More Laplace of Struve (was: Re: [Maxima] Re: [Maxima-commits] CVS: maxima/tests testsuite.lisp,1.3,1.4 rte st14.mac,1.15,1.16)
Subject: More Laplace of Struve (was: Re: [Maxima] Re: [Maxima-commits] CVS: maxima/tests testsuite.lisp,1.3,1.4 rte st14.mac,1.15,1.16)
From: Wolfgang Jenkner
Date: Fri, 04 Feb 2005 14:05:16 +0100
Raymond Toy writes:
> But, if I understand your derivation correctly, the ratio should be 1,
> not 4/3/sqrt(%pi).
Here is a better answer (I hope ;-)
>From comparing specint(hstruve[n](t)*%e^(-p*t),t) with the Laplace
transform of the power series representation
http://dlmf.nist.gov/Draft/ST/about_ST.3.1.html
of their Struve H[n] functions one finds (e.g., by applying several
times the duplication formula for the gamma function) that they could
be defined as
H[n](t):=hstruve[n](t)*gamma(n+3/2);
I checked this with a simple recursion formula for the Laplace
transform of t^n*H[n](t), which can be derived from
http://dlmf.nist.gov/Draft/ST/about_ST.6.6.html
assume(p>0);
H[n](x):=hstruve[n](x)*gamma(n+3/2);
LH[n](p):=ev(fullratsimp(specint(H[n](t)*t^n*%e^(-p*t),t)),logexpand:all);
LHsimple[0]:2/%pi/p^2;
LHsimple[n]:=2*n*LHsimple[n-1]/p^2;
LHcheck(n):=(LH[n](p),logcontract(fullratsimp(LH[n+1](p)-(2*n+1)*%%-p*diff(%%,p)-LHsimple[n])));
All of the LHcheck(n) for n=0,1,... should be 0. I checked this only
for n <= 5, though.
Wolfgang