
Bessel function
I'm having trouble with these questions.
1. If http://www.sosmath.com/CBB/latexrend...e7b31363a1.gif is a nonnegative 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 that http://www.sosmath.com/CBB/latexrend...7e7674623f.gif is an entire function of http://www.sosmath.com/CBB/latexrend...b808451dd7.gif.
An entire function is also called an integral function, is a complexvalued function that is holomorphic over the whole complex plane in complex analysis.
But how do I show it?
b) Show that http://www.sosmath.com/CBB/latexrend...0507866be8.gif satisfies 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.

a) Is...
$\displaystyle \displaystyle J_{n} (z) = (\frac{z}{2})^{n}\ \sum_{k=0}^{\infty} (1)^{k}\ \frac{(\frac{z}{2})^{2k}}{k!\ (n+k)!}$ (1)
... and if we compare the series in (1) with the series...
$\displaystyle \displaystyle e^{(\frac{z}{2})^{2}} = \sum_{k=0}^{\infty} (1)^{k} \ \frac{(\frac{z}{2})^{2k}}{k!}$ (2)
... that converges for all values of z we conclude that that holds also for the series in (1), so that both $\displaystyle J_{n} (z)$ and $\displaystyle e^{(\frac{z}{2})^{2}}$ are entire functions...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$