All Questions
Tagged with legendre-polynomials integration
103
questions
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
1
answer
42
views
Integration of Legendre polynomials with their derivatives
I need the results of $\int_{-1}^1 p_i(x) p_j'(x) dx$, where $p_i$ is the classical Legendre polynomials in $[-1,1]$. Using the matlab, I get the following result:
i \ j
0
1
2
3
4
0
0
2
0
2
0
1
0
...
- 47
3
votes
0
answers
66
views
Calculating the behaviour of an integral with Legendre polynomials of large order [closed]
I need to calculate the following integral:
$$\int_{\theta, \phi \in S^2} \left [ P_\ell(1-2\sin ^2\theta \sin^2\phi) \right ]^2 \sin\theta\, d\theta\, d\phi$$
where $S^2$ represents the unit sphere ...
0
votes
0
answers
37
views
Product of d-dimensional Legendre polynomials
Let $P_n:\mathbb{R}\rightarrow\mathbb{R}$ be the $d$-dimensional Legendre polynomials, that is they are orthonormal polynomials w.r.t the probability measure $\mu_d$ on $[-1,1]$ given by $\mu_d= \...
- 993
0
votes
0
answers
49
views
Legendre Polynomial Triple product with different arguments
I'm trying to integrate this: $$f_{jkl}\langle{\hat{a},\hat{b},\hat{c}}\rangle=\frac{1}{4\pi} \int d{\Omega}_n P_j(\hat{a}.\hat{n})P_k(\hat{b}.\hat{n})P_l(\hat{c}.\hat{n})$$ where $\hat{n}$ is a unit ...
0
votes
1
answer
90
views
Norm of Legendre Polynomials $P_m(x)$
While studying to prove the norm of Legendre polynomials $P_m(x)$ is $\sqrt{\frac{2}{2m+1}}$, I faced $\int_{-1}^{1} [D^m (x^2-1)^m]^2 dx = (2m)! \int_{-1}^{1} (1-x^2)^m dx.$ $D^m$ stands for ...
- 21
0
votes
0
answers
83
views
Re-writing a sum of binomial coefficients as an integral of shifted Legendre polynomials
This is a question regarding the answer presented here.
In order to make this post self-contained, I am wondering if someone can explain why the sum
$$
\sum_{k=0}^{n}(-1)^{n+k}\binom{n}{k}\binom{n+k}{...
user1211323
0
votes
0
answers
78
views
"Legendre-type" integrals involving $\frac{dt}{\sqrt{t^2-2t\cos(\theta)+1}}$
Summing Legendre polynomials $P_{l}(\cos\theta)$ often leads to expressions containing $\frac{1}{\sqrt{t^2-2t\cos\theta+1}}$, as this is the generating function for the Legendre polynomials. I want to ...
- 203
1
vote
1
answer
566
views
On the vanishing of the integral $\int^{n+4}_0 dt P_{\ell}\left(1 - \frac{2t}{n+4}\right)(-t+2)_{n+1}$ for $\ell \geq n+2$
I came across the following integral
\begin{equation}
\int^{n+4}_0 dt P_{\ell}\left(1 - \frac{2t}{n+4}\right)(-t+2)_{n+1}
\, ,
\end{equation}
where in the above $P_{\ell}(x)$ is the Legendre ...
user1211323
0
votes
1
answer
204
views
Legendre polynomials: computing $\int_0^1 xP_n(x)\,dx$
I am trying to find the integration:
$$
\int_{0}^{1}{xP_n\left(x\right)}\,dx
$$
I know that I should split Rodrigues's formula up into two parts $P_{2n}\left(x\right)$ for even terms and $P_{2n+1}\...
- 1
2
votes
1
answer
89
views
Can order of summation and integral be interchanged in : $\int_{-1}^1 ( \sum_{n=0}^\infty P_n(\xi)P_n(\xi^\prime)Q_l(\xi^\prime))\text{d}\xi^\prime$?
I am wanting to know if there is a proof that the order of summation and integration can be interchanged in $$\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
22
views
Any comment to speed up the calculation of double-integral having Legendre polynomials?
I want to compute the following double integral at $t=t_{0}$ rapidly. I tried different methods, but all are time consuming for I,J,M >7.
Any comments to speed up the calculation???
$$\frac{1}{2}\...
- 29
0
votes
1
answer
119
views
Prove of a Legendre's polynomial problem
I have a question that I need to prove that for $n≥1$,
$$\frac{1}{2n}\int_{-1}^{1}x\frac{d}{dx}(P_n(x)^2)dx=\frac{2}{2n+1}$$
I have to evaluate the integral instead of using the orthogonality property ...
2
votes
1
answer
119
views
Prove $\int_{0}^{1}x^{m}P_{l}(x)dx = \frac{m! (m - l + 1)!!}{(m - l +1)!(m + l +1)!!}$
I used the Rodrigues formula and integrated by parts:
$$ \dfrac{1}{2^{l}l!}\int_{0}^{1}x^{m}\dfrac{d^{l}}{dx^{l}}(x^2 - 1)^{l}dx = \dfrac{1}{2^{l}l!}[ x^{m}\dfrac{d^{l-1}}{dx^{l-1}} (x^{2} - 1)^{l} - \...
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