All Questions
Tagged with legendre-polynomials real-analysis
33
questions
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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):
$$\sum_{n=0}^{\infty} e^{-(n+\frac{1}{2})|u|} \; ...
0
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Closed Forms for Sums of Legendre Polynomials
I am investigating a series of the form $$\sum_{n=0}^\infty \frac{1}{1 + e^{nx}}P_n(x)$$ where $P_n$ is the Legendre Polynomial of degree $n$.
Because I am dealing with many of these series, it would ...
- 93
2
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1
answer
127
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Asymptotic equality of $\frac{1}{n!}\frac{d^n}{dx^n}(x^n(1-x)^n) $
Consider the Shifted Legendre Polynomial $$\tilde P_n(x)=\frac{1}{n!}\frac{d^n}{dx^n}(x^n(1-x)^n) $$ where $n\in\mathbb{N}\cup\{0\}$
Question: What is the asymptotic equality of $\tilde P_n(x)$ as $n\...
- 910
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1
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Showing that a series that involves Legendre polynomial converges.
I am currently studying Legendre polynomials and I am trying to prove its generating function specifically using the recurrence relation
$$ (n+1)L_{n+1}(x) = x(2n+1)L_n(x) - nL_{n-1}(x), \quad \forall ...
- 1,141
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1
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89
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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 $$\int_{-1}^1 \left( \sum_{n=0}^\infty P_n(\xi)P_n(\xi^\prime)Q_l(\xi^\prime)\right)\text{d}\...
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Getting the recurrence relation for Legendre polynomials by Leibnitz rule
Exercise:
For each natural number $n$ define
$$\phi_n(x)=\frac{d^n}{d x^n}\left(x^2-1\right)^n$$
Derive the formulas
$$\phi_{n+1}^{\prime}(x)=2(n+1) x \phi_n^{\prime}(x)+2(n+1)^2 \phi_n(x)$$
$$\phi_{n+...
- 385
2
votes
1
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148
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How to prove Legendre Polynomials' recurrence relation without using explicit formula?
Assume there is an inner product on linear space $V = \{ \text{polynormials}\}$:
$$\langle f, g\rangle = \int_{-1}^1 w(t) f(t)g(t) dt$$ with $w(t) \ge 0$ and not identically zero.
Then we can ...
- 1,231
1
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How $x^n$ is linearly represented by Legendre polynomials
I recently come across a problem with respect to Legendre polynomial as follows.
Let $L^2[-1,1]$ be the Hilbert space of real valued square integrable functions on $[-1,1]$ equipped with the norm $\|f\...
- 493
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58
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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:
$$
S = \sum_{n \ge 1} \frac{2n+1}{n+1}
\left( \int_0^1 P_n(x) \, \mathrm{d}x \right)
\left( \int_0^1 x \left( 1-x^2\...
2
votes
1
answer
292
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How do I expand $\frac{1}{\sqrt{ (\boldsymbol{r-r'})^2+a} }$ in legendre polynomials (spherical harmonics)?
Using the generating function for the legendre polynomial: $$ \sum_{n=0}^{\infty} P_{n}(x) t^{n}=\frac{1}{\sqrt{1-2 x t+t^{2}}} $$ It's possible to expand the coulomb potential in a basis of legendre ...
- 419
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1
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183
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Shifted Legendre polynomials symmetry relation
I have to prove that $p_n(1-x)=(-1)^np_n(x)$ ($x\in[0,1]$) for all $n\in\mathbb{N}$, where $(p_n)_n$ is the family of Legendre polynomials on $[0,1]$: given $(x^n)_{n=0,1,\ldots}$, $(p_n)_n$ is the ...
- 187
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2
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519
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Convergence of Legendre Expansion in the Mean-Square sense.
I need to show that the Legendre Expansion converges to the function in the mean-square sense, here are some definitions.
$$L_n(x)=\frac{d^n}{dx^n}(x^2-1)^n \text{ and } \mathfrak{L}_n(x)=\frac{L_n(x)}...
- 388
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157
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Orthogonality Legendre Polynomial WITHOUT integration by parts
I'm aware of the two questions [1] and [2] but I'm trying to proof the orthongonality of the Legendre polynomials without integration by parts.
$$
\int_{-1}^1 P_\ell(x)P_{\ell'}(x) dx = \frac{2}{2\ell+...
- 4,644
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2
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Given the Rodrigues' formula for Legendre's polynomials, show that it satisfies the ODE.
The Rodrigues Formula for Legendre's Polynomials is $P_{l}(x)=\frac{1}{2^{l}l!}\frac{d^{l}}{dx^{l}}(x^2-1)^l$.
I wrote $P_{l}(x)=\frac{1}{2^{l}l!}\frac{d^l}{dx^l}\sum_{k=0}^l(-1)^{k-l}\frac{l!}{k!(l-...
- 13
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58
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Differentiate $((x^{2}-1)f_{n-1}(x))^{(n)}$.
Differentiate $((x^{2}-1)f_{n-1}(x))^{(n)}$.
Using Leibniz rule, I obtain the following:
$\frac{x^{2}-1}{2^{n}n!}f_{n-1}^{(n)}(x)+\frac{x}{2^{n-1}(n-1)!}f_{n-1}^{(n-1)}(x)+\frac{1}{2^{n}(n-2)!}f_{...
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