# Approximation to Bessel Function in Integrand

Started by July 27, 2009
```Hello,

I am trying to find a closed-form solution (or reasonable approximation)
to the following integral:

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

where p is a constant and besseli(0,2*x) is the zeroth order Bessel
function of the first kind.

Integrals, Series, and Products by Gradshteyn and Ryzhik and Handbook of
Mathematical Functions by Abramowitz and Stegun) for a closed-form solution
and approximations to the Bessel function and have attempted numerical
integration but have not had any luck determining the final result. Do any
of you have suggestions?

I would greatly appreciate any feedback and assistance.

Thank you again,

Marek

```
```On Jul 27, 10:00&#2013266080;am, "mbtrawicki" <mbtrawi...@yahoo.com> wrote:
> Hello,
>
> I am trying to find a closed-form solution (or reasonable approximation)
> to the following integral:
>
> integral(x^p*ln(x)*exp(-x^2)*besseli(0,2*x),x=0..infinity),
>
> where p is a constant and besseli(0,2*x) is the zeroth order Bessel
> function of the first kind.
>
> Integrals, Series, and Products by Gradshteyn and Ryzhik and Handbook of
> Mathematical Functions by Abramowitz and Stegun) for a closed-form solution
> and approximations to the Bessel function and have attempted numerical
> integration but have not had any luck determining the final result. Do any
> of you have suggestions?
>
> I would greatly appreciate any feedback and assistance.
>
> Thank you again,
>
> Marek

Hello Marek,

Of course I would try looking in Watson[1] as he wrote the definitive
work on Bessel functions. I would look for you, but my copy is at the
farm.

Do you need an analytic answer or will a simple numerical value be
good enough? A simple Monte Carlo simulation using Gausian distributed
samples should suffice.

IHTH,

Clay

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

```
```On 27 Jul, 16:32, Clay <c...@claysturner.com> wrote:

> Of course I would try looking in Watson[1] as he wrote the definitive
> work on Bessel functions. I would look for you, but my copy is at the
> farm.

> [1] Watson,G.N., "A Treatise on the Theory of Bessel Functions", 1922
> Cambridge University Press.

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

I think the 'wierdest' book I own is a reprint
of a 1945 vintage Dover edition of Rayleygh's
"The Theory of Sound" (1877). The most 'impressive'
stuff I can remember to have browsed hands-on, was
a 1st edition Stroustrup "The C++ Programming Language."

Correct. FAR too much free time on my hands...

Rune
```
```Rune Allnor wrote:
> On 27 Jul, 16:32, Clay <c...@claysturner.com> wrote:
>
>> Of course I would try looking in Watson[1] as he wrote the definitive
>> work on Bessel functions. I would look for you, but my copy is at the
>> farm.
>
>> [1] Watson,G.N., "A Treatise on the Theory of Bessel Functions", 1922
>> Cambridge University Press.
>
> A genuine CUP edition, not a Dover reprint?
> Impressive.
>
> I think the 'wierdest' book I own is a reprint
> of a 1945 vintage Dover edition of Rayleygh's
> "The Theory of Sound" (1877). The most 'impressive'
> stuff I can remember to have browsed hands-on, was
> a 1st edition Stroustrup "The C++ Programming Language."
>
> Correct. FAR too much free time on my hands...

A wonderful Dover reprint is (from memory) "Soap Bubbles and the Forces
that Mold Them" by C. Vernon Boys. I recommend it for anyone who wants
to know /things/ and also for those who wonder how science could
possibly have been conducted without modern instruments.

Check out the water-powered audio amplifier.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```On Jul 27, 11:22&#2013266080;am, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 27 Jul, 16:32, Clay <c...@claysturner.com> wrote:
>
> > Of course I would try looking in Watson[1] as he wrote the definitive
> > work on Bessel functions. I would look for you, but my copy is at the
> > farm.
> > [1] Watson,G.N., "A Treatise on the Theory of Bessel Functions", 1922
> > Cambridge University Press.
>
> A genuine CUP edition, not a Dover reprint?
> Impressive.
>
> I think the 'wierdest' book I own is a reprint
> of a 1945 vintage Dover edition of Rayleygh's
> "The Theory of Sound" (1877). The most 'impressive'
> stuff I can remember to have browsed hands-on, was
> a 1st edition Stroustrup "The C++ Programming Language."
>
> Correct. FAR too much free time on my hands...
>
> Rune

The real thing! My father was a Mathematician and he had a penchant
for collecting old books. Of course some of them weren't too old when
he purchased them. So I inherited quite a collection.

Clay

