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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... % Examples for the algorithmic calculation of formal % Puiseux, Laurent and power series, % % Wolfram Koepf, Freie Universitaet Berlin, Germany % (taken from the original paper and adapted to REDUCE % form by Winfried Neun, ZIB Berlin) % Formal Laurent series on errcont; fps(E^x,x); k x infsum(--------------,k,0,infinity) factorial(k) fps(E^x/(x^3),x); k x infsum(-----------------,k,0,infinity) 3 factorial(k)*x fps(x * e^(x^4),x); 4*k x *x infsum(--------------,k,0,infinity) factorial(k) fps(sin (x + y),x); 2*k k x *( - 1) *cos(y)*x infsum(-----------------------,k,0,infinity) factorial(2*k + 1) 2*k k x *( - 1) *sin(y) + infsum(---------------------,k,0,infinity) factorial(2*k) simplede (sin x,x); df(y,x,2) + y %find a DE for sin simplede (sin (x)^2,x,w); df(w,x,3) + 4*df(w,x) % DE in w and x fps(asin x,x); 2*k x *factorial(2*k)*x infsum(------------------------------,k,0,infinity) 2*k 2 2 *factorial(k) *(2*k + 1) fps((asin x)^2,x); 2*k 2*k 2 2 x *2 *factorial(k) *x infsum(----------------------------,k,0,infinity) factorial(2*k + 1)*(k + 1) fps(e^(asin x),x); 2*k k 2 x *2 *prod(2*j - 2*j + 1,j,1,k)*x infsum(--------------------------------------,k,0,infinity) factorial(2*k + 1) 2*k 2 x *prod(4*j - 8*j + 5,j,1,k) + infsum(---------------------------------,k,0,infinity) factorial(2*k) fps(e^(asinh x),x); 2*k k - x *( - 1) *factorial(2*k) infsum(--------------------------------,k,0,infinity) + x k 2 4 *factorial(k) *(2*k - 1) fps((x + sqrt(1+x^2))^A,x); 2*k k 2*k - a a x *( - 1) *2 *pochhammer(------,k)*pochhammer(---,k) 2 2 infsum(----------------------------------------------------------,k,0,infinity) factorial(2*k) 2*k k 2*k - a + 1 a + 1 x *( - 1) *2 *pochhammer(----------,k)*pochhammer(-------,k)*a*x 2 2 + infsum(----------------------------------------------------------------------, factorial(2*k + 1) k,0,infinity) fps(e^(x^2)*erf x,x); 2*k 2*k 2*x *sqrt(pi)*2 *factorial(k)*x infsum(-------------------------------------,k,0,infinity) factorial(2*k + 1)*pi fps(e^x - 2 e^(-x/2) * cos(sqrt(3) * x/2 -pi/3),x); 3*k 2 9*x *x *(k + 1) infsum(--------------------,k,0,infinity) factorial(3*k + 3) % fps(int(e^(-a^2*t^2) * cos(2*x*t),t,0,infinity),x) % not yet % fps(4/x * int(e^(t^2)*erf(t),t,0,sqrt(x)/2),x); fps(sin x * e^x,x); k k/2 k*pi x *2 *sin(------) 4 infsum(---------------------,k,0,infinity) factorial(k) fps(cos x * e^(2*x),x); k k/2 1 x *5 *cos(atan(---)*k) 2 infsum(--------------------------,k,0,infinity) factorial(k) fps(1/(x-x^3),x); k k k x *( - 1) - x infsum(-----------------,k,0,infinity)*x + 1 k 2*( - 1) ---------------------------------------------- x fps(1/(x^2 + 3 x + 2),x); k k k k 2*x *( - 2) - x *( - 1) infsum(---------------------------,k,0,infinity) k k 2*( - 1) *( - 2) fps(x/(1-x-x^2),x); x fps(--------------,x,0) 2 (1 - x) - x % Logarithmic singularities and Puisieux series fps(sin sqrt x,x); (2*k + 1)/2 k x *( - 1) infsum(----------------------,k,0,infinity) factorial(2*k + 1) fps(((1 + sqrt x)/x)^(1/3),x); (6*k + 1)/6 2 x *pochhammer(---,2*k) 3 infsum(----------------------------------,k,0,infinity) 3*factorial(2*k + 1) k - 1 x *pochhammer(------,2*k) 3 + infsum(---------------------------,k,0,infinity) 1/3 x *factorial(2*k) fps(asech x,x); ***** Problem using limit operator % some more (Wolfram Koepf, priv. comm.) fps((1+x)^alpha,x); k k x *( - 