How to Solve an Integral Involving Modified Bessel Functions?

[phy07]

Homework Statement

This is my homework. I couldn't solve the integral.

Homework Equations

K_v(z) being a modified Bessel function.
β=1/kT
k: Boltzmann Constant

The Attempt at a Solution

I made p=m c sinh θ transformation and obtained an integral form as follows

∫exp(-mc²coshθ/kT)(cosh²θ-1)coshθ dθ

but i couldn't forward more.

 

[dextercioby]

You should use the integral representation of the modified Bessel function K as given by

$$K_{\nu} (z) = \int\limits_{0}^{\infty} e^{-z\cosh t}\cosh \nu t {}{} ~ dt$$

 

[phy07]

thanks for your help.now i found a result likes the modified Bessel function.
$$\int\limits_{0}^{\infty} e^{-a\cosh θ}\cosh \3 θ {}{} ~ dθ -\int\limits_{0}^{\infty} e^{-a\cosh θ}\cosh \ θ {}{} ~ dθ$$
a:constant
but i don't know how to resolve the bessel function.can you show a way?

 

[dextercioby]

What do you mean by 'resolve the Bessel function' ?

 

[phy07]

ok sorry.i must write the equation in terms of bessel function.thanks for all your helps... :)