Thank you for the hint with performing the numerical integration on the
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
Thank you again for the advice and supporting documents,
Reply by mbtrawicki●July 28, 20092009-07-28
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):
(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
even if Maple and/or Mathematica do not provide one.
Thank you again,
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
>>  Watson,G.N., "A Treatise on the Theory of Bessel Functions", 1922
>> Cambridge University Press.
Rune> A genuine CUP edition, not a Dover reprint?
It was reprinted (not Dover) recently. I have a copy.