Evaluate Integral of Bessel K Function

[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:

∫x^λ * cos(2ax) * [K_v(x)]² dx

where K_v(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.

 

[dextercioby]

Have you tried plugging it into Mathematica ?

 

[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