Artifact 724da2f1133a866a1ff3ef59a0156cfda328d6e89c98b6b36dcb479ff40a85a4:
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r36/src/ztrans.red
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2011-09-02 18:13:33
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module ztrans; % calculation of Z transformation and inverse. % Authors: Wolfram Koepf, Lisa Temme. % Version 1.0, April 1995. % ZTRANS: Z transformation, see % Bronstein, Semendjajew: Taschenbuch der Mathematik, 4.4.4 create!-package('(ztrans),'(contrib misc)); !*precise := nil; % Needed for this module at the moment. % auxiliary functions symbolic procedure newrederr(u); <<terpri!* t; prin2!* "***** "; if eqcar(u,'list) then foreach xx in cdr u do newrederr1(xx) else newrederr1 u; terpri!* nil; erfg!* := t; error1()>>; symbolic procedure newrederr1(u); if not atom u and atom car u and cdr u and atom cadr u and null cddr u then <<prin2!* car u; prin2!* " "; prin2!* cadr u>> else maprin u; flag('(newrederr),'opfn); %******************************************************************** %Ztrans procedure algebraic operator ztrans_aux; algebraic operator !~f,!~g,!~summ,binomial; algebraic procedure ztrans(f,n,z); begin scalar tmp,!*factor,!*exp; off factor; tmp := ztrans_aux(f,n,z); if part(tmp,0)=ztrans_aux then << on factor; tmp := ztrans_aux(f,n,z); off factor; >>; if part(tmp,0)=ztrans_aux then return lisp mk!*sq((list((car fkern list('ztrans,f,n,z) . 1) . 1)) . 1) else return tmp; end$ %******************************************************************** % Ztrans ruleset % by Lisa Temme algebraic << ztransrules := { ztrans_aux(1,~n,~z) => z/(z-1), ztrans_aux(binomial(~n+~~k,~m),~n,~z) => z^(k+1)/(z-1)^(m+1) when (freeof(k,n) and freeof(m,n)), ztrans_aux(factorial(~n)/(factorial(~n-~k)*factorial(~k)),~n,~z) => ztrans(binomial(n,k),n,z) when freeof(k,n), ztrans_aux(1/(~n+~~k),~n,~z) => z^(k-1)*(z*log(z/(z-1)) - sum(1/((j+1)*z^j),j,0,k-2)) when (freeof(k,n) and fixp(k) and k>0), ztrans_aux(~a^(~n+~~k),~n,~z) => a^k*z/(z-a) when (freeof(a,n) and freeof(k,n)), ztrans_aux(1/~a^(~n+~~k),~n,~z) => sub(a=1/a,ztrans(a^(n+k),n,z)) when (freeof(a,n) and freeof(k,n)), ztrans_aux(e^(~n*~~a),~n,~z) => -z/(e^a-z) when freeof(a,n), ztrans_aux(e^((~n+~~k)*~~a),~n,~z) => e^(a*k)*-z/(e^a-z) when (freeof(a,n) and freeof(k,n)), ztrans_aux(1/factorial(~n),~n,~z) => e^(1/z), ztrans_aux(1/factorial(2*~n+~~k),~n,~z) => z^((k-1)/2)*(sqrt(z)*sinh(1/sqrt(z)) - sum(1/(factorial(2*j+1)*z^j),j,0,(k-3)/2)) when (freeof(k,n) and fixp((k+1)/2) and