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Questions tagged [legendre-functions]

This tag is for questions relating to Legendre Functions (or Legendre Polynomials), solutions of Legendre's differential equation (generalized or not) with non-integer parameters.

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Derivation of the associated Legendre Polynomials

I have been struggling to find a proper derivation of the associated Legendre Polynomials and a derivation of $$P_l^{-m}(\mu)=(-1)^m\frac{(l-m)!}{(l+m)!}P_l^m(\mu)$$ Can someone point to a proper ...
Lukas Kretschmann's user avatar
1 vote
2 answers
129 views

Calculation for negative integer order Associated Legendre Function

I am currently engaging with the following hypergeometric function as a result of attempting to find a solution for this probability problem for $n$ number of dice: $$_2F_1\left (\frac{n+k}{2}, \frac{...
Lee Davis-Thalbourne's user avatar
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26 views

Legendre functions at the end points

Legendre equation is $$ (1-x^2) y'' - 2xy' + \lambda y=0$$ We are interested in finding solutions in the range $[-1,1]$. We seek solutions around the ordinary point $x=0$ $$ \sum_{n=0}^\infty c_n x^n $...
Marc Navarro's user avatar
1 vote
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27 views

$\sum_{n=0}^{\infty} c_{2n}$ and $\sum_{n=0}^{\infty} c_{2n+1}$ given $c_{n+2} = \frac{n(n+1)-\lambda}{(n+2)(n+1)}$

Legendre equation is $$ (1-x^2) y'' - 2xy' + \lambda y=0$$ We are interested in finding solutions in the range $[-1,1]$. We seek solutions around the ordinary point $x=0$ $$ \sum_{n=0}^\infty c_n x^n $...
Marc Navarro's user avatar
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50 views

Legendre functions of the second kind with negative integer degree

I have been recently reading about properties of Legendre functions in several sources and cannot seem to find any properties of Legendre functions of the second kind with negative integer degree. For ...
Lawford Hatcher's user avatar
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31 views

Help with an integral involving associated Legendre functions?

In the last few days I've come across this nice integral involving the associated Legendre functions $$ {\large\int_{-1}^{1}} \frac{P_{\ell}^m\left(u\right) P_{\lambda}^{m}\left(u\right)}{\sqrt{1 - u^...
Rafael Benevides's user avatar
1 vote
0 answers
23 views

Integral of Squared Spherical Harmonics

The following integral comes out of an expression $\langle |Y_{l,m}(\theta, \phi)|^2\rangle$ over a orientation probability distribution: $$\int_{0}^{2\pi} \int_{0}^{\pi} Y_{lm}^2(\theta, \phi)Y_{l'm'}...
MkFlash's user avatar
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First derivative of the associated Legendre function at x=1

I'm trying to find the first derivative of the associated Legendre function at $x=1$. The form I have for the first derivative is divergent at x=1: \begin{equation} \frac{d P_l^m(x)}{d \theta}=\frac{l ...
Laren's user avatar
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2 votes
1 answer
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Seeking the generating function of $\left( 1+\epsilon^2 - 2\epsilon x \right)^{-1}$

Many physical problems can be effectively solved using series expansions, such as the utilization of Legendre polynomials. Consider the function $\left( 1+\epsilon^2 - 2\epsilon x \right)^\frac{1}{2}$....
Siegfriedenberghofen's user avatar
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2 answers
277 views

Does this Fourier Legendre series inverting $\frac65\frac{x^x}{x+1}-\frac35$ diverge or it is a software issue past $\approx 25$ series terms?

$\def\P{\operatorname P}$ The goal of the Fourier Legendre series is to find an exact explicit series solution to $x^x=x+1$, not just $x=a+b+c+\dots$ with unknown series coefficients. Lagrange ...
Тyma Gaidash's user avatar
1 vote
0 answers
65 views

Asymptotic expansion of Legendre functions at $x=-1$

Let $P_{\nu}(z),\,-1<z\leq1,\,\nu\in\mathbb{C}$ Legendre functions of the first kind defined as $$P_{\nu}(z)=\,_{2}F_{1}\left(-\nu,\nu+1;1;\frac{1-z}{2}\right).$$ I found this formula on Wolfram ...
Yep's user avatar
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1 answer
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Integration of a square of Conical (Mehler) function

I want to evaluate $$\int_{\cos\theta}^1 \left( P_{-1/2+i\tau}(x) \right)^2 dx,$$ where $P$ is the Legendre function of the first kind, $i$ is the imaginary unit, and $\tau$ is a real number. Are ...
r-nishi's user avatar
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2 votes
1 answer
386 views

Derivation of an integral containing the complete elliptic integral of the first kind

This is a repost of mathoverflow to draw broader attentions. https://mathoverflow.net/questions/439770/derivation-of-an-integral-containing-the-complete-elliptic-integral-of-the-first I found the ...
r-nishi's user avatar
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How to integrate products of Legendre functions over the interval [0,1]

The associated Legendre polynomials are known to be orthogonal in the sense that $$ \int_{-1}^{1}P_{k}^{m}(x)P_{l}^{m}(x)dx=\frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{k,l} $$ This is intricately linked to ...
Chris's user avatar
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3 votes
1 answer
414 views

Proof of Bonnet's Recursion Formula for Legendre Functions of the Second Kind?

