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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Orthogonality of the Legendre polynomials with respect to $L_2$ (Integration by parts)

I want to show, that the Legendre polynonmials are orthogonal with respect to scalar product in $L_2[-1,1]$. The Legendre polynomials are defined as follows: $$P_n(x) = \left( \frac{2n+1}{2}\right)^{\...
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Partial wave decomposition of a function with Gamma functions

I have the following function $$f(x,s) = \frac{\Gamma(1-\frac{s}{2}(x-1))}{\Gamma(1+\frac{s}{2}(x-1))}$$ where $s>0$, $x \in [-1,1] $ and $\Gamma(z)$ are Gamma functions. I would like to decompose ...
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Derive Jackson Equation 3.26

I want to derive equation 3.26 from jackson's book, classical electrodynamics. $(2l+1)\int_{0}^{1}P_l(x)dx=(-\frac{1}{2})^{(l-1)/2}\dfrac{(2l+1)(l-2)!!}{2(\dfrac{l+1}{2})!}$ where l is odd, using the ...
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1 vote
2 answers
68 views

Legendre polynomial property

How do i show from the Legendre's Polynomial equation $$P_n(x)=\sum_{k=0}^N \dfrac{(-1)^k(2n-2k)!}{2^nk!(n-k)!(n-2k)!}x^{n-2k}$$ where $N=n/2$ for even $n$ and $ N=(n-1)/2$ for odd $n$. Using just ...
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Calculation of the integral of the Legendre polynomial of the second kind

Please tell me the possible options for calculating the integral of the form $\int\limits_{-\infty}^a\frac{Q_n(x)}{(x+b)^{n+2}}dx$, where $a\in[-2,-\infty)$; $b\in(-a,-\infty)$; $Q_n(x)$ - Legendre ...
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Creating a spanning set out of Legendre polynomials

In the book "An Introduction to Partial Differential Equations with MATLAB" by Matthew P Coleman (2nd edition), exercise 2 in chapter 11.3 states Show that the functions $$\phi_n(x) = P_n(x)...
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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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3 votes
1 answer
36 views

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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2 votes
1 answer
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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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Prove that the Legendre polynomial recurrence relationship satisfies the defining differential equation

I am trying to show that from this recurrent relationship $$ (n+1)P_{n+1}(x) = (2n+1)xP_n(x) - nP_{n-1}(x) $$ that the Legendre polynomial $P_n(x)$ satisfies the differential equation $$ (1-x^2)P'' - ...
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Show that the next function is always positive

I need to show that the next function is always positive for any $j\in\{0,1,2,3,...\}$ where $\Gamma(\cdot)$ is the gamma function, $m>1/2$, $K>0$, $0\leq\Delta\leq 1$, and $i=\sqrt{-1}$. ...
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  • 33
1 vote
1 answer
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Legendre-Gauss-Lobatto Nodes via Newton-Raphson

I am trying to understand the code given here, which calculates the Legendre-Gauss-Lobatto nodes via the Newton-Raphson method. These nodes are given by the zeros of $(1 - x^2)\,P'_{N}(x)$ where $P_N(...
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1 vote
1 answer
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Calculating the spherical harmonic of θ=π/2

This is a very simple question, yet I'm not sure how to approach it. I want to calculate the spherical harmonic: $$ Y_{l m}^{*}(\theta = \pi/2, \phi) $$ I know the general formula: $$ Y_{l m}^{*}(\...
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1 vote
0 answers
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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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4 votes
1 answer
129 views

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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0 votes
1 answer
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Convergence of Legendre Polynomials

I am trying to approximate the function $(1-x)^\sqrt{2}$ using Legendre polynomials on the interval $[0,1]$. I have been using $P_n(1-2x)$ as my polynomials as I want all the polynomials to satisfy $...
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1 vote
1 answer
63 views

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\...
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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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4 votes
1 answer
133 views

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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  • 405
5 votes
2 answers
110 views

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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31 views

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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3 votes
1 answer
86 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 ...
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0 votes
1 answer
42 views

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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3 votes
1 answer
124 views

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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0 votes
0 answers
27 views

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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2 votes
0 answers
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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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1 vote
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Solid angle integral, with legendre polynomials

I've been trying to solve this question from Jon Mathews Mathematical methods for physics, and I'm honestly very lost, I was given the following hint: $$ \cos \gamma=\cos \theta \cos \theta^{\prime}+\...
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Legendre polynomials from generating function

