Re: Help with Integral

Hi all,
spent the last week on vacation in Paris...

I found an approximation for the integral
u(a,b)=int_0^oo sin(b*x)*exp(-a*sqrt(a^2+b^2)) dx =
b*exp(-a-b^2 /(2*a))*sqrt(pi/(2*a))*erfi(b/sqrt(2*a))
that works well for large a, where erfi(x) is the imaginary
error function erf(i*x)/i .

By playing a little with Raymonds expression f(a,b) I found a result
containing a sum that may be solvable:
u(a,b)=a*pi/(2*sqrt(a^2+b^2))*besselI(1,sqrt(a^2+b^2))+
a*pi/(4*sqrt(a^2+b^2))*
(struveL(-1,sqrt(a^2+b^2) + struveL(1,sqrt(a^2+b^2))+
b*exp(-a)/(a^2+b^2)-a/(2*sqrt(a^2+b^2))*
sum_{n=0}^oo ((-1)^n/n! * sqrt(a^2+b^2)^n *
beta(a^2/(a^2+b^2),1+n/2,1/2)
with beta(z,a1,a2) as incomplete beta function.
Maybe this sum can be calculated by a recursive reduction
to beta(a^2/(a^2+b^2),1,1/2) and beta(a^2/(a^2+b^2),3/2,1/2).
Any ideas ?

Andreas
.

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