# Derivation and integral of Bessel's function

Let $$f(x)$$ function define by

$$f(x)=x^m e^{-bx}K_{n+1}(ax)^{'}$$ where $$K_v(⋅)$$ is the $$v$$-th order modified Bessel function of the second kind and $$L^{'}(x)$$ is the derivation of function $$L(x)$$.

I would like to compute the following integral integral

$$I=\int_{0}^{\infty}f(x)dx=\int_{0}^{\infty}x^m e^{-bx} K_{n+1}(ax)^{'}dx.$$

So my question is what is the derivation of $$K_{n+1}(ax)$$.

I found the following formula to evaluate the integral $$\int_{0}^{\infty}x^{\mu-1}e^{-\alpha x}K_{v}(bx)dx=$$ $$\frac{\sqrt\pi (2b)^v}{(\alpha+b)^{\mu+v}} \frac{\Gamma(\mu+v)\Gamma(\mu-v)}{\Gamma(\mu+\frac{1}{2})} F\left( \mu+v,v+\frac{1}{2},\mu+\frac{1}{2};\frac{\alpha-b}{\alpha+b} \right)$$ where $$F(a,b,c;z)$$ is is the generalized hypergeometric function.

So what is the final result of Integral $$I$$.

• There is some recursion for the derivatives of the $K_n$ though I have forgotten it. Check mathworld or Abramowitz and Stegun. – Ian Jan 23 at 17:54
• I would like just the exact expression of $K_{n+1}(ax)^{'}$, just the derivation. – Monir Jan 23 at 17:57