I'm having trouble with these questions.

1. If http://www.sosmath.com/CBB/latexrend...e7b31363a1.gif is a non-negative integer, then the http://www.sosmath.com/CBB/latexrend...6fbf2064ca.gif Bessel function http://www.sosmath.com/CBB/latexrend...7e7674623f.gif is defined by

http://www.sosmath.com/CBB/latexrend...5300f281a9.gif

a) Show thathttp://www.sosmath.com/CBB/latexrend...7e7674623f.gifis an entire function ofhttp://www.sosmath.com/CBB/latexrend...b808451dd7.gif.

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 thathttp://www.sosmath.com/CBB/latexrend...0507866be8.gifsatisfies the differential equation:

http://www.sosmath.com/CBB/latexrend...2b333176dc.gif

Here, I just differentiate http://www.sosmath.com/CBB/latexrend...7e7674623f.gif then then plug it in the equation and try to work out and make it to zero? Is it possible to say...

http://www.sosmath.com/CBB/latexrend...7640d2bb63.gif

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.http://www.sosmath.com/CBB/latexrend...ec8291286a.gif,

http://www.sosmath.com/CBB/latexrend...030329ccda.gif

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

Thanks.