# Questions tagged [legendre-functions]

This tag is for questions relating to Legendre Functions (or Legendre Polynomials), solutions of Legendre's differential equation (generalized or not) with non-integer parameters.

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### Should Hobson, pg183, be corrected, in particular, should an occurrence of $(t^2−1)^n(t−\mu)^{-n-m-1}$ be replaced by $(t^2−1)^{n+1}(t−\mu)^{-n-m-2}$?

The material in this question concerns substitution of a Schlaefli type integral into a differential equation. My answer to the question Showing Schlaefli integral satisfies Legendre equation should ...
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### recurrence relation associated Legendre functions

I need a little help to find the recurrence relation $$\sqrt{1-x^2}P_l^m(x) = \frac{1}{2l+1} (P_{l-1}^{m+1}-p_{l+1}^{m+1})$$ Using the identity $$(2l+1)P_l(x) = \frac{d}{dx}(P_{l+1}(x)-P_{l-1}(x))$$ I ...
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### Stuck on nasty integral regarding associated Legendre polynomials and spherical Bessel functions.

I'm preparing notes for an undergrad physics course I'm going to be teaching soon. Unfortunately, this sort of stuff was taught to me only in a very handwavy sort of way ("you take these physical ...
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### Solve $\sum_\limits{n=-\infty}^0\mathrm P_n^n(z)$ with Associated Legendre P functions of type 1

Here is a simple looking sum which should have an alternate form since it is just a double hypergeometric series with the associated Legendre P function of type 1 $\mathrm P_a^b(z)$ The definitions ...
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### How to compute this integral involving associated Legendre function?

My goal is to compute the following integral, $$\int_{-1}^{1}P^\mu_\nu(x)(1-x^2)^{\mu/2+1/2}dx.$$ I tried to compute this using Wolfram Alpha and Maple but unfortunately did not get any result. ...
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### How to find this limit involving Legendre functions?

I am working with an ODE whose general solution is of the form, $$f(\theta) = \sin(\theta)^{-k}\left[c_1 P_\nu^\mu (\cos(\theta)) + c_2 Q_\nu^\mu(\cos(\theta))\right]$$ where $\mu,\nu\in \mathbb{R}$ (...
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### Intuition behind Legendre convex function

I came across the definition of Legendre functions and Legendre transformations in my studies (in the sense of convex analysis) and I started searching about it. I found a definition in Rockefellar's ...
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### Expand square of an associated Legendre polynomial in terms of simple associated Legendre Polynomials

I have an associated Legendre Polynomial $\left(P_l^m(\cos(\theta))\right)^2$ (where $l$ and $m$ are nonnegative integers). I need to find a way to express it in terms of simple associated Legendre ...
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### How to calculate the integral of a Legendre polynomial

I would like to show that $$\int_{0}^{1}P_{l}(1-2u^{2})e^{2i\alpha u}du=i\alpha j_{l}(\alpha)h_{l}(\alpha)$$ where $P_{l}(x)$ are the Legendre polynomials, $\alpha$ is a positive constant and $j_{l}$...
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### Series involving product of Legendre polynomials

I need to compute the following sum: $$\sum_{n=0}^{\infty} (4n+3) P_{2n+1}(x)P_{2n+1}(y)$$ where $P_n(x)$ are the Legendre polynomials. Can anyone help me?
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First I took the Legendre equation: $$(1-x^2)\frac{d^2P_n(x)}{dx^2}-2x\frac{dP_n(x)}{dx}+n(n+1)P_n(x)=0$$ Then I wrote: $$P_n(x)=\sum_{k=0}^{n}a_{n, k} x^k$$ Where $a_{n, k}$ just gives the ...