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Tagged with legendre-polynomials legendre-functions
27
questions
1
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0
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33
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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 ...
1
vote
2
answers
129
views
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:
$$_2F_1\left (\frac{n+k}{2}, \frac{...
1
vote
0
answers
23
views
Integral of Squared Spherical Harmonics
The following integral comes out of an expression $\langle |Y_{l,m}(\theta, \phi)|^2\rangle$ over a orientation probability distribution:
$$\int_{0}^{2\pi} \int_{0}^{\pi} Y_{lm}^2(\theta, \phi)Y_{l'm'}...
- 19
0
votes
1
answer
97
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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 ...
- 33
2
votes
0
answers
85
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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 ...
- 469
3
votes
1
answer
414
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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, $P_n(x)$.
$$(n+1)P_{n+1}(x) = (1+2n)xP_n(x) - nP_{n-...
1
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1
answer
148
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recurrence relation associated Legendre functions
I need a little help to find the recurrence relation
$$\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))$$
I ...
- 47
1
vote
0
answers
127
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Expand square of an associated Legendre polynomial in terms of simple associated Legendre Polynomials
I have an associated Legendre Polynomial $\left(P_l^m(\cos(\theta))\right)^2$ (where $l$ and $m$ are nonnegative integers). I need to find a way to express it in terms of simple associated Legendre ...
- 167
18
votes
0
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884
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How to calculate the integral of a product of a spherical Hankel function with associated Legendre polynomials
By experimenting in Mathematica, I have found the following expression for the integral:
$$
\int_{b-a}^{b+a}\sigma h_{n}^{(1)}(\sigma)P_{n}^{m}\left(\frac{\sigma^{2}-a^{2}+b^{2}}{2b\sigma}\right)P_{n'}...
- 469
5
votes
2
answers
458
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Integral of a product of Legendre polynomials
I would like to show that
$$
\int_{-1}^{1}P_{n}^{1}(x)P_{n'}^{0}(x)\frac{x}{\sqrt{1-x^{2}}}\,\mathrm dx
=
\begin{cases}
-\frac{2n}{2n+1},&n=n'>0\\
-2,&n>n'\text{ and } n-n' \text{ even}\\...
- 469
3
votes
1
answer
496
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How to calculate the integral of a Legendre polynomial
I would like to show that
$$
\int_{0}^{1}P_{l}(1-2u^{2})e^{2i\alpha u}du=i\alpha j_{l}(\alpha)h_{l}(\alpha)
$$
where $P_{l}(x)$ are the Legendre polynomials, $\alpha$ is a positive constant and $j_{l}$...
- 469
1
vote
1
answer
212
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Series involving product of Legendre polynomials
I need to compute the following sum:
$$\sum_{n=0}^{\infty} (4n+3) P_{2n+1}(x)P_{2n+1}(y)$$
where $P_n(x)$ are the Legendre polynomials. Can anyone help me?
- 33
1
vote
0
answers
109
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Finding the Legendre Polynomial formula from the Legendre equation
First I took the Legendre equation:
$$(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$$
Where $a_{n, k}$ just gives the ...
- 510
4
votes
0
answers
94
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Series of Legendre Polynomials and Harmonic numbers. $\sum_{n=1}^{\infty} P_n (z) \frac{H(n)}{n+k}$
I would like to compute sums of the type
\begin{equation}
\sum_{n=1}^{\infty} P_n (z) \frac{H(n)}{n+k}
\end{equation}
where $P_n(z)$ are Legendre polynomials, $H(n)$ are harmonic numbers and $k = 0, 1,...
2
votes
1
answer
151
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Prove $1+\sum_{k=1}^{p} \frac{(-1)^k.n(n-1)(n-2)\cdots(n-2k+1)}{2^k.k!.(2n-1)(2n-3)\cdots(2n-2k+1)}=\frac{2^n(n!)^2}{(2n)!}$
Prove that $$
a_n\bigg[1-\frac{n(n-1)}{2(2n-1)}+\frac{n(n-1)(n-2)(n-3)}{2\cdot4\cdot(2n-1)(2n-3)}-\cdots+\frac{n(n-1)(n-2)\cdots(n-2k+1)}{2\cdot4\cdots 2k\cdot(2n-1)(2n-3)\cdots(2n-2k+1)}\bigg]=1\\
\...
- 7,674