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
Approximation to Bessel Function in Integrand
Started by ●July 27, 2009
Reply by ●July 27, 20092009-07-27
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, > > MarekHello 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.
Reply by ●July 27, 20092009-07-27
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
Reply by ●July 27, 20092009-07-27
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. �����������������������������������������������������������������������
Reply by ●July 27, 20092009-07-27
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... > > RuneThe 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
Reply by ●July 27, 20092009-07-27
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
Reply by ●July 27, 20092009-07-27
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
Reply by ●July 27, 20092009-07-27
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/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, > > MarekAre you confusing J0(x) with I0(x)?
Reply by ●July 27, 20092009-07-27
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, > > MarekHere 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
Reply by ●July 27, 20092009-07-27
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
