12 - 1397 12 3*pi + *kummerm(----,---------,----)*sin(------) - + 5 10 5 10 +48689401707698071401637346296148709517635305466315546784369062686038955387470672028214421172492426257970894859785608060010798787187515790195560655355904 + 2/5 + *2 + + 7/10 1431 1431 1417 12 + *3 *gamma(------)*kummerm(------,------,----)*pi)/( + 10 10 10 5 +143492962746861268062589909328887411843098822848155215027923792270314606867032125592231750488281250 + 7/10 1431 1417 12 3*pi + *5 *gamma(------)*gamma(------)*gamma(----)*sin(------)) + 10 10 5 10 + + 12 - 1397 - 1397 12 - 1397 - 1397 +KummerM(----,---------,---------) = kummerm(----,---------,---------) + 5 10 10 5 10 10 + + 12 - 1397 - 1397 +KummerU(----,---------,---------) = (1397*( - + 5 10 10 +152159982579315173600331019038628152238885433889572221061155552992026662790488679161680024820756008730757778143037042693923932846444113056473446432154278794079710763645820340853851602689287025084815217783532480956517472235723731181252401977856952544597761994384769996636245679210281354768553595796003820698711669598622602752713703151447293557592795912091076094345980353443096817710446904761297306235868715700448137787799653833690839289133 + 7/10 1431 + *( - 1397) *gamma(------) + 10 + + 1431 1417 - 1397 + *kummerm(------,------,---------)*pi + + 10 10 10 +10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + 7/10 + *10 + + 1417 1397 12 12 - 1397 - 1397 + *gamma(------)*gamma(------)*gamma(----)*kummerm(----,---------,---------) + 10 10 5 5 10 10 + + 3*pi + *sin(------)))/( + 10 +100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + 7/10 + *10 + + 1431 1417 12 3*pi + *gamma(------)*gamma(------)*gamma(----)*sin(------)) + 10 10 5 10 + + - 1397 - 1397 +KummerM(---------,1,1) = kummerm(---------,1,1) + 10 10 + + - 1397 - 1397 +KummerU(---------,1,1) = kummeru(---------,1,1) + 10 10 + + - 1397 12 - 1397 12 +KummerM(---------,1,----) = kummerm(---------,1,----) + 10 5 10 5 + + - 1397 12 - 1397 12 +KummerU(---------,1,----) = kummeru(---------,1,----) + 10 5 10 5 + + - 1397 - 1397 - 1397 - 1397 +KummerM(---------,1,---------) = kummerm(---------,1,---------) + 10 10 10 10 + + - 1397 - 1397 - 1397 - 1397 +KummerU(---------,1,---------) = kummeru(---------,1,---------) + 10 10 10 10 + + - 1397 12 - 1397 12 +KummerM(---------,----,1) = kummerm(---------,----,1) + 10 5 10 5 + + - 1397 12 +KummerU(---------,----,1) = ( + 10 5 + + 1411 - 1397 12 pi + 7055*gamma(------)*kummerm(---------,----,1)*sin(----)*pi + 2794 + 10 10 5 10 + + 1397 12 2 - 1411 - 2 3*pi + *gamma(------)*gamma(----)*gamma(---)*kummerm(---------,------,1)*sin(------) + 10 5 5 10 5 10 + + 2*pi 12 2*pi + *sin(------))/(50*gamma(----)*sin(------)*pi) + 5 5 5 + + - 1397 12 12 - 1397 12 12 +KummerM(---------,----,----) = kummerm(---------,----,----) + 10 5 5 10 5 5 + + - 1397 12 12 2/5 1397 12 2 +KummerU(---------,----,----) = (1397*5 *gamma(------)*gamma(----)*gamma(---) + 10 5 5 10 5 5 + + - 1411 - 2 12 3*pi 2*pi + *kummerm(---------,------,----)*sin(------)*sin(------) + 10 5 5 10 5 + + 4/5 2/5 1411 - 1397 12 12 pi + + 8466*2 *3 *gamma(------)*kummerm(---------,----,----)*sin(----)*pi)/( + 10 10 5 5 10 + + 4/5 2/5 12 2*pi + 60*2 *3 *gamma(----)*sin(------)*pi) + 5 5 + + - 1397 12 - 1397 - 1397 12 - 1397 +KummerM(---------,----,---------) = kummerm(---------,----,---------) + 10 5 10 10 5 10 + + - 1397 12 - 1397 +KummerU(---------,----,---------) = ( + 10 5 10 + + 2/5 1411 - 1397 12 - 1397 pi + 1411*1397 *gamma(------)*kummerm(---------,----,---------)*sin(----)*pi - 4 + 10 10 5 10 10 + + 2/5 1397 12 2 + *10 *gamma(------)*gamma(----)*gamma(---) + 10 5 5 + + - 1411 - 2 - 1397 3*pi 2*pi 2/5 + *kummerm(---------,------,---------)*sin(------)*sin(------))/(10*1397 + 10 5 10 10 5 + + 12 2*pi + *gamma(----)*sin(------)*pi) + 5 5 + + - 1397 - 1397 +KummerM(---------,---------,1) = e + 10 10 + + - 1397 - 1397 +KummerU(---------,---------,1) = + 10 10 + + 1397 1417 1417 + 1397*gamma(------)*(gamma(------)*e - kummerm(1,------,1)) + 10 10 10 +------------------------------------------------------------ + 1417 + 10*gamma(------) + 10 + + - 1397 - 1397 12 2/5 2 +KummerM(---------,---------,----) = e *e + 10 10 5 + + - 1397 - 1397 12 1397 +KummerU(---------,---------,----) = (1397*gamma(------)*( + 10 10 5 10 +71746481373430634031294954664443705921549411424077607513961896135157303433516062796115875244140625 + 2/5 7/10 1417 2 + *e *5 *gamma(------)*e - + 10 +24344700853849035700818673148074354758817652733157773392184531343019477693735336014107210586246213128985447429892804030005399393593757895097780327677952 + 2/5 + *2 + + 7/10 1417 12 + *3 *kummerm(1,------,----)))/( + 10 5 +717464813734306340312949546644437059215494114240776075139618961351573034335160627961158752441406250 + 7/10 1417 + *5 *gamma(------)) + 10 + + - 1397 - 1397 - 1397 1 +KummerM(---------,---------,---------) = ------------ + 10 10 10 7/10 139 + e *e + + - 1397 - 1397 - 1397 1397 +KummerU(---------,---------,---------) = (1397*gamma(------)*( - + 10 10 10 10 +212567495663303297519662433596963528677722951143732392822434307529861247918312684788866994674596144196868616065822748643411734186482425939893404665719527475329355936813211016172830688956933974043486859243594875896254908713306052460209605563066162704803073506155523685300835213856763052611669373327017337516100202429275776045541043302571869099957135889191233303801334553760006254341494325951532336811508595833526048489556116405666102486918801 + 7/10 7/10 + *e *( - 1397) + + 1417 - 1397 139 + *kummerm(1,------,---------)*e + + 10 10 +100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + 7/10 + *10 + + 1417 + *gamma(------)))/( + 10 +1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + 7/10 7/10 + *e *10 + + 1417 139 + *gamma(------)*e ) + 10 + + +on rounded; + + +precision 30; + + +20 + +t!*k!*c := 7; + + +t*k*c := 7 + + +% To test each and every possible combination of +% three arguments from t!*kummer!*b would take too +% long, but we want the possibility of trying most +% special cases. Compromise: test every seventh +% possibility. + +for each a in t!*kummer!*b do + for each b in t!*kummer!*b do + for each z in t!*kummer!*b do + << if t!*k!*c = 7 + then << write "KummerM(", a, ",", b, ",", z, ") = ", + kummerm(a,b,z); + write "KummerU(", a, ",", b, ",", z, ") = ", + kummeru(a,b,z); + t!*k!*c := 0 >>; + t!*k!*c := t!*k!*c + 1 >>; + + +KummerM(0,0,0) = 1 + +KummerU(0,0,0) = kummeru(0,0,0) + +KummerM(0,1,3.14159265358979323846264338328) = 1 + +KummerU(0,1,3.14159265358979323846264338328) = kummeru(0,1, + + 3.14159265358979323846264338328) + +KummerM(0,3.14159265358979323846264338328,26) = 1 + +KummerU(0,3.14159265358979323846264338328,26) = kummeru(0, + + 3.14159265358979323846264338328,26) + +KummerM(0,26,1) = 1 + +KummerU(0,26,1) = kummeru(0,26,1) + +KummerM(1,0, - 3.14159265358979323846264338328) = kummerm(1,0, + + - 3.14159265358979323846264338328) + +KummerU(1,0, - 3.14159265358979323846264338328) = kummeru(1,0, + + - 3.14159265358979323846264338328) + +KummerM(1,3.14159265358979323846264338328,0) = 1 + +KummerU(1,3.14159265358979323846264338328,0) = kummeru(1, + + 3.14159265358979323846264338328,0) + +KummerM(1, - 3.14159265358979323846264338328,3.14159265358979323846264338328) = + +2692.89480079631357528203659153 + +KummerU(1, - 3.14159265358979323846264338328,3.14159265358979323846264338328) = + +0.129554194296952806409644905659 + +KummerM(1,26,26) = 7.74565667206271943920803547594 + +KummerU(1,26,26) = kummeru(1,26,26) + +KummerM(3.14159265358979323846264338328,1,1) = 10.2259987795328570162092950355 + +KummerU(3.14159265358979323846264338328,1,1) = kummeru( + + 3.14159265358979323846264338328,1,1) + +KummerM(3.14159265358979323846264338328,3.14159265358979323846264338328, + + - 3.14159265358979323846264338328) = 0.0432139182637722497744177371717 + +KummerU(3.14159265358979323846264338328,3.14159265358979323846264338328, + + - 3.14159265358979323846264338328) = ( - 0.137891580772667438638357954721 + + 2.14159265358979323846264338328 + *( - 3.14159265358979323846264338328) + + + 1.22938452238615186004844111454)/ + + 2.14159265358979323846264338328 +( - 3.14159265358979323846264338328) + +KummerM(3.14159265358979323846264338328,26,0) = 1 + +KummerU(3.14159265358979323846264338328,26,0) = kummeru( + + 3.14159265358979323846264338328,26,0) + +KummerM( - 3.14159265358979323846264338328,0,3.14159265358979323846264338328) = + +kummerm( - 3.14159265358979323846264338328,0,3.14159265358979323846264338328) + +KummerU( - 3.14159265358979323846264338328,0,3.14159265358979323846264338328) = + +kummeru( - 3.14159265358979323846264338328,0,3.14159265358979323846264338328) + +KummerM( - 3.14159265358979323846264338328,1,26) = 6.185222265647228001735135594 + +62e+5 + +KummerU( - 3.14159265358979323846264338328,1,26) = kummeru( + + - 3.14159265358979323846264338328,1,26) + +KummerM( - 3.14159265358979323846264338328, - 3.14159265358979323846264338328,1 + +) = 2.71828182845904523536028747135 + +KummerU( - 3.14159265358979323846264338328, - 3.14159265358979323846264338328,1 + +) = 19.2419564406028465661346237348 + +KummerM( - 3.14159265358979323846264338328,26, - 3.14159265358979323846264338328 + +) = 1.42892253084220157246185820464 + +KummerU( - 3.14159265358979323846264338328,26, - 3.14159265358979323846264338328 + +) = kummeru( - 3.14159265358979323846264338328,26, + + - 3.14159265358979323846264338328) + +KummerM(26,1,0) = 1 + +KummerU(26,1,0) = kummeru(26,1,0) + +KummerM(26,3.14159265358979323846264338328,3.14159265358979323846264338328) = 3. + +91638029828702661403357541917e+5 + +KummerU(26,3.14159265358979323846264338328,3.14159265358979323846264338328) = 4. + +18995913372628050180833640475e-32 + +KummerM(26, - 3.14159265358979323846264338328,26) = 2.19471945265322419268333614 + +674e+34 + +KummerU(26, - 3.14159265358979323846264338328,26) = 0 + + +off rounded; + + +clear x, y, z, t!*k!*c; + + + +df(df(kummerM(a,b,z),z),z); + + + 2 +(kummerm(2,3,z)*z - 2*kummerm(2,3,z)*z + 2*kummerm(2,3,z) + 2*kummerm(1,3,z)*z + + 2 + - 4*kummerm(1,3,z) + 2*kummerm(0,3,z))/z + +df(kummerU(a,b,z),z); + + + kummeru(2,3,z)*z - kummeru(2,3,z) - kummeru(1,3,z) +---------------------------------------------------- + z + + +z := ((x^2 + y)^5) + (x^(x+y)); + + + x + y 10 8 6 2 4 3 2 4 5 +z := x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y + + +df(df(kummerM(a,b,z),y),x); + + + 4*x + 4*y +(x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 4*x + 4*y + *log(x) *x + x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 4*x + 4*y + *x + x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 3*x + 3*y + *y + 2*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 11 3*x + 3*y + *log(x) *x + 10*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 9 3*x + 3*y + *log(x) *x *y + 20*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 7 2 3*x + 3*y + *log(x) *x *y + 20*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 5 3 3*x + 3*y + *log(x) *x *y + 10*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 3 4 3*x + 3*y + *log(x) *x *y + 2*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 5 3*x + 3*y + *log(x) *x*y - x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 3*x + 3*y + *log(x) *x + 2*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 11 3*x + 3*y + *x + 2*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 10 3*x + 3*y + *x *y + 10*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 10 3*x + 3*y + *x + 10*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 9 3*x + 3*y + *x *y + 5*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 9 3*x + 3*y + *x + 10*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 8 2 3*x + 3*y + *x *y + 40*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 8 3*x + 3*y + *x *y + 20*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 7 2 3*x + 3*y + *x *y + 20*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 7 3*x + 3*y + *x *y + 20*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 6 3 3*x + 3*y + *x *y + 60*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 6 2 3*x + 3*y + *x *y + 20*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 5 3 3*x + 3*y + *x *y + 30*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 5 2 3*x + 3*y + *x *y + 20*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 4 4 3*x + 3*y + *x *y + 40*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 4 3 3*x + 3*y + *x *y + 10*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 3 4 3*x + 3*y + *x *y + 20*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 3 3 3*x + 3*y + *x *y + 10*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 2 5 3*x + 3*y + *x *y + 10*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 2 4 3*x + 3*y + *x *y + 2*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 5 3*x + 3*y + *x*y + 5*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 4 3*x + 3*y + *x*y - x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 3*x + 3*y + *x + 2*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 6 3*x + 3*y + *y - x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 3*x + 3*y + *y + 5*x + + x + y 10 8 6 2 4 3 2 4 5 9 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + 5 + + 3*x + 3*y + *x + + x + y 10 8 6 2 4 3 2 4 5 8 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + + + 3*x + 3*y + 20*x + + x + y 10 8 6 2 4 3 2 4 5 7 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + + + 3*x + 3*y + 20*x + + x + y 10 8 6 2 4 3 2 4 5 6 2 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + + 3*x + 3*y + + 30*x + + x + y 10 8 6 2 4 3 2 4 5 5 2 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + + 3*x + 3*y + + 30*x + + x + y 10 8 6 2 4 3 2 4 5 4 3 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + + 3*x + 3*y + + 20*x + + x + y 10 8 6 2 4 3 2 4 5 3 3 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + + 3*x + 3*y + + 20*x + + x + y 10 8 6 2 4 3 2 4 5 2 4 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + + 3*x + 3*y + + 5*x + + x + y 10 8 6 2 4 3 2 4 5 4 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x*y + + + 3*x + 3*y + 5*x + + x + y 10 8 6 2 4 3 2 4 5 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*y + + + 3*x + 3*y + x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + 2 + + 3*x + 3*y + *x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 3*x + 3*y + *log(x) *x + 2*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 3*x + 3*y + *x + 2*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*log(x) + + 2*x + 2*y + *y + x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 21 2*x + 2*y + *log(x) *x + 10*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 19 2*x + 2*y + *log(x) *x *y + 45*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 