# 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.

65 questions
Filter by
Sorted by
Tagged with
1 vote
33 views

### Derivation of the associated Legendre Polynomials

I have been struggling to find a proper derivation of the associated Legendre Polynomials and a derivation of $$P_l^{-m}(\mu)=(-1)^m\frac{(l-m)!}{(l+m)!}P_l^m(\mu)$$ Can someone point to a proper ...
1 vote
129 views

1 vote
222 views

• 7,674
1 vote
62 views

### Deriving normalization for the shifted associated Legendre function

Where can I find a solution for this integral: $\int_{a}^{b} P^m_l(c_1x + c_2b)P^{m'}_l(c_1x + c_2b)\,d(c_1x + c_2b)$, most solutions only solves for the interval [-1,1]. Of course I am looking for ...
67 views

### How to find the associated Legendre functions

Reading the Courant-Hilbert Methods of Mathematical Physics (p.326) we encounter: "...If we differentiate equation $$\left[(1-x^2)u')\right]'+\lambda u=0$$ with respect to $x$, we obtain a ...
• 63.1k
66 views

• 171
207 views

### Showing a summation identity for $1$, possibly tied to Legendre polynomials

The Problem: Consider the sign function on $(-1,0)\cup(0,1)$ defined by  \sigma(x) := \left. \text{sgn}(x) \right|_{(-1,0)\cup(0,1)} = \begin{cases} 1 & x \in (0,1) \\ -1 & x \in (-1,0) \end{...
• 45.9k
92 views

### Does Associated Legendre Function of Second Kind Give Delta Function?

Associated Legendre Function of Second Kind is singular at $x=\pm 1$. So I am wondering whether it satisfies the corresponding differential equation everywhere or there is a hidden functional of delta ...
• 21
How do I express $cos(3\theta)$ and $sin^{2}(\theta)$ in Legendre's polynomials, knowing that $x=cos\theta$? I know that $f(x)=\sum a_{n}P_{n}(x)$ and \$P_{n}=\frac{(-1)^{n}}{2^{n}n!}\frac{d^{n}}{dx^{n}...