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Bessel Functions

  1. Feb 27, 2006 #1
    The Bessel function can be written as a generalised power series:
    [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 :biggrin:
    (And I hope the LaTex works :tongue2: )
     
  2. jcsd
  3. Feb 27, 2006 #2

    benorin

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