17 2 2*x + 2*y + *log(x) *x *y + 120*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 15 3 2*x + 2*y + *log(x) *x *y + 210*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 13 4 2*x + 2*y + *log(x) *x *y + 252*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 11 5 2*x + 2*y + *log(x) *x *y + 210*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 9 6 2*x + 2*y + *log(x) *x *y + 120*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 7 7 2*x + 2*y + *log(x) *x *y + 45*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 5 8 2*x + 2*y + *log(x) *x *y + 10*x + + x + y 10 8 6 2 4 3 2 4 5 + *kummerm(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 2 3 9 2*x + 2*y + *log(x) *x *y + 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5*x *y + y )*x + + 2 + *y + 90 + + x + y 10 8 6 2 4 3 2 4 5 4 8 + *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + + x + y 10 8 6 2 4 3 2 4 5 4 + - 40*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + + 3 + *y + 10 + + x + y 10 8 6 2 4 3 2 4 5 2 9 + *kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + + x + y 10 8 6 2 4 3 2 4 5 2 + - 10*kummeru(2,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + + 4 + *y - 10 + + x + y 10 8 6 2 4 3 2 4 5 10 + *kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x - + + x + y 10 8 6 2 4 3 2 4 5 8 + 40*kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + + x + y 10 8 6 2 4 3 2 4 5 + - 60*kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y ) + + 6 2 + *x *y - 40 + + x + y 10 8 6 2 4 3 2 4 5 4 3 + *kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x *y + + x + y 10 8 6 2 4 3 2 4 5 2 + - 10*kummeru(1,3,x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )*x + + 4 x + y 10 8 6 2 4 3 2 4 5 + *y )/(x*(x + x + 5*x *y + 10*x *y + 10*x *y + 5*x *y + y )) + + + + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + +% 9. Bessel functions + +% =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + + +% Lengthy test of the Bessel functions. This isn't even +% remotely exhaustive of the special cases -- though a +% real person with lots of time could no doubt come up +% with a better lot of tests than this automated rubbish. +% Again, compromise by only actually doing one in five or +% nine. If you want a really thorough test, you can +% easily change this to get it; but it'll take hours to +% run. + +clear p, q; + + + +hankel1(p,q); + + +hankel1(p,q) + +r := df(ws,q); + + + hankel1(p - 1,q)*q - hankel1(p,q)*p +r := ------------------------------------- + q + + +on complex; + + + +r; + + + besselj(p - 1,q)*q - besselj(p,q)*p + i*bessely(p - 1,q)*q - i*bessely(p,q)*p +------------------------------------------------------------------------------- + q + +p:=3/8; + + + 3 +p := --- + 8 + +r; + + + - 5 3*pi - 5 3*pi +(8*i*besselj(------,q)*cos(------)*q + 8*besselj(------,q)*sin(------)*q + 8 8 8 8 + + - 3 5 + + 3*i*besselj(------,q) + 8*i*besselj(---,q)*q + 8 8 + + 3 3*pi 3 3*pi + - 3*i*besselj(---,q)*cos(------) - 3*besselj(---,q)*sin(------))/(8 + 8 8 8 8 + + 3*pi + *sin(------)*q) + 8 + +q := pi; + + +q := pi + +r; + + + - 5 3*pi - 5 3*pi +(8*i*besselj(------,pi)*cos(------)*pi + 8*besselj(------,pi)*sin(------)*pi + 8 8 8 8 + + - 3 5 + + 3*i*besselj(------,pi) + 8*i*besselj(---,pi)*pi + 8 8 + + 3 3*pi 3 3*pi + - 3*i*besselj(---,pi)*cos(------) - 3*besselj(---,pi)*sin(------))/(8 + 8 8 8 8 + + 3*pi + *sin(------)*pi) + 8 + + +on rounded; + + +*** Domain mode complex changed to complex-rounded + + +r; + + + - 0.119366207318921501826662822529 + +*hankel1(0.375,3.14159265358979323846264338328) + + + hankel1( - 0.625,3.14159265358979323846264338328) + + +off complex, rounded; + + +*** Domain