All Questions
Tagged with legendre-polynomials sequences-and-series
55
questions
5
votes
2
answers
105
views
Simplify in closed-form $\sum_n P_n(0)^2 r^n P_n(\cos \theta)$
Simplify in a closed form the sum $$S(r,\theta)=\sum_{n=0}^{\infty} P_n(0)^2 r^n P_n(\cos \theta)$$ where $P_n(x)$ is the Legendre polynomial and $0<r<1$. Note that $P_n(0)= 0$ for odd $n$ and $...
- 1,258
0
votes
1
answer
53
views
Morse and Fesbach identity $\sum_{n=0}^{\infty} e^{-(n+\frac{1}{2})|u|} \; P_n(x) =(2\cosh u - 2x)^{-1/2}$
In book called Methods of theoretical physics from Morse and Feshbach, there is identity, which I wanted to prove ($P_n(x)$ are Legendre polynomials):
$$\sum_{n=0}^{\infty} e^{-(n+\frac{1}{2})|u|} \; ...
0
votes
0
answers
69
views
Closed Forms for Sums of Legendre Polynomials
I am investigating a series of the form $$\sum_{n=0}^\infty \frac{1}{1 + e^{nx}}P_n(x)$$ where $P_n$ is the Legendre Polynomial of degree $n$.
Because I am dealing with many of these series, it would ...
- 93
5
votes
1
answer
217
views
How to express a Gaussian as a series of exponential? $\displaystyle e^{-x^2}=\sum_{n=1}^{\infty}c_n e^{-nx}$
Context
I would like to express the Gaussian function as a series of exponentials:
$$e^{-x^2}=\sum_{n=1}^{\infty}c_ne^{-n|x|}\qquad\forall x\in\mathbb{R}$$
For simplicity (the absolute value is added ...
0
votes
0
answers
26
views
I want to prove the proposition that the absolute value of integral expression must be monotonically decreasing
For arbitrary $r_0$ and $P_l(\text{cos}(\theta))$ be the Legendre polynomials,
$$
E_n=\int_{r0}^\infty \int_0^\pi -\text{sin}^3(\theta)(\left(
\left(\text{cot}(\theta) \sum_{l=0}^n R_l(r) \partial_\...
0
votes
1
answer
37
views
Proof of an identity of a Gaussian series over Legendre polynomials
The Legendre polynomials are a total basis (if normalized) of the real space $L^2[-1,1]$.
Let $w=w(t)=e^{-\frac{t^2}{2}}$, and $\frac{1}{2^nn!}\frac{d^n}{dt^n}[(t^2-1)^n]=P_n(t)\in L^2[-1,1]$.
$$\frac{...
1
vote
1
answer
112
views
Showing that a series that involves Legendre polynomial converges.
I am currently studying Legendre polynomials and I am trying to prove its generating function specifically using the recurrence relation
$$ (n+1)L_{n+1}(x) = x(2n+1)L_n(x) - nL_{n-1}(x), \quad \forall ...
- 1,141
2
votes
1
answer
89
views
Can order of summation and integral be interchanged in : $\int_{-1}^1 ( \sum_{n=0}^\infty P_n(\xi)P_n(\xi^\prime)Q_l(\xi^\prime))\text{d}\xi^\prime$?
I am wanting to know if there is a proof that the order of summation and integration can be interchanged in $$\int_{-1}^1 \left( \sum_{n=0}^\infty P_n(\xi)P_n(\xi^\prime)Q_l(\xi^\prime)\right)\text{d}\...
1
vote
0
answers
69
views
Series representation of the associated Legendre polynomial
I have found the following identity for the associated Legendre polynomial to be true:
$$
P_{n}^{m}(\tau)=\frac{n!(n+m)!}{2^n}\sum_{s=0}^{n-m}\frac{(-1)^{m+s}(1+\tau)^{n-m/2-s}(1-\tau)^{m/2+s}}{(n-s)!...
- 469
1
vote
1
answer
73
views
Simplification of an infinite sum consisting of Legendre polynomials
In an article about Legendre Polynomials, I encountered the following simplification.
\begin{align}
(something)\dots&=\int_{-1}^{1} \int_{-1}^{1} \left[\sum_{i=n+1}^{\infty} \sum_{j=n+1}^{\infty} ...
- 45
1
vote
0
answers
58
views
Expansion of divergent function with Legendre polynomials
Polynomials on $x\in[-1,1]$ can be written as an expansion in the Legendre polynomials $P_l(x)$. Is it possible to expand more general types of function on this interval in terms of these polynomials. ...
- 53
4
votes
1
answer
171
views
Proving $\frac{\pi}{2}=\sum^\infty_{l=0} \frac{(-1)^l}{2l+1}\big(P_{2l}(x)+\text{sgn}(x)P_{2l+1}(x)\big)$
Can someone help me in proving the following:
$$
\frac{\pi}{2}=\sum^\infty_{l=0} \frac{(-1)^l}{2l+1}(P_{2l}(x)+\text{sgn}(x)\cdot P_{2l+1}(x)),
$$
for any value of $x$, $-1\le x\le 1$? (Here $P_l(x)$ ...
5
votes
2
answers
246
views
Closed form expression for series involving Legendre polynomials
Given $-1 \leq x \leq 1$ and $0 \leq \eta \leq 1$,
I am interested in computing
$$
E(x,\eta) = \sum_{\ell = 0}^{+ \infty} |P_{\ell} (0)|^{2} \, P_{\ell} (x) \, \eta^{\ell} ,
$$
with $P_{\ell}$ the ...
- 892
0
votes
0
answers
31
views
Which of the following series converge? (With Legendre-Polynomials)
These should be quick tasks:
We know that the Legendre Polynomials satisfy $\int_{-1}^{1} P_m(x)P_{n}(x)dx= \delta_{mn}\frac{2}{2n+1}$
Which of the following series converge ( $ \forall x \in [-1,1]$ )...
- 825
0
votes
1
answer
58
views
Exact value of an infinite sum expressed in terms of a product of definite integrals involving Legendre polynomials.
In a fluid mechanics problem, one has to deal with the following infinite sum:
$$
S = \sum_{n \ge 1} \frac{2n+1}{n+1}
\left( \int_0^1 P_n(x) \, \mathrm{d}x \right)
\left( \int_0^1 x \left( 1-x^2\...
