- #1
HasuChObe
- 31
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Does anyone know how to prove this identity? I don't quote understand why the associated Legendre function is allowed to have arguments where |x|>1.
<br /> 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<br />
where
<br /> \gamma(t)=\begin{cases}<br /> \sqrt{t^2-1} & |t|\ge 1 \\<br /> -i\sqrt{1-t^2} & |t|<1<br /> \end{cases} \\<br /> \rho=\sqrt{x^2+y^2} \\<br /> r=\sqrt{\rho^2+z^2} \\<br /> \cos\theta=\frac{z}{r}<br />
and k is the wavenumber, h_n(x) is the spherical hankel function of the first kind, and K_m(x) is the modified cylindrical bessel function of the second kind.
<br /> 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<br />
where
<br /> \gamma(t)=\begin{cases}<br /> \sqrt{t^2-1} & |t|\ge 1 \\<br /> -i\sqrt{1-t^2} & |t|<1<br /> \end{cases} \\<br /> \rho=\sqrt{x^2+y^2} \\<br /> r=\sqrt{\rho^2+z^2} \\<br /> \cos\theta=\frac{z}{r}<br />
and k is the wavenumber, h_n(x) is the spherical hankel function of the first kind, and K_m(x) is the modified cylindrical bessel function of the second kind.
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