FPGA Central

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 Besse
function of the first kind.

I have already looked through the traditional resources (Table o
Integrals, Series, and Products by Gradshteyn and Ryzhik and Handbook o
Mathematical Functions by Abramowitz and Stegun) for a closed-form solutio
and approximations to the Bessel function and have attempted numerica
integration but have not had any luck determining the final result. Do an
of you have suggestions?

I would greatly appreciate any feedback and assistance.

Thank you again,

Marek

On Jul 27, 10:00*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.
>
> I have already looked through the traditional resources (Table of
> 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.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

On Jul 27, 11:22*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*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
it gets amended when you read the books they have made
comments in.

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 reall
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/treat...eory00watsuoft

I still am sifting through the entire document now.

I would prefer an analytic answer to the integral, but I am more tha
happy with a numerical value too. Does anyone have a closed-form solutio
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 approximatio
into the integral, I still am not able to obtain a closed-form solution o
it. I actually have not been able to successfully apply any numerica
analysis techniques to it either. I am not sure how to do it.

Thank you again,

Marek

On Jul 27, 1:37*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/treat...eory00watsuoft
>
> 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*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.
>
> I have already looked through the traditional resources (Table of
> 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

Hello Clay,

I verified your values for the integrand using Maple for p of 0 to 10.
envision p to be defined as p>-1. Is there a fast way to implemen
numerical integration in Matlab? I am using the Maple commands in Matlab
but my implementation runs so much slower that way than simply typing i
the integral for various p values directly into Maple. Unfortunately,
have to use Matlab and its vector capabilities. Here is my implementation:

p=2;
ftn=['x^' num2str(pp) '*ln(x)*exp(-x^2)*BesselI(0,2*x)'];
[r,s]=maple(['evalf(Int(' ftn ',x=0..infinity))']);
int_val=sym(r);

I have not had any success with the command "quad" in Matlab, and so
resorted to using the evalf(Int(*)) in Maple but with Matlab. An
suggestions on speeding up the implementation?

Thank you again,

Marek

On Jul 27, 2:48*pm, Clay <c...@claysturner.com> wrote:
> On Jul 27, 10:00*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.
>
> > I have already looked through the traditional resources (Table of
> > 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- Hide quoted text -
>
> - Show quoted text -

With the integrang changed to reflect the modified Bessel function, I
find these rough values for p==0 up to 10

-0.764
0.155
0.622
1.41
3.007
6.848
15.919
39.671
100.676
275.312
760.466

Does this help?

Clay

On Jul 27, 3:23*pm, Clay <c...@claysturner.com> wrote:
> On Jul 27, 2:48*pm, Clay <c...@claysturner.com> wrote:
>
>
>
>
>
> > On Jul 27, 10:00*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.
>
> > > I have already looked through the traditional resources (Table of
> > > Integrals, Series, and Products by Gradshteyn and Ryzhik and Handbookof
> > > 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- Hide quoted text -
>
> > - Show quoted text -
>
> With the integrang changed to reflect the modified Bessel function, I
> find these rough values for p==0 up to 10
>
> -0.764
> 0.155
> 0.622
> 1.41
> 3.007
> 6.848
> 15.919
> 39.671
> 100.676
> 275.312
> 760.466
>
> Does this help?
>
> Clay- Hide quoted text -
>
> - Show quoted text -

Disreragard the earlier values.


Now for the integrad I have

(x^p)(e^-(x*x))*I0(2x)

This yields for the p==0 up to p==10

1.558
1.344
1.739
2.682
4.86
9.573
20.183
46.424
112.976
280.073
756.176

Clay

Hello Clay,

I verified your results and have very similar numbers. I am using Maple t
compute those integrals. Is there no closed-form solution? I still have no
found one in any of my references for the integral. Do you know of a faste
implementation of Maple in Matlab? I am using Maple functions in Matla
since the integration is a major component of a bigger program, but m
Matlab is running so slowly this way. I am not sure whether there is an
way around it. I welcome all suggestions.

Thank you again,

Marek

"mbtrawicki" <[email protected]> writes:

> 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?

It depends what you accept for a closed-form. An old version of
Mathematica gives

1 + p 1 + p
Gamma[-----] Hypergeometric1F1[-----, 1, 1]
2 2
-------------------------------------------
2

as long as the real part of p is greater than -1.

Scott
--
Scott Hemphill [email protected]
"This isn't flying. This is falling, with style." -- Buzz Lightyear

On Jul 27, 3:34*pm, "mbtrawicki" <mbtrawi...@yahoo.com> wrote:
> Hello Clay,
>
> I verified your results and have very similar numbers. I am using Maple to
> compute those integrals. Is there no closed-form solution? I still have not
> found one in any of my references for the integral. Do you know of a faster
> implementation of Maple in Matlab? I am using Maple functions in Matlab
> since the integration is a major component of a bigger program, but my
> Matlab is running so slowly this way. I am not sure whether there is any
> way around it. I welcome all suggestions.
>
> Thank you again,
>
> Marek

Hello Marek,

An obvious closed form solution does not pop into my head. I'm using
MathCad but not using Maple.

A chunk of 'c' code to generate the Gaussian Quad coefs is not too
hard to come up with. I.e., the integrand can be sampled at about 100
to 200 points, and from that you would get a very precise result and
should execute in fractions of a second. If -1 < p < 20, then you
will only need to integrate from 0 to 8. The product I0(2x)*e^-(x*x)
acts alot like e^-(x/1.7)^3, so unless p is big, the exponential term
crushes x^p to 0.

Look up how to calculate the Gaussian Quadrature coefs and code that
up. The integral becomes trivial to do from there.