I have trouble to understand the following transform mentioned in Special functions and their applications by N. N. Lebedev, section 4.2 We can write the generating function of Legendre polynomials, $...
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0 votes
1 answer
119 views

Proof the Recurrence formula with Rodrigues' Formula

My University Professor gave the task to proof the recurrence formula without generating function of Legendre Polynomial , only with Rodrigues' Formula . So far, I used : $P_l\left(x\right)=\frac{1}{{...
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3 votes
1 answer
95 views

How to relate $\int_{-1}^{1}e^{ikru} P_{\ell}(u)du$ to $j_\ell(kr)$ in a simple way?

Consider the following epansion of the function $e^{ikru}$ in terms of Legendre polynomials, $P_\ell(u)$, $$e^{ikru}=\sum_{\ell=0}^{\infty}C_\ell(r)P_\ell(u)$$ where $k$ is a constant real parameter, ...
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0 votes
0 answers
48 views

Least squares function approximation using Legendre polynomials

My task is to write a program that approximates a given function as a combination of $n$ first Legendre polynomials using Least Squares method in $[-1; 1]$. I understand that I need to minimize $$\...
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1 answer
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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\...
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2 votes
1 answer
49 views

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 ...
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0 votes
0 answers
73 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 $P_l^m$ ...
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Show that the inner product of the Legendre polynomials and $p\in L^2 ([-1,1])$ is equal to zero if the polynomial has degrees less than $n$

Let $(P_n )^{\infty}_{n=0}$ be the Legendre polynomials in $L^2([-1,1])$, normalised as $||P_n||^2=\frac{2}{2n+1}$. I need to show that, for any polynomial $p\in L^2([-1,1])$ with degree less than $n$,...
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  • 174
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1 answer
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Initial guess in Newton-Raphson method.

To find roots using the Newton-Raphson method, the initial guess is very important otherwise it may take several iterations to give the value of roots. For the given Legendre polynomial ($ P _ 8 $), ...
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0 votes
0 answers
25 views

Expand square of a product of functions in terms of simple products of the same kind of functions

I have a product of the square of a spherical Bessel function and the square of an associated Legendre polynomial $\left(j_l\left(k_{l,m}r\right)P_l^m(\cos(\phi))\right)^2$ (where $l$ and $m$ are ...
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1 vote
0 answers
43 views

Expand square of an associated Legendre polynomial in terms of simple associated Legendre Polynomials

I have an associated Legendre Polynomial $\left(P_l^m(\cos(\theta))\right)^2$ (where $l$ and $m$ are nonnegative integers). I need to find a way to express it in terms of simple associated Legendre ...
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2 votes
0 answers
59 views

Sum of products of Legendre function

I was wondering if the following sum is known in closed form: $$\sum_{n=0}^{\infty} n P_{n-\frac{1}{2}}(x) P_{n-\frac{1}{2}}(y)$$ where $P_{n-\frac{1}{2}}(x)$ are Legendre functions. I know of a ...
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2 votes
1 answer
68 views

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 ...
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  • 409
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28 views

Why is $ \frac{1}{|\boldsymbol{r}-\frac{\boldsymbol{R}}{2}|} =\sum_{L=0}^{\infty} \frac{r_{<}^{L}}{r_{>}^{L+1}} P_{L}(\zeta)$

I want to show that $$ \frac{1}{\left|\boldsymbol{r}-\frac{\boldsymbol{R}}{2}\right|} =\sum_{L=0}^{\infty} \frac{r_{<}^{L}}{r_{>}^{L+1}} P_{L}(\zeta)$$ where $r_{<}\left(r_{>}\right) \text ...
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8 votes
0 answers
305 views

How to calculate the integral of a product of a spherical Hankel function with associated Legendre polynomials

From numerical experiments in Mathematica, I have found the following expression for the integral: $$ \int_{-1}^{1}h_{n}^{(1)}\left(\sqrt{a^{2}+b^{2}+2ab\tau}\right)P_{n}^{m}\left(\frac{a\tau+b}{\sqrt{...
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232 views

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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0 answers
35 views

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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1 vote
0 answers
52 views

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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3 votes
1 answer
131 views

Integral of Legendre and Chebyshev polynomials.

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, ...
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