k>0), ztrans_aux(1/factorial(2*~n+~~k),~n,~z) => z^(k/2)*(cosh(1/sqrt(z)) - sum(1/(factorial(2*j)*z^j),j,0,k/2-1)) when (freeof(k,n) and fixp(k/2) and k>=0), ztrans_aux((-1)^~n/factorial(2*~n+~~k),~n,~z) => (-z)^((k-1)/2)*(sqrt(z)*sin(1/sqrt(z)) - sum((-1)^j/(factorial(2*j+1)*z^j),j,0,(k-3)/2)) when(freeof(k,n) and fixp((k+1)/2) and k>0), ztrans_aux((-1)^~n/factorial(2*~n+~~k),~n,~z) => (-z)^(k/2)*(cos(1/sqrt(z)) - sum((-1)^j/(factorial(2*j)*z^j),j,0,k/2-1)) when (freeof(k,n) and fixp(k/2) and k>=0), ztrans_aux(sinh(~~al*~n+~~p),~n,~z) => z*(z*sinh(p) + sinh(al-p)) /(z^2 - 2*z*cosh(al) + 1) when (freeof(al,n) and freeof(p,n)), ztrans_aux(cosh(~~al*~n+~~p),~n,~z) => z*(z*cosh(p) - cosh(al-p)) /(z^2 - 2*z*cosh(al) + 1) when (freeof(al,n) and freeof(p,n)), ztrans_aux(sin(~~b*~n+~~p),~n,~z) => z*(z*sin(p) + sin(b-p)) /(z^2 - 2*z*cos(b) + 1) when (freeof(b,n) and freeof(p,n)), ztrans_aux(cos(~~b*~n+~~p),~n,~z) => z*(z*cos(p) - cos(b-p)) /(z^2 - 2*z*cos(b) + 1) when (freeof(b,n) and freeof(p,n)), ztrans_aux(e^(~~a*~n)*sin(~~b*~n),~n,~z) => z*e^a*sin(b)/(z^2-2*z*e^a*cos(b)+e^(2*a)) when (freeof(a,n) and freeof(b,n)), ztrans_aux(e^(~~a*~n)*cos(~~b*~n),~n,~z) => z*(z-e^a*cos(b))/(z^2-2*z*e^a*cos(b)+e^(2*a)) when (freeof(a,n) and freeof(b,n)), ztrans_aux(cos(~~b*(~n+~~k))/(~n+~~k),~n,~z) => z^(k-1)*(z*log(z/sqrt(z^2-2*z*cos(b)+1)) - sum(cos(b*(j+1))/((j+1)*z^j),j,0,k-2)) when (freeof(b,n) and freeof(k,n) and fixp(k) and k>0), ztrans_aux(sin(~~b*(~n+~~k))/(~n+~~k),~n,~z) => z^(k-1)*(-z*atan(sin(b)/(cos(b)-z)) - sum(sin(b*(j+1))/((j+1)*z^j),j,0,k-2)) when (freeof(b,n) and freeof(k,n) and fixp(k) and k>0), ztrans_aux((-1)^n*cos(~~b*(~n+~~k))/(~n+~~k),~n,~z) => -(-z)^(k-1)*(z*log(sqrt(z^2+2*z*cos(b)+1/z)) - sum((-1)^j*cos(b*(j+1))/((j+1)*z^j),j,0,k-2)) when (freeof(b,n) and freeof(k,n) and fixp(k)), ztrans_aux(cos(~~b*~n)/factorial(~n),~n,~z) => cos(sin(b)/z)*e^(cos(b)/z) when freeof(b,n), ztrans_aux(cos(~~b*(~n+~~k))/factorial(~n+~~k),~n,~z) => z^k*(cos(sin(b)/z)*e^(cos(b)/z) - sum(cos(b*j)/(factorial(j)*z^j),j,0,k-1)) when (freeof(b,n) and fixp(k)), ztrans_aux(sin(~~b*~n)/factorial(~n),~n,~z) => sin(sin(b)/z)*e^(cos(b)/z) when freeof(b,n), ztrans_aux(sin(~~b*(~n+~~k))/factorial(~n+~~k),~n,~z) => z^k*(sin(sin(b)/z)*e^(cos(b)/z) - sum(sin(b*j)/(factorial(j)*z^j),j,0,k-1)) when (freeof(b,n) and fixp(k)), %LINEARITY ztrans_aux(-~f,~n,~z) => -ztrans(f,n,z), ztrans_aux(~a,~n,~z) => a*ztrans(1,n,z) when freeof(a,n), ztrans_aux(~a*~f,~n,~z) => a*ztrans(f,n,z) when freeof(a,n), ztrans_aux(~f/~b,~n,~z) => ztrans(f,n,z)/b when freeof(b,n), ztrans_aux(~a/~g,~n,~z) => a*ztrans(1/g,n,z) when (freeof(a,n) and not(a=1)), ztrans_aux(~a*~f/~g,~n,~z) => a*ztrans(f/g,n,z) when freeof(a,n), ztrans_aux(~f/(~b*~g),~n,~z) => ztrans(f/g,n,z)/b when freeof(b,n), ztrans_aux((~f+~g)/~~h,~n,~z) => ztrans(f/h,n,z) + ztrans(g/h,n,z), %MULTIPLICATION ztrans_aux(~n^~~p*~~f,~n,~z) => -z*df(ztrans(n^(p-1)*f,n,z),z) when freeof(p,n) and fixp(p) and p>0, ztrans_aux(~n^~~p*~~f/~g,~n,~z) => -z*df(ztrans(n^(p-1)*f/g,n,z),z) when freeof(p,n) and fixp(p) and p>0, %Shift up ztrans_aux(~f(~n+~k),~n,~z) => z^k*(ztrans(f(n),n,z)-sum(f(n)*z^(-n),n,0,k-1)) when freeof(k,n) and fixp(k) and k>0, ztrans_aux(~f(~n+~k)/~g(~n+~k),~n,~z) => z^k*(ztrans(f(n)/g(n),n,z)- sum(f(n)/g(n)*z^(-n),n,0,k-1)) when freeof(k,n) and fixp(k) and k>0, ztrans_aux(1/~g(~n+~k),~n,~z) => z^k*(ztrans(1/g(n),n,z)- sum(1/g(n)*z^(-n),n,0,k-1)) when freeof(k,n) and fixp(k) and k>0, %Similar Expressions ztrans_aux(~a^(~n+~~k)*~f,~n,~z) => a^k*sub(z=(z/a),ztrans(f,n,z)) when freeof(a,n) and freeof(k,n), ztrans_aux(~a^(~n+~~k)*~~f/~g,~n,~z) => a^k*sub(z=(z/a),ztrans(f/g,n,z)) when freeof(a,n) and freeof(k,n), ztrans_aux(~a^(~n-~~k)*~~f/~g,~n,~z) => a^k*sub(z=(z/a),ztrans(f/g,n,z)) when freeof(a,n) and freeof(k,n), ztrans_aux(1/~a^(~n+~~k)*~f,~n,~z) => 1/a^k*sub(z=z*a,ztrans(f,n,z)) when freeof(a,n) and freeof(k,n), ztrans_aux(1/~a^(~n+~~k)*~~f/~g,~n,~z) => 1/a^k*sub(z=z*a,ztrans(f/g,n,z)) when freeof(a,n) and freeof(k,n), %Summations ztrans_aux(sum(~f(~k)*~g(~n-~k),~k,0,~n),~n,~z) => ztrans(f(n),n,z)*ztrans(g(n),n,z) when freeof(k,n), ztrans_aux(~summ(~f,~k,0,~n),~n,~z) => z*ztrans(sub(k=n,f),n,z)/(z-1) when freeof(k,n) and summ = sum, % ztrans_aux(~summ(~~f/~g,~k,0,~n),~n,~z) => % z*ztrans(SUB(k=n,f/g),n,z)/(z-1) % when (freeof(k,n) and summ = sum), ztrans_aux(~summ(~f,~k,0,(~n+~w)),~n,~z) => z*ztrans(sub(k=n,f),n,z)/(z-1) + sum(z^x*(ztrans(sub(k=n,f),n,z) - sum(sub(k=n,f)/z^n,n,0,x-1)),x,1,w) when (freeof(w,n) and fixp(w) and w>0 and summ = sum), % ztrans_aux(~summ(~~f/~g,~k,0,(~n+~w)),~n,~z) => % z*ztrans(SUB(k=n,f/g),n,z)/(z-1) + % sum(z^x*(ztrans(SUB(k=n,f/g),n,z) - % sum(SUB(k=n,f/g)/z^n),n,0,(x-1)),x,1,w) % when (freeof(w,n) and fixp(w) and w>0 % and summ = sum), ztrans_aux(~summ(~f,~k,~p,~n),~n,~z) => ztrans(sum(sub(k=k+p,f),k,0,n-p),n,z) when (freeof(p,n) and fixp(p) and p>0 and summ = sum), ztrans_aux(~summ(~f,~k,0,(~nn)),~n,~z) => ztrans(sub(k=n,f),n,z)/(z-1) - sum(1/z^y*ztrans(sub(k=n,f),n,z),y,1,((n-nn)-1)) when (freeof((nn-n),n) and fixp(nn-n) and (nn-n)<0 and summ = sum), % ztrans_aux(~summ(~~f/~g,~k,0,(~nn)),~n,~z) => % ztrans(SUB(k=n,f/g),n,z)/(z-1) - % sum(1/z^y*ztrans(SUB(k=n,f/g),n,z),y,1,((n-nn)-1)) % when (freeof((nn-n),n) and fixp (nn-n) and % (nn-n)<0 and summ = sum), ztrans_aux(~summ(~f,~k,~p,~n),~n,~z) => ztrans(sum(sub(k=k+p,f),k,0,n+(-p)),n,z) when (freeof(p,n) and fixp(p) and p<0 and summ = sum), ztrans_aux(~summ(~f,~k,~p,~q),~n,~z) => (begin scalar r; r := q-p; return ztrans(sum(sub(k=k+p,f),k,0,r),n,z); end) when (not(p=0) and summ = sum), %Errors %====== ztrans_aux(~~f/(~n+~~k),~n,~z) => (begin newrederr{"ERROR: zero divisor in ", sum(f/((n+k)*z^n),n,0,infinity)} end) when (numberp k and k<1), ztrans_aux(~~f/factorial(~n+~~k),~n,~z) => (begin newrederr{"ERROR: zero divisor in " ,sum(f/(factorial(n+k)*z^n),n,0,infinity)} end) when (numberp k and k<0) }$ let ztransrules>>; % INVZTRANS: inverse Z transformation, see % Bronstein, Semendjajew: Taschenbuch der Mathematik, 4.4.4 load!-package 'residue; %###################################################################### % Final simplification, % by Wolfram Koepf algebraic<< ztranstrighypsimplificationrules:={ asin(sin(~xx))=>xx, acos(cos(~xx))=>xx, atan(tan(~xx))=>xx, acot(cot(~xx))=>xx, asinh(sinh(~xx))=>xx, acosh(cosh(~xx))=>xx, atanh(tanh(~xx))=>xx, acoth(coth(~xx))=>xx, (1-sin(~xx)^2)^(1/2)=>cos(xx), (1-cos(~xx)^2)^(1/2)=>sin(xx), (cosh(~xx)^2-1)^(1/2)=>sinh(xx), (1+sinh(~xx)^2)^(1/2)=>cosh(xx), (cosh(~xx)+sinh(~xx))^~nn=>cosh(nn*xx)+sinh(nn*xx), (cosh(~xx)-sinh(~xx))^~nn=>cosh(nn*xx)-sinh(nn*xx) } $ operator invztrans,invztrans_aux,invztrans1,invztrans_end; let {binomial(~n,~k)=>prod(n-i,i,0,k-1)/factorial(k) when fixp(k)}; >>; % Procedural embedding, % by Wolfram Koepf algebraic procedure do_invztrans(f,z,n); begin scalar tmp,numtmp,dentmp; tmp := invztrans1(f,z,n); numtmp:=num(tmp); dentmp:=den(tmp); numtmp:=(numtmp where ztranstrighypsimplificationrules); dentmp:=(dentmp where ztranstrighypsimplificationrules); tmp:=numtmp/dentmp; % tmp:=sub(invztrans_end=invztrans,tmp); % macht Probleme wegen Rekursivitaet, next if has no part(.0) return tmp; % if part(tmp,0)=invztrans then % return lisp mk!*sq((list((car fkern list('invztrans,f,z,n) . 1) % . 1)) . 1) % else return tmp; end$ %******************************************************************** % invztrans ruleset % by Lisa Temme put('slash, 'simpfn, 'simpiden); algebraic << invztransrules:= { %Linear rules %============ invztrans (~p,~z,~n) => !$do_invztrans!$ when freeof((!$do_invztrans!