Originally Posted by

**homemadebread** . Any other suggestion?

Thanks in advance.

My best regards, Home-made-bread

Yeah, numerically. Ok, suppose I wanted to know what the Laplace Transform of that was at $\displaystyle n=2, m=3, a=1,b=1/2,s=2$ then I'd do the following in Mathematia:

Code:

In[3]:= n = 2;
m = 3;
a = 1;
b = 1/2;
s = 2;
NIntegrate[
BesselI[n, a t] BesselJ[m, b Sqrt[t]] Exp[-s t],
{t, 0, \[Infinity]}]
Out[8]= 0.000288939

Bingo. Then I claim:

$\displaystyle \mathcal{L}\left\{I_2(t) J_3(1/2\sqrt{t})\right\}\biggr|_{s=2}\approx 0.00029$

and if I needed to, I could do this for any values of the parameters and as long as the integration is well-behaved, I could obtain the value of the transform pretty accurately and if necessary, run a fit on the data points to obtain a least-square fit function $\displaystyle f(s)$ even if $\displaystyle s$ were complex.