module specfn2; % Part 2 of the Special functions package for REDUCE.
% The special Special functions.
% Author : Victor Adamchik, Byelorussian University Minsk, Byelorussia.
% Major modifications by: Winfried Neun, ZIB Berlin.
% |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| %
% %
% Please report bugs to Winfried Neun, %
% Konrad-Zuse-Zentrum %
% fuer Informationstechnik Berlin, %
% Heilbronner Str. 10 %
% 10711 Berlin - Wilmersdorf %
% Federal Republic of Germany %
% or by email, neun@sc.ZIB-Berlin.de %
% %
% |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| %
% %
% This package provides algebraic %
% manipulations upon some special functions: %
% %
% -- Generalized Hypergeometric Functions %
% -- Meijer's G Function %
% -- to be extended %
% %
% |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| %
create!-package ('(specfn2 ghyper meijerg),
'(contrib specfn));
load_package specfn;
% Various help utilities and smacros for hypergeometric function
% simplification.
symbolic smacro procedure diff1sq(u,v);
addsq(u,negsq(v))$
symbolic smacro procedure mksqnew u;
!*p2f(car fkern(u) .* 1) ./ 1;
symbolic smacro procedure gamsq(u);
mksqnew('GAMMA . list(prepsq u));
symbolic smacro procedure multgamma u;
%u -- list of SQ.
<<for each pp in u do <<p := multsq(gamsq pp,p)>>; p>>
where p = '(1 . 1);
symbolic smacro procedure besssq(v,u);
mksqnew('BesselJ . list(prepsq v,prepsq u))$
symbolic smacro procedure bessmsq(v,u);
mksqnew('BesselI . list(prepsq v,prepsq u))$
symbolic smacro procedure simppochh(v,u);
mksqnew('Pochhammer . list(prepsq v,prepsq u))$
symbolic procedure multpochh(u,k);
<< for each pp in u do <<p := multsq (simppochh (pp,k),p)>>; p>>
where p = '(1 . 1);
symbolic smacro procedure psisq(v);
mksqnew('psi . list(prepsq v))$
symbolic smacro procedure dfpsisq(v,u);
mksqnew('dfpsi . list(prepsq v,prepsq u))$
symbolic procedure simpfunc(u,v);
% u -- name of the function, PF.
% v -- argument, SQ.
begin scalar l,v1!wq;
v1!wq:=prepsq v; l:=('!Adamchik . v1!wq);
u:=subf(car simp!* list(u,'!Adamchik),list l);
return u $
end$
algebraic operator intsin,intcos,ints,intfs,intfc$
symbolic procedure subsqnew(u,v,z);
% u,v -- SQ.
% z -- PF .
begin scalar a!1,lp,a;
a!1:=prepsq v; lp:=((z . a!1));
a:=quotsq(subf(car u,list lp),subf(cdr u,list lp));
return a;
end$
symbolic procedure expdeg(x,y);
% x,y -- SQ.
% value is the x**y.
if null car y then '(1 . 1) else
if numberp(car y) and numberp(cdr y) then
simpx1(prepsq x,car y,cdr y) else
quotsq(expdeg1(car x ./ 1 ,y),expdeg1(cdr x ./ 1 ,y))$
symbolic procedure expdeg1(x,y);
% x,y -- SQ.
% value is the x**y.
simp!*(prepsq(subsqnew(subsqnew(simp!*
'(expt a!g9 b!!g9),x,'a!g9),y,'b!!g9)))$
symbolic smacro procedure difflist(u,v);
% u -- list of SQ.
% v -- SQ.
% value is (u) - v.
for each uu in u collect addsq(uu,negsq v);
symbolic procedure listplus(u,v);
% value is (u) + v.
difflist(u,negsq v)$
symbolic smacro procedure addlist u;
% u -- list of PF.
<<for each pp in u do <<p := addsq(simp!* pp,p)>>; p>>
where p = '(nil . 1);
symbolic smacro procedure listsq(u);
% u - list of PF.
% value is list of SQ.
for each uu in u collect simp!* uu;
symbolic smacro procedure listmin(u);
% u - list.
% value is (-u).
for each uu in u collect negsq uu;
symbolic smacro procedure multlist(u);
<< for each pp in u do <<p := multsq(pp,p)>>; p>> where p = '(1 . 1);
symbolic procedure parfool u;
% u -- SQ.
% value is T if u = 0,-1,-2,...
if null car u then t
else
if and(numberp car u,eqn(cdr u,1),lessp(car u,0)) then t
else nil$
symbolic procedure znak u;
% u -- SQ.
if numberp u then
if u > 0 then T else NIL
else
if numberp car u then
if car u > 0 then T else NIL
else
if not null cdar u then T
else
if numberp cdaar u then
if cdaar u > 0 then T else NIL
else
znak(cdaar u ./ 1)$
symbolic procedure sdiff(a,b,n);
% value is (1/b*d/db)**n(a) .