$ := do_invztrans(p,z,n)),lisp 'fail), invztrans1(~p,~z,~n) => p*invztrans1(1,z,n) when freeof(p,z) and not (p=1), invztrans1(~p*~f,~z,~n) => p*invztrans1(f,z,n) when freeof(p,z), invztrans1(~f/~q,~z,~n) => invztrans1(f,z,n)/q when freeof(q,z), invztrans1(~p/~g,~z,~n) => p*invztrans1(1/g,z,n) when freeof(p,z) and not(p=1), invztrans1(~p*~f/~g,~z,~n) => p*invztrans1(f/g,z,n) when freeof(p,z), invztrans1(~f/(~q*~g),~z,~n) => invztrans1(f/g,z,n)/q when freeof(q,z), invztrans1(-~f,~z,~n) => -invztrans1(f,z,n), invztrans1((~f+~g)/~~h,~z,~n) => invztrans1(f/h,z,n) + invztrans1(g/h,z,n), %********************************************************************** %For trigonometric/hyperbolic rational %input goto ruleset invztrans_aux %===================================== invztrans1(~f/~g,~z,~n) => invztrans_aux(f,g,z,n) when ( not(freeof(f/g,sin)) or not(freeof(f/g,cos)) or not(freeof(f/g,sinh)) or not(freeof(f/g,cosh)) ), %If not a trig/hyperbolic rational %input goto ruleset invztrans_end %(ie. all remaining inputs) %================================= invztrans1(~f,~z,~n) => invztrans_end(f,z,n) %, % invztrans1(~f,~z,~n) => % (begin % return lisp mk!*sq((list((car fkern list('invztrans1,reval 'f, % reval 'z,reval 'n) . 1) %. 1)) . 1); % end) };let invztransrules; %###################################################################### invztrans_auxrules := { %Linearity %========= invztrans_aux(~f,-~~x*~z^2+~~w*~z-~y,~z,~n) => -invztrans_aux(f,x*z^2-w*z+y,z,n), invztrans_aux(~f+~h,~g,~z,~n) => invztrans_aux(f,g,z,n) + invztrans_aux(h,g,z,n), %Rules to match trigonometric/hyperbolic %rational inputs. %======================================= invztrans_aux(~z,(~~x*~z^2-~~w*~z+~y),~z,~n) => sub(srx=sqrt(x), srw=sqrt(w), sry=sqrt(y), 2*sry^n*sin(acos(srx*w/(2*sry*x))*n) / ( srx^n*sqrt(4*x*y-w^2) ) ) when (numberp(x) and numberp(w) and numberp(y) and y>0 and w>0 and (w^2)<(4*x*y)) or (numberp(x) and numberp(w) and not(numberp(y)) and w>0) or (not(numberp(x) and numberp(w) and numberp(y)) and freeof((w/x),cosh)), invztrans_aux(~z,(~~x*~z^2+~~w*~z+~y),~z,~n) => sub(srx=sqrt(x), srw=sqrt(w), sry=sqrt(y), -2*sry^n*(-1)^n*sin(acos(srx*w/(2*sry*x))*n) / ( srx^n*sqrt(4*x*y-w^2) ) ) when (numberp(x) and numberp(w) and numberp(y) and y>0 and w>0 and (w^2)<(4*x*y)) or (numberp(x) and numberp(w) and not(numberp(y)) and w<0) or (not(numberp(x) and numberp(w) and numberp(y)) and freeof((w/x),cosh)), invztrans_aux(~z,(~~x*~z^2-~~w*~z+~y),~z,~n) => sub(srx=sqrt(x), srw=sqrt(w), sry=sqrt(y), 2*sry^n*sinh(acosh(srx*w/(2*sry*x))*n) / ( srx^n*sqrt(w^2-4*x*y) ) ) when (numberp(x) and numberp(w) and numberp(y) and y>0 and (w^2)>(4*x*y)) or (not(numberp(x) and numberp(w) and