I'm having trouble with these questions.

1. If is a non-negative integer, then the Bessel function is defined by

a) Show thatis an entire function of.

An entire function is also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane in complex analysis.

But how do I show it?

b) Show thatsatisfies the differential equation:

Here, I just differentiate then then plug it in the equation and try to work out and make it to zero? Is it possible to say...

and then find the derivates from there? Is there anything I show look out for when do this? Because before I tried it, and I just got a huge fraction that doesn't simplify. It is way too long to type it in here.

c) Show that.,

I'm having the most trouble with this question. Don't know where to start basically!!

Thanks.