% a,n--SQ b--PF .
if null car n then a else
if and(numberp(car n),numberp(cdr n),eqn(cdr n,1),not lessp(car n,0))
then
multsq(invsq(simp!* b), diffsq(sdiff(a,b,diff1sq(n, '(1 . 1))),b))
else
rerror('specialf,130,"***** error parameter in sdiff")$
symbolic procedure derivativesq(a,b,n);
% a -- SQ.
% b -- ATOM.
% n -- order, SQ.
if null n or null car n then a
else derivativesq(diffsq(a,b),b,diff1sq(n,'(1 . 1)))$
symbolic procedure addend( u,v,x);
% u,v -- lists of SQ.
% x -- SQ.
cons(diff1sq(car u,x),difflist(v,diff1sq(car u,x)))$
symbolic procedure parlistfool(u,v);
%v -- list.
%value is the T if u-(v)=0,-1,-2,...
if null v then nil else
if parfool(diff1sq(u,car v)) then t else
parlistfool(u,cdr v)$
symbolic procedure listparfool(u,v);
%u -- list.
%value is the T if (u)-v=0,-1,-2,...
if null u then nil else
if parfool(diff1sq(car u,v)) then t else
listparfool(cdr u,v)$
symbolic procedure listfool u;
%u -- list.
%value is the T if any two of the terms (u)
%differ by an integer or zero.
if null cdr u then nil else
if parlistfool(car u,cdr u) or listparfool(cdr u,car u)
then t else listfool(cdr u)$
symbolic procedure listfooltwo(u,v);
%u,v -- lists.
%value is the T if (u)-(v)=0,-1,-2,...
if null u then nil else
if parlistfool(car u,v) then t else
listfooltwo(cdr u,v)$
symbolic smacro procedure pdifflist(u,v);
% u -- SQ.
% v -- list of SQ.
%value is a list: u-(v).
for each vv in v collect diff1sq(u,vv);
symbolic procedure redpar1(u,n);
% value is a paire, car-part -- sublist of the length n
% cdr-part -- .
begin scalar bm;
while u and not(n=0) do begin
bm:=cons (car u,bm);
u:=cdr u;
n:=n-1;
end;
return cons(reverse bm,u);
end$
symbolic procedure redpar (l1,l2);
begin scalar l3;
while l2 do <<
if member(car l2 , l1) then l1 := delete(car l2,l1)
else l3 := (car l2) . l3 ;
l2 := cdr l2 >>;
return list (l1,reverse l3);
end;
algebraic operator Lommel,Heaviside;
symbolic smacro procedure heavisidesq(u);
mksqnew('Heaviside . list(prepsq u));
symbolic smacro procedure StruveLsq(v,u);
mksqnew('StruveL . list(prepsq v,prepsq u));
symbolic smacro procedure StruveHsq(v,u);
mksqnew('StruveH . list(prepsq v,prepsq u));
symbolic smacro procedure besssq(v,u);
mksqnew('BesselJ . list(prepsq v,prepsq u));
symbolic smacro procedure bessmsq(v,u);
mksqnew('BesselI . list(prepsq v,prepsq u));
symbolic smacro procedure neumsq(v,u);
mksqnew('BesselY . list(prepsq v,prepsq u));
symbolic smacro procedure simppochh(v,u);
mksqnew('Pochhammer . list(prepsq v,prepsq u));
symbolic smacro procedure psisq(v);
mksqnew('psi . list(prepsq v));
symbolic smacro procedure dfpsisq(v,u);
mksqnew('Polygamma . list(prepsq u,prepsq v));
symbolic smacro procedure Lommel2sq (u,v,w);
mksqnew('Lommel2 . list(prepsq u,prepsq v,prepsq w));
symbolic smacro procedure tricomisq (u,v,w);
mksqnew('KummerU . list(prepsq u,prepsq v,prepsq w));
symbolic smacro procedure macdsq (v,u);
mksqnew('BesselK . list(prepsq v,prepsq u));
fluid '(v1!wq,a!g9,b!!g9);
symbolic smacro procedure sumlist u;
% u -- list of the PF
<<for each pp in u do <<p := addsq(simp pp,p)>>; p>>
where p = '(nil . 1);
symbolic smacro procedure difflist(u,v);
% u -- list of SQ.
% v -- SQ.
% value is (u) - v.
for each uu in u collect addsq(uu,negsq v);
symbolic smacro procedure addlist u;
% u -- list of PF.
<<for each pp in u do <<p := addsq(simp!* pp,p)>>; p>>
where p = '(nil . 1);
symbolic smacro procedure diff1sq(u,v);
addsq(u,negsq(v));
symbolic smacro procedure listsq(u);
% u - list of PF.
% value is list of SQ.
for each uu in u collect simp!* uu;
symbolic smacro procedure listmin(u);
% u - list.
% value is (-u).
for each uu in u collect negsq uu;
symbolic smacro procedure multlist(u);
<< for each pp in u do <<p := multsq(pp,p)>>; p>> where p = '(1 . 1);
symbolic smacro procedure pdifflist(u,v);
% u -- SQ.
% v -- list of SQ.
%value is a list: u-(v).
for each vv in v collect diff1sq(u,vv);
symbolic smacro procedure listprepsq(u);
for each uu in u collect prepsq uu;
endmodule;
end;