numberp(y))), invztrans_aux(~z,(~~x*~z^2+~~w*~z+~y),~z,~n) => sub(srx=sqrt(x), srw=sqrt(w), sry=sqrt(y), -2*(-sry)^n*sinh(acosh(srx*w/(2*sry*x))*n) / ( srx^n*sqrt(w^2-4*x*y) ) ) when (numberp(x) and numberp(w) and numberp(y) and y>0 and (w^2)>(4*x*y)) or (not(numberp(x) and numberp(w) and numberp(y))), invztrans_aux(~z^2,(~~x*~z^2-~~w*~z+~y),~z,~n) => sub(srx=sqrt(x), srw=sqrt(w), sry=sqrt(y), (sry^n*(sqrt(4*x*y-w^2)*cos(acos(srx*w/(2*sry*x))*n) + sin(acos(srx*w/(2*sry*x))*n)*w)) / ( srx^n*sqrt(4*x*y-w^2)*x ) ) when (numberp(x) and numberp(w) and numberp(y) and y>0 and w>0 and (w^2)<(2*x*y)) or (numberp(x) and numberp(w) and w>0) or (not(numberp(x) and numberp(w) and numberp(y)) and freeof((w/x),cosh)), invztrans_aux(~z^2,(~~x*~z^2+~~w*~z+~y),~z,~n) => sub(srx=sqrt(x), srw=sqrt(w), sry=sqrt(y), (sry^n*(-1)^n*(sqrt(4*x*y-w^2)*cos(acos(srx*w/(2*sry*x))*n) + sin(acos(srx*w/(2*sry*x))*n)*w)) / ( srx^n*sqrt(4*x*y-w^2)*x ) ) when (numberp(x) and numberp(w) and numberp(y) and y>0 and w>0 and (w^2)>(4*x*y)) or (numberp(x) and numberp(w) and w<0) or (not(numberp(x) and numberp(w) and numberp(y)) and freeof((w/x),cosh)), invztrans_aux(~z^2,(~~x*~z^2-~~w*~z+~y),~z,~n) => sub(srx=sqrt(x), srw=sqrt(w), sry=sqrt(y), (sry^n*(sqrt(w^2-4*x*y)*cosh(acosh(srx*w/(2*sry*x))*n) + sinh(acosh(srx*w/(2*sry*x))*n)*w)) / ( srx^n*sqrt(w^2-4*x*y)*x ) ) when (numberp(x) and numberp(w) and numberp(y) and y>0 and w>(4*x*y)) or (not(numberp(x) and numberp(w) and numberp(y))), invztrans_aux(~z^2,(~~x*~z^2+~~w*~z+~y),~z,~n) => sub(srx=sqrt(x), srw=sqrt(w), sry=sqrt(y), ((-sry)^n*(sqrt(w^2-4*x*y)*cosh(acosh(srx*w/(2*sry*x))*n) + sinh(acosh(srx*w/(2*sry*x))*n)*w)) / ( srx^n*sqrt(w^2-4*x*y)*x ) ) when (numberp(x) and numberp(w) and numberp(y) and y>0 and w>(4*x*y)) or (not(numberp(x) and numberp(w) and numberp(y))), invztrans_aux(~f,~g,~z,~n) => invztrans_end(f/g,z,n) };let invztrans_auxrules; %###################################################################### invztrans_endrules := { %Rules to match other %trigonometric inputs %==================== invztrans_end(~z*atan(sin(~b)//(cos(~b)-~z)),~z,~n) => -sin(b*(n+1))/(n+1) when numberp(b) or (freeof(b,z) and not(numberp(b))), invztrans_end(~z*atan(sin(~b)//(~z+cos(~b))),~z,~n) => (-1)^n*sin(b*(n+1))/(n+1) when numberp(a) or (freeof(a,z) and not(numberp(a))), invztrans_end(~z*log(~z/sqrt(~z^2-~a*~z+1)),~z,~n) => cos(acos(a/2)*(n+1))/(n+1) when (numberp(a) and a>0 and a<=-2) or (freeof(a,z) and not(numberp(a))), invztrans_end(~z*log(~z/sqrt(~z^2+~a*~z+1)),~z,~n) => cos(acos(-a/2)*(n+1))/(n+1) when (numberp(a) and a<0 and a>=-2) or (freeof(a,z) and