I'm doing some self-study on Legendre's Equation. I have seen and understand the proof of Bonnet's Recursion Formula for the Legendre Polynomials, $P_n(x)$. $$(n+1)P_{n+1}(x) = (1+2n)xP_n(x) - nP_{n-...
Chris Duerschner's user avatar
2 votes
0 answers
43 views

Should Hobson, pg183, be corrected, in particular, should an occurrence of $(t^2−1)^n(t−\mu)^{-n-m-1}$ be replaced by $(t^2−1)^{n+1}(t−\mu)^{-n-m-2}$?

The material in this question concerns substitution of a Schlaefli type integral into a differential equation. My answer to the question Showing Schlaefli integral satisfies Legendre equation should ...
user151522's user avatar
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An integral formula involving Bessel functions and an associated Legendre funtion

Table of Integrals, Series, and Products (8th edition)(I.S.Gradshteyn,I.M.Ryzhik,Daniel Zwillinger, and Victor Moll) contains the following formula.(Page 693) 6.578 $\,$6.11 $$\int_{0}^{\infty} x^{\mu+...
hitsu's user avatar
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Finding a closed form for $\int_0^\pi (z+\sqrt{z^2-1} \cos x)^q \cos(nx)\ dx$

I wish to find a closed form expression for the integral: $$\int_0^\pi (z+\sqrt{z^2-1} \cos x)^q \cos(nx)\ dx$$ where $z > 1$ and $q,n \in \mathbb{R}$. Thankfully, there is a closed form result if $...
Patrick.B's user avatar
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a little problem about Rolle theorem

As Rolle theorem goes,if $f(x)$ is continuous and well-defined in $[a,b]$, derivable in $(a,b)$, and $f'(x)$ is bounded, $f(a)=f(b)$, then there exists $c$ ($a<c<b$), which satisfies $f'(c)=0$....
xinyi's user avatar
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1 vote
0 answers
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Linear Elliptic PDE Variable Coefficients Non-Separating Variables

I am trying to obtain Analytical solutions for the following Linear Elliptic PDE in the dependent variable U(x,y) having variable coefficients. 'x' is a (pseudo) radial coordinate, and 'y' is an ...
Prakash_S's user avatar
1 vote
1 answer
652 views

Prove that the Dirac delta function can be presented in terms of the Legendre Polynomials

I tried everything I could, but I can't find a derivation for this equation. Can I get some hints on how to approach to prove the following result? $$\delta(1-x)=\sum_{n=0}^{\infty}{\frac{(2n+1)P_n(x)}...
Luiz Guerra's user avatar
1 vote
1 answer
148 views

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 ...
epselonzero's user avatar
4 votes
1 answer
274 views

Stuck on nasty integral regarding associated Legendre polynomials and spherical Bessel functions.

I'm preparing notes for an undergrad physics course I'm going to be teaching soon. Unfortunately, this sort of stuff was taught to me only in a very handwavy sort of way ("you take these physical ...
Rain's user avatar
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2 votes
1 answer
233 views

Solve $\sum_\limits{n=-\infty}^0\mathrm P_n^n(z)$ with Associated Legendre P functions of type 1

Here is a simple looking sum which should have an alternate form since it is just a double hypergeometric series with the associated Legendre P function of type 1 $\mathrm P_a^b(z)$ The definitions ...
Тyma Gaidash's user avatar
1 vote
0 answers
74 views

Proving that : $Q_0(z)=\frac{1}{2}\ln\left(\frac{z+1}{z-1}\right)$ is single valued.

The Legendre function of second kind $Q_\nu(z)$ has branch points at $z=\pm 1$. The branch points are joined by a cut along the real axis. Show that $$Q_0(z)=\frac{1}{2}\ln\left(\frac{z+1}{z-1}\right)$...
Young Kindaichi's user avatar
3 votes
1 answer
166 views

Help showing the classical Legendre equation has limit circle boundary points.

I am following this paper by Krall and Zettl. I am trying to use the results of Sturm-Louiville (SL) theory to study eigen functions of the classical Legendre equation: $$ \tag1 \frac{d}{dx}\left((1-x^...
valcofadden's user avatar
1 vote
0 answers
42 views

How to compute this integral involving associated Legendre function?

