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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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)}...
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21 views

Discontinuity in Legendre Function of the Second Kind

The Legendre function of the first kind, $P_{\nu }(z)$, is usually defined in a way that it has a branch cut along the segment ($-\infty <z\leq -1$] while the Legendre function of the second kind,$...
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1 answer
35 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 ...
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0 votes
0 answers
33 views

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
4 votes
1 answer
90 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 ...
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40 views

Solve $\sum_{n=0}^\infty \text P_{-n}^{-n}(z)$ and $\sum_{n=0}^\infty \mathsf P_{-n}^{-n}(z$) with Associated Legendre P functions of type $1$ and $3$

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 the First (aka Second) Type $\text P_a^b(z)$ ...
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  • 5,259
1 vote
0 answers
47 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)$...
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3 votes
1 answer
68 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^...
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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. ...
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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 $\mu,\nu\in \mathbb{R}$ (...
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2 votes
1 answer
62 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 ...
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0 answers
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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
46 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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1 vote
0 answers
53 views

Generating function for Legendre functions of second kind with half integer order

Good morning, I am looking for a generating function $f(t,z)$ for the Legendre functions of second kind $Q_{n-\frac{1}{2}}(z)$. $$f(t,z) = \sum_{n=0}^{\infty} t^n Q_{n-\frac{1}{2}}(z) $$ Can anyone ...
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2 votes
0 answers
60 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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  • 31
8 votes
0 answers
307 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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  • 239
5 votes
1 answer
133 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(...
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0 votes
0 answers
31 views

Associated Legendre Function Limit Issue

I've been attempting to implement the associated Legendre functions and I was using the recurrence relations shown below. $(n-m+1)P_{n+1}^{m}(\cos(\theta) )-(2n+1)\cos(\theta )P_{n}^{m}(\cos(\theta) )+...
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  • 101
1 vote
0 answers
37 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}...
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  • 65
5 votes
2 answers
256 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}\\...
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  • 239
3 votes
1 answer
251 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 $j_{l}$...
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  • 239
1 vote
1 answer
62 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?
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  • 33
1 vote
0 answers
56 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 ...
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  • 472
4 votes
0 answers
76 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,...
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1 vote
2 answers
76 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 \...
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  • 5,030
2 votes
1 answer
134 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\\ \...
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  • 6,801
1 vote
0 answers
36 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 ...
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2 votes
0 answers
38 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 ...
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2 votes
0 answers
65 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} |...
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  • 453
1 vote
0 answers
28 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, $...
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2 votes
0 answers
130 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 ...
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1 vote
0 answers
41 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)^{...
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5 votes
1 answer
140 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{...
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2 votes
1 answer
42 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 ...
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  • 21
0 votes
1 answer
563 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}...
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2 votes
1 answer
366 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 ...
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  • 35
0 votes
0 answers
43 views

Monotonicity of Legendre functions (of first kind) with respect to theirs order

I will denote by $P^m_l(x)$ the Legendre first kind function of order $m$ and of degree $l$, $m$ is a positive integer and $l$ is real number. I fix the value of $x\in[0,1]$ and $l=-0.5$. Is there any ...
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  • 41
0 votes
1 answer
108 views

How to prove that Spherical Harmonics must have integer order $m$ and degree $n$?

Spherical harmonics : \begin{equation} Y_{m}^{n}(\theta,\phi) = N_{}\mathcal{P}_{m}^{n}(\cos \phi) e^{in\theta}, \end{equation} are the solution of the two angular equations (Legendre associated eq ...
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0 votes
1 answer
244 views

Convex envelope of a function

I am trying to calculate the convex envelope of a function, which is calculated by doing the Legendre transform twice (right?). Therefore, I was trying to calculare the convex envelope of $f(x)=(x^2-1)...
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2 votes
1 answer
64 views

By changing the variable in the φ equation to x = cos φ, derive the self adjoint form of the Legendre equation