mode complex-rounded changed to rounded + + +df(df(besselj(pp,qq)+rr * hankel1(pp*2,qq) * + bessely(pp-qq,qq),qq),qq); + + + 2 +(besselj(pp - 2,qq)*qq - 2*besselj(pp - 1,qq)*pp*qq + besselj(pp - 1,qq)*qq + + 2 + + besselj(pp,qq)*pp + besselj(pp,qq)*pp + + 2 + + bessely(pp - qq,qq)*hankel1(2*pp - 2,qq)*qq *rr + + - 4*bessely(pp - qq,qq)*hankel1(2*pp - 1,qq)*pp*qq*rr + + + bessely(pp - qq,qq)*hankel1(2*pp - 1,qq)*qq*rr + + 2 + + 4*bessely(pp - qq,qq)*hankel1(2*pp,qq)*pp *rr + + + 2*bessely(pp - qq,qq)*hankel1(2*pp,qq)*pp*rr + + 2 + + df(bessely(pp - qq,qq),qq,2)*hankel1(2*pp,qq)*qq *rr + + 2 + + 2*df(bessely(pp - qq,qq),qq)*hankel1(2*pp - 1,qq)*qq *rr + + 2 + - 4*df(bessely(pp - qq,qq),qq)*hankel1(2*pp,qq)*pp*qq*rr)/qq + + +% Possible values for real args +t!*bes!*vr := { 1, pi, -pi, 26 }$ + + + +% Possible values for real and imaginary parts of complex args +t!*bes!*vc := { 0, 3, -41/2 }$ + + + +array s!*bes(4)$ + + + +s!*bes(1) := "BesselJ"$ + + +s!*bes(2) := "BesselY"$ + + +s!*bes(3) := "BesselI"$ + + +s!*bes(4) := "BesselK"$ + + + +pre := 16; + + +pre := 16 + +precision pre; + + +30 + +preord := 10**pre; + + +preord := 10000000000000000 + +t!*b!*c := 3; + + +t*b*c := 3 + + +algebraic procedure do!*one!*bessel(s,n,z); + (if s = 1 then besselj(n,z) + else if s = 2 then bessely(n,z) + else if s = 3 then besseli(n,z) + else besselk(n,z)); + + +do*one*bessel + + +algebraic procedure pr!*bessel(s,n,z,k); + << if t!*b!*c = k + then + << on rounded; + bes1 := do!*one!*bessel(s,n,z); + precision(pre+5); + bes2 := do!*one!*bessel(s,n,z); + if bes1 neq 0 + then disc := floor abs(100*(bes2-bes1)*preord/bes1) + else disc := 0; + precision pre; + write s!*bes(s), "(", n, ",", z, ") = ", bes1; + if not numberp disc then + << precom := !*complex; + on complex; + disc := disc; + if not precom then off complex >>; + if disc neq 0 + then write " (discrepancy ", disc, "% of one s.f.)"; + if numberp disc and disc > 200 then + << write "***** WARNING Significant Inaccuracy."; + write " Lower precision result:"; + write " ", bes1; + write " Higher precision result:"; + precision(pre+5); write " ", bes2; precision pre >>; + off rounded; + t!*b!*c := 0 >>; + t!*b!*c := t!*b!*c + 1 >>; + + +pr*bessel + + +% About to begin Bessel test. We have a list of possible +% values, and we test every Bessel, with every value on the +% list as both order and argument. Every Bessel is computed +% twice, to different precisions (differing by 3), and any +% discrepancy is reported. The value reported is the diff- +% erence between the two computed values, expressed as a +% percentage of the unit of the least significant displayed +% digit. A discrepancy between 100% and 200% means that the +% last digit of the displayed value was found to differ at +% higher precision; values greater than 200% are cause for +% concern. An ideal discrepancy would be between about 1% +% and 20%; if the value is found to be zero, no discrepancy +% is reported. + +off msg; + + + +for s := 1:4 do + << write(" ... Testing ", s!*bes(s), " for real domains ... "); + for each n in t!*bes!*vr do + for each z in t!*bes!*vr do + pr!