The Bessel function can be written as a generalised power series:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

J_m(x) = \sum_{n=0}^\infty \frac{(-1)^n}{ \Gamma(n+1) \Gamma(n+m+1)} ( \frac{x}{2})^{2n+m}

[/tex]

Using this show that:

[tex]

\sqrt{\frac{ \pi x}{2}} J_{1/2}(x)=\sin{x} [/tex]

where

[tex]

\Gamma(p)=\int_{0}^{\infty} x^{p-1}e^{-x}dx

[/tex]

and therefore

[tex] \Gamma(p+1)=p\Gamma(p) [/tex]

Wea re also given that:

[tex] \Gamma(3/2)=\frac{\sqrt{\pi}}{2} [/tex]

My answer so far goes something like:

We are obviously trying to get the series expansion for the Bessel function into the form of the Taylor series for sin:

[tex] \sin{x} = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} [/tex]

Simplyfing the expression for J by replacing m by 1/2 and Gamma(n+1) by n! I ca get reasonably close to the Taylor series but I'm having trouble getting rid of the Gamma(n+3/2)

Any help would be much appreciated!

Also as this is my first post: Hello Everybody

(And I hope the LaTex works :tongue2: )

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# Bessel Functions

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