# Thread: Laplace transform of Bessel functions

1. ## 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.

2. 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

3. 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

4. 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 $n=2, m=3, a=1,b=1/2,s=2$ 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:

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

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 $f(s)$ even if $s$ were complex.