# Questions tagged [legendre-polynomials]

For questions about Legendre polynomials, which are solutions to a particular differential equation that frequently arises in physics.

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### 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 ...
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### $I(x) = -\int_0^1 \frac{1}{z}\ln\left(\frac{1-x z + \sqrt{1-2 x z+ z^2}}{2}\right)\,dz$

Is there a closed form integral for $$I(x) = -\int_0^1 \frac{1}{z}\ln\left(\frac{1-x z + \sqrt{1-2 x z+ z^2}}{2}\right)\,dz$$ for $-1 < x < 1$? This integral is related to Legendre polynomials ...
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### recurrence relation associated Legendre functions

I need a little help to find the recurrence relation $$\sqrt{1-x^2}P_l^m(x) = \frac{1}{2l+1} (P_{l-1}^{m+1}-p_{l+1}^{m+1})$$ Using the identity $$(2l+1)P_l(x) = \frac{d}{dx}(P_{l+1}(x)-P_{l-1}(x))$$ I ...
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### Evaluating an integral with derivatives of Associated Legendre polynomials

I came across the following integral $$\int_{-1}^{+1} (1-x^{2}) \frac{\partial P_{lm}(x)}{\partial x} \frac{\partial P_{km}(x)}{\partial x} dx$$ where $P_{lm}(x)$ is an associated Legendre polynomial, ...
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### How to evaluate the following sum: $\sum_{n=1}^\infty \frac{P_n(x) - P_{n-1}(x)}{ n } \cos(nt)$

I am trying to find a closed form expression for the following sum, $$F(x,t)= \frac{1}{\log\left(\frac{1+x}{2}\right)}\sum_{n=1}^\infty \frac{P_n(x) - P_{n-1}(x)}{ n } \cos(nt) ~,$$ where $P_n(x)$ ...
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### Legendre equation solution at the regular singular point, infinity

Consider the Legendre equation, $$(1-x^2)y''-2xy'+\alpha (\alpha +1)y=0$$ Let $x=1/t$, then $$\frac{dy}{dx}=-t^2\frac{dy}{dt}$$ $$\frac{d^2y}{dx^2}=t^4\frac{d^2y}{dt^2}+2t^3\frac{dy}{dt}$$ The ...
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### Proving $\frac{\pi}{2}=\sum^\infty_{l=0} \frac{(-1)^l}{2l+1}\big(P_{2l}(x)+\text{sgn}(x)P_{2l+1}(x)\big)$

Can someone help me in proving the following: $$\frac{\pi}{2}=\sum^\infty_{l=0} \frac{(-1)^l}{2l+1}(P_{2l}(x)+\text{sgn}(x)\cdot P_{2l+1}(x)),$$ for any value of $x$, $-1\le x\le 1$? (Here $P_l(x)$ ...
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### Are these both Legendre's Equations?

Equation 1 : $(2x + 3)^2 y'' + (2x+3)y' - 12y = 6x$; can be solved by the substitutions used for Cauchy-Euler Equations. Equation 2 : $(1-x^2)y'' - 2xy' + p(p+1)y = 0$; which only has power series ...
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### The values of $P_n(x)$ at the zeros of $P'_n(x)$

I recently come across a problem with respect to Legendre polynomial as follow. For any $n \in \mathbb{N}$, $P_n(x) := \frac{1}{2^n n!}\frac{{\rm d}^n (x^2-1)^n}{ {\rm d} x^n}$ is the classical ...
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### Closed form expression for series involving Legendre polynomials

Given $-1 \leq x \leq 1$ and $0 \leq \eta \leq 1$, I am interested in computing $$E(x,\eta) = \sum_{\ell = 0}^{+ \infty} |P_{\ell} (0)|^{2} \, P_{\ell} (x) \, \eta^{\ell} ,$$ with $P_{\ell}$ the ...
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### Which of the following series converge? (With Legendre-Polynomials)

These should be quick tasks: We know that the Legendre Polynomials satisfy $\int_{-1}^{1} P_m(x)P_{n}(x)dx= \delta_{mn}\frac{2}{2n+1}$ Which of the following series converge ( $\forall x \in [-1,1]$ )...
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### Lear recursive algorithm for associated Legendre polynomials

I have recently been working on reconstructing the Earth gravitational field with spherical harmonics. Part of the process involves the calculation of associated Legendre polynomials up to relatively ...
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### 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 ...
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### Prove for general form of function at -x containing derivatives of order n

I have stumbled across multiple casses of functions (explicitly Hermit and Legendre polynomials) for which I wanted to prove the symmetry. While doing so I always ended up with the following equations:...
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### There is no continuous function $f$ so that $\int_a^b f(x)x^n dx$, $0\leq a<b$ is positive for all even $n$ and negative for odd $n$

This was CIIM 2021 problem 6. I'm trying to find a solution to this problem considering the inner product $$\left<f,g\right>=\int_a^b f(x)g(x)dx.$$ Let's work on the case $a=0$, $b=1$ first. Our ...
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### How to prove $(l+1)P_{l+1}(x) = (2l+1)xP_l(x) - lP_{l-1}(x)$?

How to prove $(l+1)P_{l+1}(x) = (2l+1)xP_l(x) - lP_{l-1}(x)$? Hint: Try to derivate the generating function by $t$, and subtract from the generating function. I will be glad if someone can help me, I'...
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### Problem with Legendre-Fourier series for sinx when the number of terms approaches infinity

After I learned about Fourier series expansion, I understand orthogonality of trigonometric functions was the key when I calculate the coefficients of Fourier series. As I knew that Legendre ...
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### How can I modify this recurrence relation for the Normalized Associated Legendre Polynomial to use the full normalization instead of spherical?

I'm implementing an algorithm for a numerically stable normalized associated Legendre polynomial, but I need a different normalization factor than the source I'm using. The source is here and uses a ...
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### Expectation of x/√(x²+2px+1) under Normal distribution

I'm need to find (or at least approximate) as a function of $p$, the expectation under $x \sim Normal(0,1)$ of: $$f(x) = \frac{x}{\sqrt{x^2+2px+1}},\hspace{1em}\textrm{where}-1<p<1$$ Wolfram ...
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### Understanding the dimensionality of Legendre polynomials.

I have been looking at the Laplace equation $\nabla^2 f = 0$ in various dimensions. In 3 dimensions, the angular equation leads to the well-known spherical harmonics, defined up to normalisation as \...
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### a quadratic to approximate a greatest integer function on an interval

I came across this problem: Define $\mathrm{f}: [0,4] \rightarrow \mathbb{R}$ such that $f(x)=[[x]]$ if $x \in [0,4)$ and $f(4)=3$. Find the closest quadratic approximation to $f$ using the set of ...
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I am trying to expand Legendre polynomials into Chebyshev polynomials, shown as: $$P_{n}(x)=\sum_{k=0}^{n}a_{k}T_{k}(x),$$ where $P_{n}$ is Legendre polynomials and $T_{k}$ is Chebyshev polynomials, ...