Thread: Laplace transform of Bessel functions

Dear all,

I would like to ask if anyone of you knows where to get the Laplace transform of the following function:

f(t) = I_n(a*t) * J_m(b*sqrt{t})

With

I_n() = Modified Bessel function of the first kind, of integer order n
J_m() = Bessel function of the first kind, of integer order m
a,b = real numbers



I was trying to look up in the book

A.P. Prudnikov, "Integrals and series, vol. 4 Laplace transforms"

but is not available in my library until march 8th!


I thank your kind help,
Home-made-bread.

Originally Posted by homemadebread

Dear all,

I would like to ask if anyone of you knows where to get the Laplace transform of the following function:

f(t) = I_n(a*t) * J_m(b*sqrt{t})

With

I_n() = Modified Bessel function of the first kind, of integer order n
J_m() = Bessel function of the first kind, of integer order m
a,b = real numbers



I was trying to look up in the book

A.P. Prudnikov, "Integrals and series, vol. 4 Laplace transforms"

but is not available in my library until march 8th!


I thank your kind help,
Home-made-bread.

I don't know the legal status of Library Genesis but they have a copy of volume 4 of Prudnikov in djvu format (for which you can download a free reader).

So google for Library Genesis and take it from there.

CB

Originally Posted by CaptainBlack

I don't know the legal status of Library Genesis but they have a copy of volume 4 of Prudnikov in djvu format (for which you can download a free reader).

So google for Library Genesis and take it from there.

CB

Hello everybody,

I first thank CaptainBlack so much for answering, I even did not know about Library Genesis. I think I will use it a lot in the future!


However, Prudnikov's book does not give the result I am looking for. Any other suggestion?


Thanks in advance.
My best regards, Home-made-bread

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 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:



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 even if were complex.