# How to calculate integral involving Bessel function?

$$\int_0^{\infty}y^{1+\frac{m+c}{2}}K_{c-m}(2b\sqrt{y})dy,$$

How to calculate this integral? Note that, $K_{c-m}$ is the modified Bessel function of the second kind.

• May we assume $b$ is positive? – David H Jan 16 '15 at 20:53
• Yes, you can assume that. – Kira Jan 16 '15 at 22:04

Let $t = 2b\sqrt{y}$, then $\frac{t}{2b^2} dt = dy$, so $$\int_0^\infty \left(\frac{t}{2b}\right)^{2+m+c} K_{c-m}(t) \frac{t}{2b^2} dt$$ $$\frac{(2b)^{-3-m-c}}{b} \int_0^\infty t^{3+m+c} K_{c-m}(t) dt$$ Now applying the integral identity here, $$\frac{(2b)^{-3-m-c}}{b} 2^{2+m+c}\Gamma(m+2)\Gamma(c+2)$$ $$\frac{b^{-4-m-c}}{2} \Gamma(m+2)\Gamma(c+2)$$