Artifact 7f6c60bd27f52a63b9d550371dfb20622aee1feb25a7a084618a80f95002f069:
- Executable file
r37/packages/ztrans/ztrrules.red
— part of check-in
[f2fda60abd]
at
2011-09-02 18:13:33
on branch master
— Some historical releases purely for archival purposes
git-svn-id: https://svn.code.sf.net/p/reduce-algebra/code/trunk/historical@1375 2bfe0521-f11c-4a00-b80e-6202646ff360 (user: arthurcnorman@users.sourceforge.net, size: 25643) [annotate] [blame] [check-ins using] [more...]
- Executable file
r38/packages/ztrans/ztrrules.red
— part of check-in
[f2fda60abd]
at
2011-09-02 18:13:33
on branch master
— Some historical releases purely for archival purposes
git-svn-id: https://svn.code.sf.net/p/reduce-algebra/code/trunk/historical@1375 2bfe0521-f11c-4a00-b80e-6202646ff360 (user: arthurcnorman@users.sourceforge.net, size: 25643) [annotate] [blame] [check-ins using]
module ztrrules; % Ztrans ruleset. % Author: 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)}; let {binomial(~n,~k)=> (for i:=0:k-1 product n-i)/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;