not(numberp(a))), invztrans_end(~z*log(sqrt(~z^2-~a*~z+1)/~z),~z,~n) => (-1)^n* cos(acos(-a/2)*(n+1))/(n+1) when (numberp(a) and a<0 and a>=-2) or (freeof(a,z) and not(numberp(a))), invztrans_end(~z*log(sqrt(~z^2+~a*~z+1)/~z),~z,~n) => (-1)^n*cos(acos(a/2)*(n+1))/(n+1) when (numberp(a) and a>0 and a<=-2) or (freeof(a,z) and not(numberp(a))), invztrans_end(cos(~a/~z)*e^(sqrt(1-~a^2)/~z),~z,~n) => cos(asin(a)*n)/factorial(n) when (numberp(a) and a<=1 and a>=-1) or (freeof(a,z) and not(numberp(a))), %********************************************************************** %Rule to calculate the Residues and hence %determine the invztrans of a rational input %=========================================== % by Wolfram Koepf invztrans_end(~f,~z,~n)=> (begin scalar denominator, result, solutionset, solution, !*fullroots; on fullroots; denominator:=den(f); solution:=solve(denominator,z); if not freeof(solution,root_of) then rederr("denominator could not be factorized"); solutionset:= for i:=1:length(solution) collect(part(part(solution,i),2)); result:= for each a in solutionset sum(residue(f*z^(n-1),z,a)); return(result) end) when type_ratpoly(f,z), %********************************************************************** %Rules to match non-rational inputs %================================== %(Binomial) %---------- invztrans(~z^~~k/(z+~~a)^~~m,~z,~n) => binomial(n+k-1,m-1)*(-a)^(n+k)/(-a)^m when freeof(k,z) and freeof(m,z) and freeof(a,z) and (not(numberp k) or (numberp k and fixp k)) and (not(numberp m) or (numberp m and fixp m)), %(over n!) %--------- invztrans_end(e^(~k/~z),~z,~n) => k^n/factorial(n) when freeof(k,z), invztrans_end(e^(~k/~z)/~z,~z,~n) => n/k*k^n/factorial(n) when freeof(k,z), invztrans_end(1/e^(~k/~z),~z,~n) => (-k)^n/factorial(n) when freeof(k,z), invztrans_end(1/(e^(~k/~z)*~z),~z,~n) => -n/k*(-k)^n/factorial(n) when freeof(k,z), invztrans_end(e^(~k/(~~j*~z)),~z,~n) => (k/j)^n/factorial(n) when freeof(k,z) and freeof(j,z), invztrans_end(e^(~k/(~~j*~z))/~z,~z,~n) => n/(k/j)*(k/j)^n/factorial(n) when freeof(k,z) and freeof(j,z), invztrans_end(1/e^(~k/(~~j*~z)),~z,~n) => (-k/j)^n/factorial(n) when freeof(k,z) and freeof(j,z), invztrans_end(1/(e^(~k/(~~j*~z))*~z),~z,~n) => n/(-k/j)*(-k/j)^n/factorial(n) when freeof(k,z) and freeof(j,z), invztrans_end(cos(sin(~~b)/~z)*e^(cos(~~b)/~z),~z,~n) => cos(b*n)/factorial(n) when freeof (b,z), invztrans_end(sin(sin(~~b)/~z)*e^(cos(~~b)/~z),~z,~n) => sin(b*n)/factorial(n) when freeof (b,z), %(over 2n!) %---------- invztrans_end(cosh(~k/sqrt(~z)),~z,~n) => k^(2*n)/factorial(2*n) when