1) *pochhammer( - alpha,k) infsum(-----------------------------------,k,0,infinity) factorial(k) fps((1+sqrt(1+x))^beta,x); k k beta x *( - 1) *2 *pochhammer( - beta,2*k) infsum(---------------------------------------------,k,0,infinity) 2*k 2 *factorial(k)*pochhammer( - beta + 1,k) fps(sin(x)^2+cos(x)^2,x); 1 fps(sin(x)^2*cos(x)^2,x); 2*k k 2*k 2 x *( - 4) *2 *x infsum(----------------------------,k,0,infinity) factorial(2*k + 1)*(k + 1) fps(sin(x)*cos(x^2),x); 2 fps(sin(x)*cos(x ),x,0) fps((x-1)^(-1),x); k infsum( - x ,k,0,infinity) fps(atan(x+y),x); fps(atan(x + y),x,0) fps((1-x^5)^6,x); 30 25 20 15 10 5 x - 6*x + 15*x - 20*x + 15*x - 6*x + 1 fps(asec x,x); ***** Problem using limit operator fps(besseli(0,x),x); 2*k x *besseli(0,0) infsum(--------------------,k,0,infinity) 2*k 2 2 *factorial(k) fps(besseli(1,x),x); 2*k x *besseli(0,0)*x infsum(--------------------------------------,k,0,infinity) 2*k 2*2 *factorial(k + 1)*factorial(k) fps(exp(x^(1/3)),x); (3*k + 1)/3 x infsum(--------------------,k,0,infinity) factorial(3*k + 1) k x + infsum(----------------,k,0,infinity) factorial(3*k) (3*k + 2)/3 3*x *(k + 1) + infsum(------------------------,k,0,infinity) factorial(3*k + 3) fps(log(1-x),x); k - x *x infsum(---------,k,0,infinity) k + 1 fps(exp x*sinh x,x); k k x *2 *x infsum(------------------,k,0,infinity) factorial(k + 1) fps(atan x,x); 2*k k x *( - 1) *x infsum(----------------,k,0,infinity) 2*k + 1 fps(sin x+sinh x,x); 4*k 2*x *x infsum(--------------------,k,0,infinity) factorial(4*k + 1) fps(sin x*sinh x,x); 4*k k 2*k 2 x *( - 1) *2 *x infsum(------------------------------,k,0,infinity) factorial(4*k + 1)*(2*k + 1) fps(int(erf(x),x),x); *** ci already defined as operator *** si already defined as operator 2*k k - x *sqrt(pi)*( - 1) infsum(---------------------------,k,0,infinity) factorial(k)*pi*(2*k - 1) fps(sqrt(2-x),x); k - x *sqrt(2)*factorial(2*k) infsum(------------------------------,k,0,infinity) k 2 8 *factorial(k) *(2*k - 1) fps(sqrt(1+x)+sqrt(1-x),x); 2*k - 2*x *factorial(4*k) infsum(--------------------------------,k,0,infinity) 2*k 2 4 *factorial(2*k) *(4*k - 1) fps(exp(a+b*x)*exp(c+d*x),x); k a + c k x *e *(b + d) infsum(--------------------,k,0,infinity) factorial(k) fps(1/cos(asin x),x); 2*k x *factorial(2*k) infsum(---------------------,k,0,infinity) 2*k 2 2 *factorial(k) fps(sqrt(1-x^2)+x*asin x,x); 2*k x *factorial(2*k) infsum(-----------------------------------,k,0,infinity) k 2 2 4 *factorial(k) *(4*k - 4*k + 1) fps(sqrt(1-sqrt(x)),x); (2*k + 1)/2 - x *factorial(4*k) infsum(------------------------------------------,k,0,infinity) 4*k 2*2 *factorial(2*k + 1)*factorial(2*k) k - x *factorial(4*k) + infsum(--------------------------------,k,0,infinity) 2*k 2 4 *factorial(2*k) *(4*k - 1) fps(cos(n*acos x),x); 2*k 2*k n*pi - n n x *2 *cos(------)*pochhammer(------,k)*pochhammer(---,k) 2 2 2 infsum(--------------------------------------------------------------,k,0, factorial(2*k) infinity) + infsum( 2*k 2*k - n + 1 n + 1 n*pi x *2 *pochhammer(----------,k)*pochhammer(-------,k)*sin(------)*n*x 2 2 2 --------------------------------------------------------------------------,k, factorial(2*k + 1) 0,infinity) fps(cos x+I*sin x,x); k k x *( - 1) infsum(-----------------,k,0,infinity) k i *factorial(k) fps(cos(3*asinh x),x); 2*k k 2 x *( - 1) *prod(4*j - 8*j + 13,j,1,k) infsum(------------------------------------------,k,0,infinity) factorial(2*k) fps(cos(n*asinh x),x); 2*k k 2*k - i*n i*n