My goal is to compute the following integral, $$\int_{-1}^{1}P^\mu_\nu(x)(1-x^2)^{\mu/2+1/2}dx.$$ I tried to compute this using Wolfram Alpha and Maple but unfortunately did not get any result. ...
Student's user avatar
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1 vote
0 answers
9 views

How to find this limit involving Legendre functions?

I am working with an ODE whose general solution is of the form, $$f(\theta) = \sin(\theta)^{-k}\left[c_1 P_\nu^\mu (\cos(\theta)) + c_2 Q_\nu^\mu(\cos(\theta))\right]$$ where $\mu,\nu\in \mathbb{R}$ (...
Student's user avatar
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4 votes
1 answer
538 views

Intuition behind Legendre convex function

I came across the definition of Legendre functions and Legendre transformations in my studies (in the sense of convex analysis) and I started searching about it. I found a definition in Rockefellar's ...
YetAnotherUsr's user avatar
1 vote
0 answers
127 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 ...
edgardeitor's user avatar
18 votes
0 answers
884 views

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

By experimenting in Mathematica, I have found the following expression for the integral: $$ \int_{b-a}^{b+a}\sigma h_{n}^{(1)}(\sigma)P_{n}^{m}\left(\frac{\sigma^{2}-a^{2}+b^{2}}{2b\sigma}\right)P_{n'}...
Chris's user avatar
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5 votes
1 answer
188 views

How to deduce this Fourier cosine transform of the product of modified Bessel function

I want to know how to deduce $$ \int_0^\infty K_\nu(ax)I_\nu(bx)\cos cxdx=\frac{1}{2\sqrt{ab}}Q_{\nu-\frac12}(\frac{a^2+b^2+c^2}{2ab}) $$ My attempt: I have evaluated $$ \int_0^\infty J_\nu(ax)J_\nu(...
数理课代表蕾米莉亚's user avatar
1 vote
0 answers
84 views

Legendre's Complete Elliptic Integral of the 1st Kind - Calculating the argument

Given the definition of the complete elliptic integral of the 1st kind (see this link here), I am interested in finding a particular value of $k$ such that $$K(k) \equiv \int_0^1\dfrac{\operatorname{d}...
Brad's user avatar
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5 votes
2 answers
457 views

Integral of a product of Legendre polynomials

I would like to show that $$ \int_{-1}^{1}P_{n}^{1}(x)P_{n'}^{0}(x)\frac{x}{\sqrt{1-x^{2}}}\,\mathrm dx = \begin{cases} -\frac{2n}{2n+1},&n=n'>0\\ -2,&n>n'\text{ and } n-n' \text{ even}\\...
Chris's user avatar
  • 469
3 votes
1 answer
496 views

How to calculate the integral of a Legendre polynomial

I would like to show that $$ \int_{0}^{1}P_{l}(1-2u^{2})e^{2i\alpha u}du=i\alpha j_{l}(\alpha)h_{l}(\alpha) $$ where $P_{l}(x)$ are the Legendre polynomials, $\alpha$ is a positive constant and $j_{l}$...
Chris's user avatar
  • 469
1 vote
1 answer
212 views

Series involving product of Legendre polynomials

I need to compute the following sum: $$\sum_{n=0}^{\infty} (4n+3) P_{2n+1}(x)P_{2n+1}(y)$$ where $P_n(x)$ are the Legendre polynomials. Can anyone help me?
ashtar's user avatar
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1 vote
0 answers
109 views

Finding the Legendre Polynomial formula from the Legendre equation

First I took the Legendre equation: $$(1-x^2)\frac{d^2P_n(x)}{dx^2}-2x\frac{dP_n(x)}{dx}+n(n+1)P_n(x)=0$$ Then I wrote: $$P_n(x)=\sum_{k=0}^{n}a_{n, k} x^k$$ Where $a_{n, k}$ just gives the ...
Asv's user avatar
  • 510
4 votes
0 answers
94 views

Series of Legendre Polynomials and Harmonic numbers. $\sum_{n=1}^{\infty} P_n (z) \frac{H(n)}{n+k}$

I would like to compute sums of the type \begin{equation} \sum_{n=1}^{\infty} P_n (z) \frac{H(n)}{n+k} \end{equation} where $P_n(z)$ are Legendre polynomials, $H(n)$ are harmonic numbers and $k = 0, 1,...
Siddhartha Morales's user avatar
1 vote
2 answers
222 views

How can I derive the Legendre function of first kind in terms of the hypergeometric function?

