A Bessel related question - What is the M(u,v,phi) function?

Question

I have came across a function, written as M(u,v,phi), where it is defined as: $$ = 1/2 \pi * \int_0^\infty e^{(u \cos(\theta)} * e^{(v \cos(2(\theta + \phi)} d(\theta) $$

To my knowledge, this M(.,.,.) function presents properties related to the Bessel functions, where M(u,0,phi) = I_0(u) , and M(0,v,phi) = I_0(v). and I_0(.) is the modified Bessel function of the first kind and order zero.

My wonders are whether is there any resource I can read to get to deal with this function; What is it ? How to Integrate it ? Any more information or references would be useful.

Thank You in Advance,

Answer 1

I guess this "sort" of functions gives some generalization of modified Bessel functions.
"Sort", because I'm not sure that you copied it correctly: guess the limits of the integral should be slightly different (maybe $\int_0^{\frac{\pi}{2}}\cdot \ \mathrm d\theta$).
Anyways, I you are interested in different modifications of Bessel functions then you should resort to so table of integrals and special functions handbooks, like:

1. Gradshteyn and Ryzhik's Table of Integrals, Series, and Products.
2. Frank W. J. Olver NIST Handbook of Mathematical Functions
3. A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev Integrals & Series (V.1-3)
4. Milton Abramowitz, Irene A. Stegun Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables
5. NIST Digital Library of Mathematical Functions
6. G. N. Watson A Treatise on the Theory of Bessel Functions

And many, many more.