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Approximation to Bessel Function in Integrand

Started by mbtrawicki 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,

I really appreciate your response about the closed-form solution. Here is
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. 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[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