Bessel function

**lpd** · September 14th 2010, 10:35 AM

I'm having trouble with these questions.

1. If

is a non-negative integer, then the

Bessel function

is defined by

a) Show that

is 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 that

satisfies 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.

**chisigma** · September 14th 2010, 11:36 AM

a) Is...

$\displaystyle J_{n} (z) = (\frac{z}{2})^{n}\ \sum_{k=0}^{\infty} (-1)^{k}\ \frac{(\frac{z}{2})^{2k}}{k!\ (n+k)!}$ (1)

... and if we compare the series in (1) with the series...

$\displaystyle e^{-(\frac{z}{2})^{2}} = \sum_{k=0}^{\infty} (-1)^{k} \ \frac{(\frac{z}{2})^{2k}}{k!}$ (2)

... that converges for all values of z we conclude that that holds also for the series in (1), so that both $J_{n} (z)$ and $e^{-(\frac{z}{2})^{2}}$ are entire functions...

Kind regards

$\chi$ $\sigma$

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