We have $$\frac{d}{d\phi}(sin\phi \frac{dP}{d\phi}) + \lambda sin\phi P =0 $$ By changing the variable in the equation to x = cos $\phi$, derive the self adjoint form of the Legendre equation: $$\...
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1 vote
1 answer
67 views

Can't retrieve Legendre equation from Laplace equation

I am tasked with the above, converting the 2-dimensional polar form of the Laplace transform: $$ \frac{\partial}{\partial r}\Bigl(r^2 \frac{\partial z}{\partial r}\Bigr) + \frac{1}{\sin(\phi)} \frac{...
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0 votes
1 answer
165 views

derivative of normalized associated Legendre function at the limits of x = +/-1

I'm trying to compute the derivative of the normalized associated Legendre function of $x=\cos \theta$ and I'm having issues when $\cos \theta =\pm 1$ which causes the denominator to go to $0$. I've ...
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  • 101
2 votes
0 answers
39 views

find $a_L$ in $ \frac{Y_{\ell}^{m}Y_{\ell'}^{-m}}{\sin^2\theta}=\sum_{L=0}^{\ell+\ell'}a_LY_L^0$

Can you prove that $$ \frac{Y_{\ell}^{m}Y_{\ell'}^{-m}}{\sin^2\theta}=\sum_{L=0}^{\ell+\ell'}a_LY_L^0~, $$ where $Y_{\ell}^{m}$ denotes the spherical harmonics of degree $\ell$ and order $m$ (both are ...
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  • 121
1 vote
1 answer
102 views

How to convert a hypergeom function to the Legendre function?

Anyone can help me to convert the following maple pdsolve expressed by the hypergeom function to the $LegendreP(n,b,x)$ or $Q$ function? \begin{equation} dsolve\Big( (1-x^2)\cdot \frac{d^2 y(x)}{dx^...
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  • 55
0 votes
1 answer
78 views

Transforming an ODE into Legendre's Equation

I am trying to transform the ODE $$\frac{1}{\sin(\theta)}\frac{d}{d\theta}\left(\sin(\theta)\frac{dS}{d\theta}\right)+\lambda S=0,$$ in to Legendre's equation $$(1-\mu^2)\frac{d^2S}{d\mu^2}-2\mu\frac{...
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0 votes
1 answer
245 views

Integrating multiple products of order 1 Legendre functions

I would like a closed form expression for the integrals of some products like the following: $$ \int_{-1}^1 P^1_j P^1_k P^1_l P^1_m dx $$ and $$ \int_{-1}^1 P^0_j P^0_k P^1_l P^1_m dx, $$ where $P^...
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3 votes
0 answers
113 views

An upper bound of the first eigenvalue of Laplacian on a Riemannian manifold.

I'm reading the Cheng's thesis ""Eigenvalue Comparison Theorems and Its Geometric Applications," and the author obtains an estimate of eigenvalues of the Laplacian based upon his theorem: If $M$ is $...
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  • 111
1 vote
1 answer
385 views

Series Legendre Polynomial

We have to prove $$\ln \Big(\frac{x+1}{1-x}\Big)=\sum_{n≥0} \frac{x^{n+1}}{n+1}P_n(x)$$ using the generatrix function of Legendre polynoms. I don't know if it is useful, but $$\int_{-1}^{1}\frac{1}{...
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1 vote
0 answers
38 views

Equivalence of spherical harmonic definitions

In Griffith's "Introduction to quantum mechanics", the spherical harmonics are defined for integers $l$ and $-l\leq m\leq l$ as $$Y_{l}^m(\theta,\phi) = \epsilon \sqrt{\frac{2l+1}{4\pi}\frac{(l-|m|)!}...
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  • 598
1 vote
1 answer
441 views

Sum of associated Legendre functions

I want to find the sums of two expressions involving the Schmidt-normalized associated Legendre functions. They are defined by \begin{align} S_l^0(x) &= P_l^0(x) \\ S_l^m(x) &= \sqrt{2 \frac{(...
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