# Approximation to Bessel Function in Integrand

Started by July 27, 2009
```Hello Clay,

Thank you for the hint with performing the numerical integration on the
integral

integral(x^p*ln(x)*exp(-x^2)*besseli(0,2*x),x=0..infinity)

using Gaussian Quadrature. I know that Matlab has some build in functions
("quad" and variants), but I do not think that they have Gaussian
Quadrature. I am using Maple commands in Matlab to perform the integration,
but I probably have to code it up myself. Is there any way to simplify
maybe parts of the integrand (not necessarily using the Bessel function
approximations for large inputs)? I use the approximation

Io(y) ~ (1/sqrt(2*pi*y))*exp(y) for large y

but still have issues finding the integral in a table. I wonder whether
somehow substituting in another simplification for any combination of
functions in the integral would then allow me to use an entry in a table.
The problems are naturally with the ln(x) and/or Bessel function. I have
even tried integration by parts but ran into trouble with the ln(x) in u*v
and corresponding integral for dv in the relationship

integral(u*v) = u*v - integral(v*du),

where the two integrals and u*v are all evaluated from 0 to infinity. I do
not see really any way to find a closed-form solution but am still working
on it.

Thank you again for the advice and supporting documents,

Marek
```
```Hello Scott,

my results from Maple using on that integral (without the ln(x) function):

> f:=int(x^p*exp(-x^2)*BesselJ(0,sqrt(-1)*2*x),x=0..infinity);

GAMMA(p/2 + 1/2) (p + 1) LaguerreL(- p/2 + 1/2, 1)
f := 1/2 --------------------------------------------------
p - 1

GAMMA(p/2 + 1/2) LaguerreL(- p/2 + 1/2, 1, 1)
- ---------------------------------------------
p - 1

> combine(convert(f,hypergeom));

(1/4 hypergeom([p/2 - 1/2], , 1) p

+ 1/4 hypergeom([p/2 - 1/2], , 1)

+ 1/4 hypergeom([p/2 - 1/2], , 1) p

- 3/4 hypergeom([p/2 - 1/2], , 1)) GAMMA(p/2 - 1/2)

I am sure that my result simplifies even further into your result too. Are
there maybe any mathematical papers that possibly deal with integrals
involving ln(x)? I just really believe that there has to be a closed-form
solution for the integral

integral(x^p*ln(x)*exp(-x^2)*besseli(0,2*x),x=0..infinity)

even if Maple and/or Mathematica do not provide one.

Thank you again,

Marek
```
```>>>>> "Rune" == Rune Allnor <allnor@tele.ntnu.no> writes:

Rune> On 27 Jul, 16:32, Clay <c...@claysturner.com> wrote:
>> Of course I would try looking in Watson as he wrote the definitive
>> work on Bessel functions. I would look for you, but my copy is at the
>> farm.

>>  Watson,G.N., "A Treatise on the Theory of Bessel Functions", 1922
>> Cambridge University Press.

Rune> A genuine CUP edition, not a Dover reprint?
Rune> Impressive.

It was reprinted (not Dover) recently.  I have a copy.

Ray
```