All Questions
Tagged with legendre-polynomials definite-integrals
52
questions
1
vote
0
answers
28
views
Proof involving integrals, binomial coefficients and Legendre polynomials
I'm currently working on a research paper involving L-moments statistics. This is the first time for me working on that subject.
I stumbled upon this article mentioning a very important equality, ...
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
1
vote
0
answers
26
views
Tripe integral involving the square of associated Legendre polynomials and a derivative of a Legendre polynomial
I encountered the following integral in the physics literature
$$
\int_{-1}^{1}P_{\ell}^m(x)^2P_{n}^\prime(x){\rm d}x
$$
where $P_{\ell}^m(x)$ is an associated Legendre polynomial of degree $\ell$ and ...
- 369
1
vote
0
answers
64
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Integral of product of Legendre polynomial and exponential function
Kindly help me with the following integral :
$
I_l(a) = \int_{-1}^{+1} dx\, e^{a x} P_l(x) \quad
$
($a$ is real and positive).
I thought to use the following relation given in Gradshteiyn and also ...
- 65
1
vote
0
answers
18
views
Integrals of Legendre polynomial and a rational function
Is there are analytic expression of the following definite integral?
$$
\int_{-1}^1 x^\alpha (1-x^2)^{\beta} P_l(x) P_m(x) \text{d}x
$$
0
votes
0
answers
26
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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,
$$
E_n=\int_{r0}^\infty \int_0^\pi -\text{sin}^3(\theta)(\left(
\left(\text{cot}(\theta) \sum_{l=0}^n R_l(r) \partial_\...
0
votes
0
answers
78
views
"Legendre-type" integrals involving $\frac{dt}{\sqrt{t^2-2t\cos(\theta)+1}}$
Summing Legendre polynomials $P_{l}(\cos\theta)$ often leads to expressions containing $\frac{1}{\sqrt{t^2-2t\cos\theta+1}}$, as this is the generating function for the Legendre polynomials. I want to ...
- 203
2
votes
0
answers
235
views
Integral of associated Legendre polynomials over the unit interval
I am looking for a closed-form expression for the integral of the associated Legendre polynomial $P_l^m$ over the unit interval ($l \ge m$ non-negative integers),
$$
I_l^m = \int_{0}^{1} P_l^m(x) \, ...
- 73
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
views
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
5
votes
2
answers
142
views
Convergence of an integral with Legendre polynomials
Let's consider the following integral
$$
I(\ell) = \int_{-1}^1 dx P_\ell(x) A(x)
$$
where $ P_\ell(x) $ is the $\ell$-th Legendre polynomial and $A(x) = \frac{1}{1-\lambda x}$ with $0\leq \lambda<1$...
- 640
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:
$$
S = \sum_{n \ge 1} \frac{2n+1}{n+1}
\left( \int_0^1 P_n(x) \, \mathrm{d}x \right)
\left( \int_0^1 x \left( 1-x^2\...
2
votes
1
answer
361
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Weighting for Gauss-Legendre Quadrature
The textbook I am reading shows that the weighting of Gauss-Legendre Quadrature is
\begin{align*}
w(x_i) = \frac{1}{P_n'(x_i)}\int_{-1}^1 \frac{P_n(x)}{x-x_i} dx
\end{align*}
which is evaluated ...
- 286
0
votes
0
answers
140
views
Integral of Associated Legendre Functions
Does anyone know how to compute the following integrals:
$$
\int_{-1}^1 \frac{P_l^m(x) P_n^m(x)}{\sqrt{1 - x^2}} dx
$$
and
$$
\int_{-1}^1 \frac{x}{\sqrt{1-x^2}} P_l^m(x) P_n^m(x) dx
$$
where $P_l^m$ ...
- 1,026
18
votes
0
answers
884
views
How to calculate the integral of a product of a spherical Hankel function with associated Legendre polynomials
By experimenting in Mathematica, I have found the following expression for the integral:
$$
\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