*bessel(s, n, z, 5) >>; + + + ... Testing BesselJ for real domains ... + +BesselJ(1, - 3.141592653589793) = - 0.2846153431797528 + +BesselJ(3.141592653589793,26) = - 0.006989220174690161 + + (discrepancy 5% of one s.f.) + +BesselJ(26,1) = 3.660826744416803e-35 + + ... Testing BesselY for real domains ... + +BesselY(1,3.141592653589793) = 0.358872916776719 + +BesselY(3.141592653589793, - 3.141592653589793) = + + 6.283185307179586 + 0.1545613960392598*( - 1.570796326794897) - 4.829362563540275 +-------------------------------------------------------------------------------- + 3.141592653589793 + ( - 1.570796326794897) + +BesselY( - 3.141592653589793,26) = - 0.1386083623177915 + + ... Testing BesselI for real domains ... + +BesselI(1,1) = 0.565159103992485 + +BesselI(3.141592653589793,3.141592653589793) = 1.011423335928613 + +BesselI( - 3.141592653589793, - 3.141592653589793) = - 0.8856101155917482 + + + 0.4221616153281286*i + +BesselI(26,26) = 68397.86776155122 + + ... Testing BesselK for real domains ... + +BesselK(3.141592653589793,1) = 9.025908765806763 + +BesselK( - 3.141592653589793,3.141592653589793) = 0.1107526602738113 + + (discrepancy 1% of one s.f.) + +BesselK(26, - 3.141592653589793) = besselk(26, - 3.141592653589793) + + +on complex; + + + +for s := 1:3 do + << write (" ... Testing ", s!*bes(s), " for complex domains ... "); + for each nr in t!*bes!*vc do + for each ni in t!*bes!*vc do + for each zr in t!*bes!*vc do + for each zi in t!*bes!*vc do + pr!*bessel(s, nr+ni*i, zr+zi*i, 9) >>; + + + ... Testing BesselJ for complex domains ... + +BesselJ(0, - 20.5 + 3.0*i) = 1.05389016561334 + 1.410918160335249*i + +BesselJ(3*i, - 20.5 + 3.0*i) = 0.01225787392170983 + 0.01066256817009466*i + +BesselJ( - 20.5*i, - 20.5 + 3.0*i) = - 6.607837931625446e+38 + + + 7.203284455482089e+38*i + +BesselJ(3, - 20.5 + 3.0*i) = 1.568613483726435 - 0.7011991107137573*i + +BesselJ(3 + 3*i, - 20.5 + 3.0*i) = 0.007904103001381543 - 0.006566520928092784*i + +BesselJ(3.0 - 20.5*i, - 20.5 + 3.0*i) = - 7.069920310202644e+37 + + - 1.753271554229047e+37*i + +BesselJ( - 20.5, - 20.5 + 3.0*i) = 0.1758742246345278 - 0.332739860634916*i + +BesselJ( - 20.5 + 3.0*i, - 20.5 + 3.0*i) = 0.08815299110072903 + + - 0.1369698556512304*i + +BesselJ( - 20.5 - 20.5*i, - 20.5 + 3.0*i) = - 5.364748129151297e+46 + + + 2.608178375230083e+47*i + + ... Testing BesselY for complex domains ... + +BesselY(0, - 20.5 + 3.0*i) = - 1.404746667469566 + 1.060048452645186*i + +BesselY(3*i, - 20.5 + 3.0*i) = 0.4973091982659732 + 0.7985114801567726*i + +BesselY( - 20.5*i, - 20.5 + 3.0*i) = - 7.203284455482089e+38 + + - 6.607837931625446e+38*i + + (discrepancy 10% of one s.f.) + +BesselY(3, - 20.5 + 3.0*i) = 0.6963128100601111 + 1.576222640523309*i + +BesselY(3 + 3*i, - 20.5 + 3.0*i) = - 1.117333163968302 + 0.9789575771194796*i + +BesselY(3.0 - 20.5*i, - 20.5 + 3.0*i) = 1.753271554229047e+37 + + - 7.069920310202644e+37*i + +BesselY( - 20.5, - 20.5 + 3.0*i) = 0.2353954565395826 + 0.144691313932682*i + +BesselY( - 20.5 + 3.0*i, - 20.5 + 3.0*i) = - 0.1527215881543493 + + + 0.2371137974094512*i + +BesselY( - 20.5 - 20.5*i, - 20.5 + 3.0*i) = - 2.608178375230083e+47 + + - 5.364748129151297e+46*i + + ... Testing BesselI for complex domains ... + +BesselI(0, - 20.5 + 3.0*i) = - 6.891185608459107e+7 - 1.506065792318474e+7*i + +BesselI(3*i, - 20.5 + 3.0*i) = - 6879.511500081906 - 1745.250262122384*i + +BesselI( - 20.5*i, - 20.5 + 3.0*i) = 4.756052726395977e+40 + + - 1.836844915663626e+40*i + +BesselI(3, - 20.5 + 3.0*i) = 5.56813934269752e+7 + 1.026348768124686e+7*i + +BesselI(3 + 3*i, - 20.5 + 3.0*i) = 5917.619873410601 - 1360.623977225443*i + +BesselI(3.0 - 20.5*i, - 20.5 + 3.0*i) = 1.125266362182384e+40 + + - 3.487618645712341e+39*i + +BesselI( - 20.5, - 20.5 + 3.0*i) = - 0.0005935206748136158 + + - 0.001045512507192035*i + +BesselI( - 20.5 + 3.0*i, - 20.5 + 3.0*i) = 0.0001367212493519011 + + - 0.00008899048591346719*i + +BesselI( - 20.5 - 20.5*i, - 20.5 + 3.0*i) = - 4.705541285798881e+45 + + + 6.968045629188714e+45*i + + +off complex; + + + +on msg; + + + +write (" ..."); + + + ... + +write ("Bessel test complete."); + + +Bessel test complete. + + +end; +(TIME: specfn 121310 136641)