IHTH,
Clay

Scott Hemphill <[email protected]> writes:

> "mbtrawicki" <[email protected]> writes:
>
>> 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?
>
> It depends what you accept for a closed-form. An old version of
> Mathematica gives
>
> 1 + p 1 + p
> Gamma[-----] Hypergeometric1F1[-----, 1, 1]
> 2 2
> -------------------------------------------
> 2
>
> as long as the real part of p is greater than -1.

However, this integrand doesn't contain the factor "ln(x)", which is
what you were really looking for.

Scott
--
Scott Hemphill [email protected]
"This isn't flying. This is falling, with style." -- Buzz Lightyear

Clay <[email protected]> writes:

> On Jul 27, 2:48Â*pm, Clay <c...@claysturner.com> wrote:
>> On Jul 27, 10:00Â*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.
>>
>> > I have already looked through the traditional resources (Table of
>> > 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 Â*(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- Hide quoted text -
>>
>> - Show quoted text -
>
> With the integrang changed to reflect the modified Bessel function, I
> find these rough values for p==0 up to 10
>
> -0.764
> 0.155
> 0.622
> 1.41
> 3.007
> 6.848
> 15.919
> 39.671
> 100.676
> 275.312
> 760.466

Mathematica's numbers are:

-0.76932709667515764264
0.14908684058079851859
0.63074368868670269879
1.4073145953911196549
3.0499225104518270595
6.8393124552132027185
16.055428433081164515
39.484039092403305545
101.50192031004547648
271.98733252729141392
757.55019687159582342

Scott
--
Scott Hemphill [email protected]
"This isn't flying. This is falling, with style." -- Buzz Lightyear

Clay <[email protected]> writes:

> On Jul 27, 3:23Â*pm, Clay <c...@claysturner.com> wrote:
>> On Jul 27, 2:48Â*pm, Clay <c...@claysturner.com> wrote:
>>
>>
>>
>>
>>
>> > On Jul 27, 10:00Â*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.
>>
>> > > I have already looked through the traditional resources (Table of
>> > > 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 Â*(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- Hide quoted text -
>>
>> > - Show quoted text -
>>
>> With the integrang changed to reflect the modified Bessel function, I
>> find these rough values for p==0 up to 10
>>
>> -0.764
>> 0.155
>> 0.622
>> 1.41
>> 3.007
>> 6.848
>> 15.919
>> 39.671
>> 100.676
>> 275.312
>> 760.466
>>
>> Does this help?
>>
>> Clay- Hide quoted text -
>>
>> - Show quoted text -
>
> Disreragard the earlier values.
>
>
> Now for the integrad I have
>
> (x^p)(e^-(x*x))*I0(2x)
>
> This yields for the p==0 up to p==10
>
> 1.558
> 1.344
> 1.739
> 2.682
> 4.86
> 9.573
> 20.183
> 46.424
> 112.976
> 280.073
> 756.176
>
> Clay

And Mathematica's numbers for these are:

1.5538993500654319234
1.3591409142295226177
1.7423093452556455141
2.7182818284590452354
4.8384531982505785615
9.5139863996066583238
20.272069964427690401
46.210791083803769001
111.66415726192771680
284.06045107397022710
756.64455829311024375

Scott
--
Scott Hemphill [email protected]
"This isn't flying. This is falling, with style." -- Buzz Lightyear

On Jul 27, 4:40*pm, Clay <c...@claysturner.com> wrote:
> On Jul 27, 3:34*pm, "mbtrawicki" <mbtrawi...@yahoo.com> wrote:
>
> > Hello Clay,
>
> > I verified your results and have very similar numbers. I am using Mapleto
> > compute those integrals. Is there no closed-form solution? I still havenot
> > found one in any of my references for the integral. Do you know of a faster
> > implementation of Maple in Matlab? I am using Maple functions in Matlab
> > since the integration is a major component of a bigger program, but my
> > Matlab is running so slowly this way. I am not sure whether there is any
> > way around it. I welcome all suggestions.
>
> > Thank you again,
>
> > Marek
>
> Hello Marek,
>
> An obvious closed form solution does not pop into my head. I'm using
> MathCad but not using Maple.
>
> A chunk of 'c' code to generate the Gaussian Quad coefs is not too
> hard to come up with. I.e., the integrand can be sampled at about 100
> to 200 points, and from that you would get a very precise result and
> should execute in fractions of a second. If *-1 < p < 20, then you
> will only need to integrate from 0 to 8. The product I0(2x)*e^-(x*x)
> acts alot like *e^-(x/1.7)^3, so unless p is big, the exponential term
> crushes x^p to 0.
>
> Look up how to calculate the Gaussian Quadrature coefs and code that
> up. The integral becomes trivial to do from there.
>
> IHTH,
> Clay

See here for how to quickly integrate this function. There is a graph
of the integral for p = -1 up to 5, just so you can see how it looks.
This MathCad sheet ran in under a second with 1000 values for p. It
uses Gaussian Quadrature.

http://www.claysturner.com/dsp/x.pdf

IHTH,

Clay

Hello Clay,

Thank you for the hint with performing the numerical integration on th
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 function
("quad" and variants), but I do not think that they have Gaussia
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 simplif
maybe parts of the integrand (not necessarily using the Bessel functio
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 whethe
somehow substituting in another simplification for any combination o
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 hav
even tried integration by parts but ran into trouble with the ln(x) in u*
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 d
not see really any way to find a closed-form solution but am still workin
on it.

Thank you again for the advice and supporting documents,

Marek

Hello Scott,

I really appreciate your response about the closed-form solution. Here i
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], 1) p

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

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

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

I am sure that my result simplifies even further into your result too. Ar
there maybe any mathematical papers that possibly deal with integral
involving ln(x)? I just really believe that there has to be a closed-for
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 <[email protected]> writes:

Rune> 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.

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

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

Ray