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Associated Legendre Polynomial Identity

  1. Jul 15, 2014 #1
    Does anyone know how to prove this identity? I don't quote understand why the associated Legendre function is allowed to have arguments where [itex]|x|>1[/itex].
    [tex]
    h_n(kr)P_n^m(\cos\theta)=\frac{(-i)^{n+1}}{\pi}\int_{-\infty}^\infty e^{ikzt}K_m(k\rho\gamma(t))P_n^m(t)\,dt
    [/tex]

    where
    [tex]
    \gamma(t)=\begin{cases}
    \sqrt{t^2-1} & |t|\ge 1 \\
    -i\sqrt{1-t^2} & |t|<1
    \end{cases} \\
    \rho=\sqrt{x^2+y^2} \\
    r=\sqrt{\rho^2+z^2} \\
    \cos\theta=\frac{z}{r}
    [/tex]

    and [itex]k[/itex] is the wavenumber, [itex]h_n(x)[/itex] is the spherical hankel function of the first kind, and [itex]K_m(x)[/itex] is the modified cylindrical bessel function of the second kind.
     
    Last edited: Jul 15, 2014
  2. jcsd
  3. Jul 15, 2014 #2

    Dr Transport

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    Science Advisor
    Gold Member

    could not find it in my copy of Watson, can you give a reference where you found this integral to start??
     
  4. Jul 16, 2014 #3
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