Associated Legendre Polynomial Identity

In summary, the conversation is about proving an identity involving the associated Legendre function and various other functions. The integral in question can be found in the paper "One- and two-dimensional lattice sums for the three-dimensional Helmholtz equation" on ScienceDirect.
  • #1
HasuChObe
31
0
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:
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  • #2
could not find it in my copy of Watson, can you give a reference where you found this integral to start??
 
  • #3

What is the Associated Legendre Polynomial Identity?

The Associated Legendre Polynomial Identity is a mathematical formula that expresses the relationship between associated Legendre polynomials and their derivatives. It is used in various fields of physics and engineering, such as quantum mechanics and electromagnetism.

How is the Associated Legendre Polynomial Identity derived?

The Associated Legendre Polynomial Identity is derived from the differential equation satisfied by associated Legendre polynomials, which is a special case of the Legendre differential equation. It can also be derived using the addition theorem for associated Legendre functions.

What is the significance of the Associated Legendre Polynomial Identity?

The Associated Legendre Polynomial Identity is significant because it allows for the simplification of complex mathematical expressions involving associated Legendre polynomials. It also allows for the calculation of integrals involving these polynomials, making it a valuable tool in various areas of science and engineering.

How is the Associated Legendre Polynomial Identity used in physics?

The Associated Legendre Polynomial Identity is used in physics to solve problems related to spherical symmetry, such as those involving electric and magnetic fields, quantum states, and angular momentum. It is also used in the study of celestial mechanics and astrophysics.

Are there any applications of the Associated Legendre Polynomial Identity outside of science and engineering?

Yes, the Associated Legendre Polynomial Identity has applications in other fields such as statistics, signal processing, and computer graphics. It is also used in music theory to analyze the harmonics of musical instruments.

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