Solve Integral Homework Equation w/ Defined Fns

[EngWiPy]

Homework Statement

I need to find the follwoing integral in closed form using defined functions.

Homework Equations

$$\int_0^{\infty}\gamma^{a}(\gamma^2+\gamma)^{\frac{n+1}{2}}\,\mbox{exp}[-\gamma(s+\zeta)]\,K_{n+1}\left(2\,\sqrt{\frac{(\gamma^2+\gamma)(j+1)}{\rho}}\right)\,d\gamma$$

where $$K_v(.)$$ is the modified Bessel function of the second kind and $$v^{th}$$ order.

The Attempt at a Solution

I didn't find a similar expression in the table of integrals.

 

[mighty2000]

However, I suggest using integration by parts to simplify the integral and possibly express it in terms of known functions. First, let's rewrite the integral using the definition of the modified Bessel function:

\int_0^{\infty}\gamma^{a}(\gamma^2+\gamma)^{\frac{n+1}{2}}\,\mbox{exp}[-\gamma(s+\zeta)]\,\frac{1}{2}\left(\frac{2\,\sqrt{\frac{(\gamma^2+\gamma)(j+1)}{\rho}}}{\gamma}\right)^{n+1}\,d\gamma

Now, we can use integration by parts with u = \gamma^a(\gamma^2+\gamma)^{\frac{n+1}{2}} and dv = \mbox{exp}[-\gamma(s+\zeta)]\,\frac{1}{2}\left(\frac{2\,\sqrt{\frac{(\gamma^2+\gamma)(j+1)}{\rho}}}{\gamma}\right)^{n+1}\,d\gamma. This will simplify the integral and possibly lead to an expression in terms of known functions. Good luck!