freeof(k,z), invztrans_end(cos(~k/sqrt(~z)),~z,~n) => (-(k^2))^n/factorial(2*n) when freeof(k,z), invztrans_end(cosh(~k/(~~j*sqrt(~z))),~z,~n) => (k/j)^(2*n)/factorial(2*n) when freeof(k,z) and freeof(j,z), invztrans_end(cos(~k/(~~j*sqrt(~z))),~z,~n) => (-(k/j)^2)^n/factorial(2*n) when freeof(k,z) and freeof(j,z), %(over (2n+1)!) %-------------- invztrans_end(sqrt(~z)*sinh(~k/sqrt(~z)),~z,~n) => k*k^(2*n)/factorial(2*n+1) when freeof (k,z), invztrans_end(sqrt(~z)*sinh(~k/sqrt(-~z)),~z,~n) => i*k*(-k^2)*n/factorial(2*n+1) when freeof (k,z), invztrans_end(sqrt(~z)*sin(~k/sqrt(~z)),~z,~n) => k*(-k^2)^n/factorial(2*n+1) when freeof (k,z), invztrans_end(sqrt(-~z)*sinh(~k/sqrt(~z)),~z,~n) => sqrt(-k^2)*k^(2*n)/factorial(2*n+1) when freeof (k,z), invztrans_end(sqrt(-~z)*sin(~k/sqrt(~z)),~z,~n) => k*(-k^2)^n/(i*factorial(2*n+1)) when freeof (k,z), invztrans_end(sqrt(-~z)*sinh(~k/sqrt(-~z)),~z,~n) => k*(-k^2)*n/factorial(2*n+1) when freeof (k,z), invztrans_end(sqrt(-~z)*sin(~k/sqrt(~z)),~z,~n) => k*(-k^2)*n/(i*factorial(2*n+1)) when freeof (k,z), invztrans_end(sqrt(~z)*sinh(~k/(~~j*sqrt(~z))),~z,~n) => (k/j)*(k/j)^(2*n)/factorial(2*n+1) when freeof (k,z) and freeof(j,z), invztrans_end(sqrt(-~z)*sinh(~k/(~~j*sqrt(~z))),~z,~n) => (k/j)*(k/j)^(2*n)/factorial(2*n+1) when freeof (k,z) and freeof(j,z), invztrans_end(sqrt(-~b*~z)*sinh(~k/(sqrt(~b)*sqrt(~z))),~z,~n) => sqrt(-k^2)*(k^2/b)^n/factorial(2*n+1) when freeof (k,z) and freeof(j,z), invztrans_end(sqrt(~z)*sin(~k/(~~j*sqrt(~z))),~z,~n) => (sqrt(-k^2)/j)*(-k^2)^n/j^(2*n)/(i*factorial(2*n+1)) when freeof (k,z) and freeof(j,z), invztrans_end(sqrt(-~z)*sin(~k/(~~j*sqrt(~z))),~z,~n) => (k/j)*(k/j)^(2*n)/(i*factorial(2*n+1)) when freeof (k,z) and freeof(j,z), invztrans_end(sqrt(-~b*~z)*sin(~k/(sqrt(~b)*sqrt(~z))),~z,~n) => k*(-k^2/b)^n/(i*factorial(2*n+1)) when freeof (k,z) and freeof(b,z), invztrans_end(sqrt(~z)*sinh(~k/(~~j*sqrt(-~z))),~z,~n) => i*(k/j)*(k/j)^(2*n)/factorial(2*n+1) when freeof (k,z) and freeof(j,z), invztrans_end(sqrt(~z)*sin(~k/(~~j*sqrt(-~z))),~z,~n) => (sqrt(-k^2)/j)*(sqrt(-k^2)/j)^(2*n)/factorial(2*n+1) when freeof (k,z) and freeof(j,z), %(over n+1) %---------- invztrans_end(~z*log(~~b*~z/(~~b*~z+~a)),~z,~n) => (-a/b)^(n+1)/(n+1) when freeof(a,z) and freeof(b,z), invztrans_end(~z*log((~~b*~z+~a)/(~~b*~z)),~z,~n) => -invztrans1(z*log(b*z/(b*z+a)),z,n) when freeof(a,z) and freeof(b,z), %If input has not matched any rules %return INVZTRANS(~f,~z,~n) %================================== invztrans_end(~f,~z,~n) => lisp 'fail };let invztrans_endrules; >>; endmodule; end;