I have been given a modified Bessel function and it it defined by the integral

I have been asked to show that

(a)

(b)

Firstly i have started by trying to integrate the integral. Using and . Therefore and

Using integration by parts

I ended up with

knowing that

Therefore

I'm not sure if i'm on the right track to solving this problem. Any ideas of what i can do next? How do i then show that

(a)

(b)