Questions tagged [legendre-polynomials]
For questions about Legendre polynomials, which are solutions to a particular differential equation that frequently arises in physics.
200
questions with no upvoted or accepted answers
18
votes
0
answers
884
views
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:
$$
\int_{b-a}^{b+a}\sigma h_{n}^{(1)}(\sigma)P_{n}^{m}\left(\frac{\sigma^{2}-a^{2}+b^{2}}{2b\sigma}\right)P_{n'}...
- 469
10
votes
0
answers
259
views
Evaluate $\int_{0}^{1} \operatorname{Li}_3\left [ \left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ] \text{d}x$
Possibly evaluate the integral?
$$
\int_{0}^{1} \operatorname{Li}_3\left [
\left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ]
\text{d}x.
$$
I came across this when playing with Legendre polynomials, ...
- 5,370
9
votes
0
answers
620
views
Trying to prove that $\pi$ is irrational using Legendre Polynomials.
Unfortunately, numerical data sugggest that is not possible to show that $\pi$ is irrational with the polynomials below. I've to search for another polynomial... But I've little faith that such ...
- 2,691
6
votes
0
answers
229
views
Can even degree Legendre polynomials have roots in common?
I'm wondering whether the Legendre polynomials $P_m(x)$ and $P_{m+2k}(x)$, with $m$ even and $k \in \mathbb{N}^+$, can have common roots.
For $k=1$ it is straightforward to show that there are no ...
- 1,192
5
votes
0
answers
387
views
Pointwise convergence of a Legendre polynomial expansion
$\langle\cdot,\cdot\rangle$ is the dot product on the real vector space $\mathcal C ([0,1],\mathbb R)$ defined by $\langle f,g\rangle = \int_{-1}^1 fg$, and $(L_n)$ is the family of normalised ...
- 435
4
votes
0
answers
127
views
A closed-form formula involving Legendre polynomials for the "information potential" of the binomial distribution
This is a follow-up of this recent question to which I have given an answer.
It is mainly the "Edit" part of this answer that has an interest here with the following formula I have found (...
- 83.9k
4
votes
0
answers
90
views
Alternative orthogonality relations between associated Legendre polynomials
The usual orthogonality relations quoted for associated Legendre polynomials is:
$$
\int_{-1}^{1}P_{n}^{m}(x)P_{n'}^{m}(x)dx=\frac{2(n+m)!}{(2n+1)(n-m)!}\delta_{n,n'}
$$
However, I have come across ...
- 469
4
votes
0
answers
94
views
Series of Legendre Polynomials and Harmonic numbers. $\sum_{n=1}^{\infty} P_n (z) \frac{H(n)}{n+k}$
I would like to compute sums of the type
\begin{equation}
\sum_{n=1}^{\infty} P_n (z) \frac{H(n)}{n+k}
\end{equation}
where $P_n(z)$ are Legendre polynomials, $H(n)$ are harmonic numbers and $k = 0, 1,...
4
votes
0
answers
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views
The Distribution of Abscissae and Sums of Weights in Gaussian Quadrature
Let $n$ be a positive integer, and for $i = 1, 2, \ldots, n$, let
$x_i$ be the $i^\text{th}$ abscissa for $n$-point Gaussian
quadrature, and $w_i$ the associated weight, so that $-1 < x_1 <
x_2 &...
- 12.3k
4
votes
0
answers
992
views
Derivative of Legendre Polynomial
I am given with a relation
\begin{equation}
\frac{d^2}{dx^2}(P_l(x))=\frac{1}{2}\sum_{n=(0,1),2}^{l-2}(l-n)(l+n+1)(2n+1)P_n(x).
\end{equation}
The above sum starts with 0 for even end point, and 1 for ...
- 504
4
votes
0
answers
94
views
Axysimmetric Poisson equation solution
I'm struggling to find the solution to the follow Poisson problem in spherical coordinates:
$$ \Delta\, f\left(r,\theta\right) = \sum_{l=1}^{4}k_{l} \left(r\right) P_{l}^{1}\left( \theta \right)$$
...
- 563
3
votes
0
answers
212
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Solving PDE with Legendre functions
I'm studying a paper which has a PDE of the form
$$\frac{\partial p}{\partial L}(L,n) = A\bigg((n^2-1)\frac{\partial p}{\partial n}(L,n)\bigg),\quad n>1,$$
with a Dirac delta initial condition $p(...
- 457
3
votes
0
answers
54
views
Solving an inequality containing Legendre polynomials
Let it be given that $A$ is a real valued $m$ by $n$ matrix with entries $p_j\left(\frac{i}{m-1}\right)_{i,j=0}^{m-1,n-1}$, where $p_j$ is a Legendre polynomial. Show that $$\frac{\|{x}\|_2}{2} \leq \...
- 31
3
votes
0
answers
316
views
Closed form solution of a contour integral involving Bessel function integral representation
I have the following integral for which I want a closed-form solution:
$$
I_n(kr,\cos\theta) = \int_{C} \frac{e^{ikrt}P_n(t)}{t-\cos\theta} dt,
$$
where $P_n$ is a Legendre polynomial and $C$ is a ...
- 131
3
votes
0
answers
2k
views
What is the derivative of the associated Legendre Polynomials at the end points?
I have been searching for different solutions for the derivatives of associated Legendre polynomials at the end points. The associated Legendre polynomial is defined as:
$$P^m_l(x)=(-1)^m(1-x^2)^{m/2}\...
- 269