# Evaluate Integral of Bessel K Function

1. Jul 9, 2013

### kschau

Hey All

Got a tough one and I'm just not seeing the path here. I need to find the close form expression of:

The integral from zero to infinity:

xλ * cos(2ax) * [Kv(x)]2 dx

where Kv(x) is the modified Bessel function of the second kind of order v and argument x. If it helps, the value of v=1/3 and the value of λ=2/3

The result will have the form of a hypergeometric function 2F1

I've just been racking my brain for too long with this one. If anyone has some experience with Bessel functions, any help would be appreciated.

2. Jul 10, 2013

### dextercioby

Have you tried plugging it into Mathematica ?

3. Jul 10, 2013

### Bill Simpson

In[1]:= Integrate[x^(2/3)*Cos[2 a x]*BesselK[1/3, x]^2, {x, 0, Infinity}]

Out[1]= ConditionalExpression[(Pi^2*Hypergeometric2F1[5/6, 7/6, 4/3, -a^2])/(4*Gamma[1/3]), Abs[Im[a]] <= 1]

so if the absolute value of the imaginary component of a is less than or equal to 1 then you have your hypergeometric as expected.

Verify this independently several different ways before you depend on it.

http://reference.wolfram.com/mathematica/ref/Hypergeometric2F1.html
http://reference.wolfram.com/mathematica/ref/Gamma.html
http://reference.wolfram.com/mathematica/ref/ConditionalExpression.html

Last edited: Jul 10, 2013