x *( - 1) *2 *pochhammer(--------,k)*pochhammer(-----,k) 2 2 infsum(--------------------------------------------------------------,k,0, factorial(2*k) infinity) fps(sin(n*log(x+sqrt(1+x^2))),x); 2*k k 2*k - i*n + 1 i*n + 1 infsum((x *( - 1) *2 *pochhammer(------------,k)*pochhammer(---------,k)*n*x 2 2 )/factorial(2*k + 1),k,0,infinity) fps(sqrt(1+x^2)*asinh x-x,x); 2*k k 2*k 3 2*x *( - 1) *2 *factorial(k + 1)*factorial(k)*x infsum(------------------------------------------------------,k,0,infinity) factorial(2*k + 3) fps(int(erf(x)/x,x),x); 2*k k 2*x *sqrt(pi)*( - 1) *x infsum(----------------------------------,k,0,infinity) 2 factorial(k)*pi*(4*k + 4*k + 1) erf(x) + sub(x=0,int(--------,x)) x fps(asin(x)^2/x^4,x); 2*k 2*k 2 x *2 *factorial(k) infsum(-------------------------------,k,0,infinity) 2 factorial(2*k + 1)*x *(k + 1) % we had problems here: fps(cos(asin x),x); 2*k - x *factorial(2*k) infsum(----------------------------,k,0,infinity) k 2 4 *factorial(k) *(2*k - 1) fps(sinh(log x),x); fps(sinh(log(x)),x,0) fps(atan(cot x),x); Could not find the limit of: atan(cot(x)),x,0 ***** problem using limit operator % we can cure this one by defining the limit: let limit(atan(cot ~x),x,0) => pi/2; fps(atan(cot x),x); pi - 2*x ---------- 2 fps(exp(nnn*x)*cos(mmm*x),x); k 2 2 infsum((x *((impart(mmm) + 2*impart(mmm)*repart(nnn) + impart(nnn) 2 2 k - 2*impart(nnn)*repart(mmm) + repart(mmm) + repart(nnn) )**--- 2 impart(nnn) - repart(mmm) 2 *cos(atan(---------------------------)*k) + (impart(mmm) impart(mmm) + repart(nnn) 2 - 2*impart(mmm)*repart(nnn) + impart(nnn) 2 2 k + 2*impart(nnn)*repart(mmm) + repart(mmm) + repart(nnn) )**--- 2 impart(nnn) + repart(mmm) 2 *cos(atan(---------------------------)*k) - (impart(mmm) impart(mmm) - repart(nnn) 2 - 2*impart(mmm)*repart(nnn) + impart(nnn) 2 2 k + 2*impart(nnn)*repart(mmm) + repart(mmm) + repart(nnn) )**--- 2 impart(nnn) + repart(mmm) 2 *sin(atan(---------------------------)*k)*i + (impart(mmm) impart(mmm) - repart(nnn) 2 + 2*impart(mmm)*repart(nnn) + impart(nnn) 2 2 k - 2*impart(nnn)*repart(mmm) + repart(mmm) + repart(nnn) )**--- 2 impart(nnn) - repart(mmm) *sin(atan(---------------------------)*k)*i))/(2*factorial(k)),k,0, impart(mmm) + repart(nnn) infinity) fps(sqrt(2-x^2),x); 2*k - x *sqrt(2)*factorial(2*k) infsum(--------------------------------,k,0,infinity) k 2 8 *factorial(k) *(2*k - 1) fps(ci x,x); 2*k k x *( - 1) *infinity*x ci(0) + infsum(------------------------------,k,0,infinity) factorial(2*k + 1)*(2*k + 1) fps(log(1-2*x*y+x^2),x); 2 fps(log(1 - 2*x*y + x ),x,0) FPS(sin x,x,pi); 2*k k ( - pi + x) *( - 1) *( - pi + x) infsum(------------------------------------,k,0,infinity) factorial(2*k + 1) % This one takes ages : %fps(acos(cos(x)),x); fps_search_depth := 7; fps_search_depth := 7 % does not find aa DE with the default fps(sin(x^(1/3)),x); 2*k k k k infsum(( - x *( - 1) *108 *factorial(k)*x)/(6*46656 *factorial(3*k + 1) 7 5 *factorial(2*k + 1)*pochhammer(---,k)*pochhammer(---,k)),k,0,infinity) 6 6 (6*k + 2)/3 k 2*k 3*k k + infsum((x *( - 1) *2 *3 *factorial(k + 1)*x)/(20*46656 11 *factorial(3*k + 3)*factorial(2*k + 1)*pochhammer(----,k) 6 7 *pochhammer(---,k)),k,0,infinity) + infsum( 6 (6*k + 1)/3 k 2*k 3*k x *( - 1) *2 *3 *factorial(k) --------------------------------------------------------------------------,k, k 7 5 46656 *factorial(3*k)*factorial(2*k)*pochhammer(---,k)*pochhammer(---,k) 6 6 0,infinity) end; (TIME: fps 165079 183049)