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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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 ...
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Calculation for negative integer order Associated Legendre Function
I am currently engaging with the following hypergeometric function as a result of attempting to find a solution for this probability problem for $n$ number of dice:
$$_2F_1\left (\frac{n+k}{2}, \frac{...
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Legendre functions at the end points
Legendre equation is
$$ (1-x^2) y'' - 2xy' + \lambda y=0$$
We are interested in finding solutions in the range $[-1,1]$. We seek solutions around the ordinary point $x=0$
$$ \sum_{n=0}^\infty c_n x^n $...
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$\sum_{n=0}^{\infty} c_{2n}$ and $\sum_{n=0}^{\infty} c_{2n+1}$ given $c_{n+2} = \frac{n(n+1)-\lambda}{(n+2)(n+1)}$
Legendre equation is
$$ (1-x^2) y'' - 2xy' + \lambda y=0$$
We are interested in finding solutions in the range $[-1,1]$. We seek solutions around the ordinary point $x=0$
$$ \sum_{n=0}^\infty c_n x^n $...
- 106
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Legendre functions of the second kind with negative integer degree
I have been recently reading about properties of Legendre functions in several sources and cannot seem to find any properties of Legendre functions of the second kind with negative integer degree. For ...
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Help with an integral involving associated Legendre functions?
In the last few days I've come across this nice integral involving the associated Legendre functions
$$
{\large\int_{-1}^{1}} \frac{P_{\ell}^m\left(u\right) P_{\lambda}^{m}\left(u\right)}{\sqrt{1 - u^...
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Integral of Squared Spherical Harmonics
The following integral comes out of an expression $\langle |Y_{l,m}(\theta, \phi)|^2\rangle$ over a orientation probability distribution:
$$\int_{0}^{2\pi} \int_{0}^{\pi} Y_{lm}^2(\theta, \phi)Y_{l'm'}...
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First derivative of the associated Legendre function at x=1
I'm trying to find the first derivative of the associated Legendre function at $x=1$.
The form I have for the first derivative is divergent at x=1:
\begin{equation}
\frac{d P_l^m(x)}{d \theta}=\frac{l ...
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Seeking the generating function of $\left( 1+\epsilon^2 - 2\epsilon x \right)^{-1}$
Many physical problems can be effectively solved using series expansions, such as the utilization of Legendre polynomials. Consider the function $\left( 1+\epsilon^2 - 2\epsilon x \right)^\frac{1}{2}$....
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Does this Fourier Legendre series inverting $\frac65\frac{x^x}{x+1}-\frac35$ diverge or it is a software issue past $\approx 25$ series terms?
$\def\P{\operatorname P}$
The goal of the Fourier Legendre series is to find an exact explicit series solution to $x^x=x+1$, not just $x=a+b+c+\dots$ with unknown series coefficients. Lagrange ...
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Asymptotic expansion of Legendre functions at $x=-1$
Let $P_{\nu}(z),\,-1<z\leq1,\,\nu\in\mathbb{C}$ Legendre functions of the first kind defined as $$P_{\nu}(z)=\,_{2}F_{1}\left(-\nu,\nu+1;1;\frac{1-z}{2}\right).$$ I found this formula on Wolfram ...
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Integration of a square of Conical (Mehler) function
I want to evaluate
$$\int_{\cos\theta}^1 \left( P_{-1/2+i\tau}(x) \right)^2 dx,$$
where $P$ is the Legendre function of the first kind, $i$ is the imaginary unit, and $\tau$ is a real number.
Are ...
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Derivation of an integral containing the complete elliptic integral of the first kind
This is a repost of mathoverflow to draw broader attentions.
https://mathoverflow.net/questions/439770/derivation-of-an-integral-containing-the-complete-elliptic-integral-of-the-first
I found the ...
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How to integrate products of Legendre functions over the interval [0,1]
The associated Legendre polynomials are known to be orthogonal in the sense that
$$
\int_{-1}^{1}P_{k}^{m}(x)P_{l}^{m}(x)dx=\frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{k,l}
$$
This is intricately linked to ...
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Proof of Bonnet's Recursion Formula for Legendre Functions of the Second Kind?
I'm doing some self-study on Legendre's Equation. I have seen and understand the proof of Bonnet's Recursion Formula for the Legendre Polynomials, $P_n(x)$.
$$(n+1)P_{n+1}(x) = (1+2n)xP_n(x) - nP_{n-...