I was reading in Wikipedia about Legendre's differential equation. I was particularly interested in the simple case of the equation given by $$ \left(1-x^2\right)y'' -2xy' + \lambda(\lambda+1)y = 0 \...
Robert Lee's user avatar
  • 7,273
2 votes
1 answer
151 views

Prove $1+\sum_{k=1}^{p} \frac{(-1)^k.n(n-1)(n-2)\cdots(n-2k+1)}{2^k.k!.(2n-1)(2n-3)\cdots(2n-2k+1)}=\frac{2^n(n!)^2}{(2n)!}$

Prove that $$ a_n\bigg[1-\frac{n(n-1)}{2(2n-1)}+\frac{n(n-1)(n-2)(n-3)}{2\cdot4\cdot(2n-1)(2n-3)}-\cdots+\frac{n(n-1)(n-2)\cdots(n-2k+1)}{2\cdot4\cdots 2k\cdot(2n-1)(2n-3)\cdots(2n-2k+1)}\bigg]=1\\ \...
Sooraj S's user avatar
  • 7,674
1 vote
0 answers
62 views

Deriving normalization for the shifted associated Legendre function

Where can I find a solution for this integral: $ \int_{a}^{b} P^m_l(c_1x + c_2b)P^{m'}_l(c_1x + c_2b)\,d(c_1x + c_2b)$, most solutions only solves for the interval [-1,1]. Of course I am looking for ...
Mahmoud S. M. Shaqfa's user avatar
2 votes
0 answers
67 views

How to find the associated Legendre functions

Reading the Courant-Hilbert Methods of Mathematical Physics (p.326) we encounter: "...If we differentiate equation $$ \left[(1-x^2)u')\right]'+\lambda u=0 $$ with respect to $x$, we obtain a ...
Emilio Novati's user avatar
2 votes
0 answers
66 views

Is this task impossible?

I'm dealing with what I can only assume is an impossible problem. I want to find the coefficients such that $$x^2+2y^2+3z^2=\sum_{l=0}^\infty \sum_{m=-l}^l a_{lm}\rho^l P_l^m (cos(\theta))e^{im\phi} |...
Algebraic's user avatar
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2 votes
0 answers
82 views

Convergence of a serie to obtain the eigenvalues of general Legendre operator

We have the Legendre operator, $$ \mathcal{L}u = - \frac{d}{dx}\bigg[ (1-x^2)\frac{du}{dx}\bigg], $$ and we want to find its eigenvalues and eigenfunctions, therefore we must solve the following ODE, $...
Scottish Questions's user avatar
2 votes
0 answers
453 views

Legendre Polynomials proving a relation using Rodrigue's Formula

I have been trying to prove the relation$\left (P'_n(1)=\frac{1}{2}n(n+1)\right )$ and have proved it directly from the Legendre differential equation as $P_n(x)${the Legendre polynomial} is the ...
Jack's user avatar
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1 vote
0 answers
109 views

Derivation of the relation between Associated Legendre Function and Gauss Hypergeometric Function

I was trying to derive the Hobson's relation between the associated Legendre function and Gauss Hypergeometric function. This is given by, $$P^m_n(z)=\frac{1}{\Gamma(1-m)}\left(\frac{z+1}{z-1}\right)^{...
Nothingham's user avatar
5 votes
1 answer
207 views

Showing a summation identity for $1$, possibly tied to Legendre polynomials

The Problem: Consider the sign function on $(-1,0)\cup(0,1)$ defined by $$ \sigma(x) := \left. \text{sgn}(x) \right|_{(-1,0)\cup(0,1)} = \begin{cases} 1 & x \in (0,1) \\ -1 & x \in (-1,0) \end{...
PrincessEev's user avatar
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2 votes
1 answer
92 views

Does Associated Legendre Function of Second Kind Give Delta Function?

Associated Legendre Function of Second Kind is singular at $x=\pm 1$. So I am wondering whether it satisfies the corresponding differential equation everywhere or there is a hidden functional of delta ...
tadashi's user avatar
  • 21
0 votes
1 answer
2k views

How to express in Legendre's polynomials?

How do I express $cos(3\theta)$ and $sin^{2}(\theta)$ in Legendre's polynomials, knowing that $x=cos\theta$? I know that $f(x)=\sum a_{n}P_{n}(x)$ and $P_{n}=\frac{(-1)^{n}}{2^{n}n!}\frac{d^{n}}{dx^{n}...
random name's user avatar
2 votes
1 answer
591 views

Integrating odd Legendre polynomials using generating function

I must show using generating function of Legendre polynomials, that \begin{align} \int_0^1 P_{2n+1}(x)\phantom{1}dx = (-1)^n\frac{(2n)!}{2^{2n+1}n!(n+1)!} \end{align} My attempt is to change the ...
Jean P.'s user avatar
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