All Questions
Tagged with legendre-polynomials mathematical-physics
17
questions
2
votes
0
answers
67
views
Fourier-Legendre series for $x^n$
I'm struggling to find the Legendre expansion for $x^n$ (exercise 15.1.17 from Mathematical Methods for Physicists).
I'm trying to evaluate the following integral:
$$a_m = \frac{2m+1}{2} \int_{-1}^{1} ...
- 29
-1
votes
1
answer
59
views
Legendre's Series: How can I show that $\int_{-1}^1 f(x)S(x)dx= f(1)$
Consider the series:
$$S(x)=\sum_{n=0}^{\infty}\frac{(2n+1)P_n(x)}{2}$$
Show that:
$$\int_{-1}^1 f(x)S(x)dx= f(1)$$
where $f(x)$ is any function of the interval $[−1, 1]$ on the real numbers which can ...
- 11
0
votes
1
answer
475
views
Prove that the Legendre polynomials satisfy $P_n(1) = 1$ and $P_n(-1)=(-1)^n$
Problem. Use the following relationship for the Legendre polynomials
$$ P_n(\cos\theta) = \frac{1}{n!} \frac{\partial^n}{\partial t^n} (1 - 2t\cos\theta + t^2)^{(-1/2)} |_{t=0} $$
in order to prove ...
- 151
0
votes
1
answer
204
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Legendre polynomials: computing $\int_0^1 xP_n(x)\,dx$
I am trying to find the integration:
$$
\int_{0}^{1}{xP_n\left(x\right)}\,dx
$$
I know that I should split Rodrigues's formula up into two parts $P_{2n}\left(x\right)$ for even terms and $P_{2n+1}\...
- 1
3
votes
2
answers
94
views
Integral connection with Hermite and Legendre polynomials
Show that $$\int\limits_{-\infty}^{+\infty}x^n e^{-x^2} H_n(tx) dx =\sqrt{\pi} n! P_n(t)$$
Case seems rather complex, I'm completely stuck...
- 45
0
votes
1
answer
74
views
Could someone help prove some properties of Legendre Polynomials?
I have already proved other properties of the Legendre polynomials, like:
$$P_n(-x) = (-1)^n \, P_n(x)$$
$$P_{2n+1}(0) = 0$$
$$P_n(\pm1)= (\pm1)^n$$
But I can't get this one:
$$P_{2n}(0) = \frac{(-1)^...
- 29
1
vote
1
answer
179
views
Eigenfunction expansion of a heat equation with Legendre polynomials
I am trying to solve the following PDE by performing an eigenfunction expansion:
$$
\frac{\partial p}{\partial t} = -\cos \varphi \frac{\partial p}{\partial x} + D\frac{\partial^2 p}{\partial \varphi^...
- 185
1
vote
0
answers
40
views
Verifying Legendre's equation
Use the following results:
$$(l+1)P_{l+1}(x)-x(2l+1)P_l(x)+lP_{l-1}(x)=0,$$
$$P_l(x)+2xP'_l(x)=P'_{l+1}(x)+P'_{l-1}.$$
in order to show the following recurrence relations:
$$(2l+1)P_l(x)=P'_{l+1}(x)-P'...
2
votes
1
answer
268
views
Calculate the integral where $P_{n}$ and $P_{m}$ are Legendre Polynomials
Calculate the folowing integral:
$$I_{k,m}=\int_{-1}^{1} x(1-x^2)P'_{n}(x)P'_{m} dx $$
So, my attempt to solve this consisted in:
First, I thought of manipulating the folowing relations so i could get ...
- 31
3
votes
1
answer
176
views
Convert ODE to a form of Bessel differential equation
I'm working on the solution of the equation
$$\tan^2u\partial^2_u y_2 + (2+\tan^2u)\tan u \partial_u y_2 -a^2\lambda_2y_2 - n^2(1+\cot^2u)y_2 = 0.$$
It is possible to write the above equation in terms ...
- 373
1
vote
0
answers
172
views
Spherical Harmonics Sum Identity
I'm taking a course in Quantum Mechanics and this problem is causing me some struggles. Can someone help me prove this identity?
$$\sum_{m = -l}^l m^2 |Y_{l}^{m}(\theta, \phi)|^2 = \frac{l(l+1)(2l+1)}{...
2
votes
1
answer
3k
views
Parity of spherical harmonics
I would like to prove that $Y_{\ell m}(-\mathbf{r}) = (-1)^\ell\, Y_{\ell m}(\mathbf{r})$. In this formula, $Y_{\ell m}$ are the spherical harmonics given by
\begin{equation}
Y_{\ell m}(\theta, \...
- 456
1
vote
1
answer
154
views
What are the necessary and sufficient condition for $f(x)$ to be expandable into Legendre polynomials?
In physics, we often expand functions $f(x)$ of a real variable $x$ which is piecewise continuous in the interval $(-1,1)$ along with its derivatives, in terms of Legendre polynomials $\{P_n(x)\}$ as $...
- 1,502
3
votes
1
answer
576
views
Closed form of sum of infinite series of Legendre polynomials
I was working on a research and we end up to discover that the Green's function on some domain is of the form :\begin{equation}
G = \frac{-1}{4\pi}\sum_{l=0}^{\infty}\frac{2l+1}{l(l+1)+\frac{1}{\alpha}...
- 488
2
votes
1
answer
2k
views
Integrating Associated Legendre Polynomials
As part of a derivation for the question I asked here in Physics stackexchange, I am trying to calculate the following integral, but I am not sure how to proceed:
$\int_0^{2\pi}P_l^m(\cos\theta)P_{l-...
- 235