```
```On 27 Jul, 18:17, Clay <c...@claysturner.com> wrote:
> On Jul 27, 11:22&#2013266080;am, Rune Allnor <all...@tele.ntnu.no> wrote:
>
>
>
>
>
> > On 27 Jul, 16:32, Clay <c...@claysturner.com> wrote:
>
> > > Of course I would try looking in Watson[1] as he wrote the definitive
> > > work on Bessel functions. I would look for you, but my copy is at the
> > > farm.
> > > [1] Watson,G.N., "A Treatise on the Theory of Bessel Functions", 1922
> > > Cambridge University Press.
>
> > A genuine CUP edition, not a Dover reprint?
> > Impressive.
>
> > I think the 'wierdest' book I own is a reprint
> > of a 1945 vintage Dover edition of Rayleygh's
> > "The Theory of Sound" (1877). The most 'impressive'
> > stuff I can remember to have browsed hands-on, was
> > a 1st edition Stroustrup "The C++ Programming Language."
>
> > Correct. FAR too much free time on my hands...
>
> > Rune
>
> The real thing! My father was a Mathematician and he had a penchant
> for collecting old books. Of course some of them weren't too old when
> he purchased them. So I inherited quite a collection.

I borrowed a few books from my grandfather's collection
after he passed away. One always has some impression
about a person from meeting them in real life. Then

In retrospect, I wish I had borrowed those books while
he was still around.

Rune
```
```Hello everyone,

Thank you for the information about the Watson book, Clay. I really
appreciate it. I actually found a copy of it online:

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

http://www.archive.org/details/treatiseontheory00watsuoft

I still am sifting through the entire document now.

I would prefer an analytic answer to the integral, but I am more than
happy with a numerical value too. Does anyone have a closed-form solution
for it?

I have used the Bessel function approximation I0(y) ~
(1/sqrt(2*pi*y))*exp(y) for large y but, when I plug that approximation
into the integral, I still am not able to obtain a closed-form solution of
it. I actually have not been able to successfully apply any numerical
analysis techniques to it either. I am not sure how to do it.

Thank you again,

Marek
```
```On Jul 27, 1:37&#2013266080;pm, "mbtrawicki" <mbtrawi...@yahoo.com> wrote:
> Hello everyone,
>
> Thank you for the information about the Watson book, Clay. I really
> appreciate it. I actually found a copy of it online:
>
> [1] Watson,G.N., "A Treatise on the Theory of Bessel Functions", 1922
> Cambridge University Press.
>
> http://www.archive.org/details/treatiseontheory00watsuoft
>
> I still am sifting through the entire document now.
>
> I would prefer an analytic answer to the integral, but I am more than
> happy with a numerical value too. Does anyone have a closed-form solution
> for it?
>
> I have used the Bessel function approximation I0(y) ~
> (1/sqrt(2*pi*y))*exp(y) for large y but, when I plug that approximation
> into the integral, I still am not able to obtain a closed-form solution of
> it. I actually have not been able to successfully apply any numerical
> analysis techniques to it either. I am not sure how to do it.
>
> Thank you again,
>
> Marek

Are you confusing J0(x) with I0(x)?
```
```On Jul 27, 10:00&#2013266080;am, "mbtrawicki" <mbtrawi...@yahoo.com> wrote:
> Hello,
>
> I am trying to find a closed-form solution (or reasonable approximation)
> to the following integral:
>
> integral(x^p*ln(x)*exp(-x^2)*besseli(0,2*x),x=0..infinity),
>
> where p is a constant and besseli(0,2*x) is the zeroth order Bessel
> function of the first kind.
>
> Integrals, Series, and Products by Gradshteyn and Ryzhik and Handbook of
> Mathematical Functions by Abramowitz and Stegun) for a closed-form solution
> and approximations to the Bessel function and have attempted numerical
> integration but have not had any luck determining the final result. Do any
> of you have suggestions?
>
> I would greatly appreciate any feedback and assistance.
>
> Thank you again,
>
> Marek

Here are some rough estimates for integer values of p from 0 to 10:

integral_0_infty of  (x^p)*(e^-x^2)*(ln(x))*J0(2x)

1st value is for p==0, 2nd for p==1, and so on

-0.864
-0.173
-0.079
-0.066
-0.086
-0.142
-0.259
-0.484
-0.949
-1.831
-3.51

I hope this helps. After looking at the integrand, it is not a bad one
to partition (split the integral into several smaller pieces). What
values do you envision "p" to have? This looks like it can be done by
Gaussian Quadrature easily enough. I did these estimates by Monte-
Carlo.

IHTH,

Clay

```
```Hello Clay,

I checked my references a few times and do see that

I0(y) ~ (1/sqrt(2*pi*y))*exp(y) for large y.

In my integral, the integrand is actually written as

INTEGRAL(x^p*exp(-x^2)*J0(sqrt(-1)*2*x),x=0..infinity),

which can be written as

INTEGRAL(x^p*exp(-x^2)*I0(2*x),x=0..infinity),

where

In(z)=(sqrt(-1)^-n)*Jn(sqrt(-1)*z)

or

I0(z)=J0(sqrt(-1)*z).

I take it that there is no closed-form solution to this integral?

Thank you again,

Marek
```