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.
61
questions
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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:
$$\int_{0}^{2\pi} \int_{0}^{\pi} Y_{lm}^2(\theta, \phi)Y_{l'm'}...
- 19
0
votes
0
answers
32
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First derivative of the associated Legendre function at x=1
I'm trying to find the first derivative of the associated Legendre function at $x=1$.
The form I have for the first derivative is divergent at x=1:
\begin{equation}
\frac{d P_l^m(x)}{d \theta}=\frac{l ...
- 1
2
votes
1
answer
64
views
Seeking the generating function of $\left( 1+\epsilon^2 - 2\epsilon x \right)^{-1}$
Many physical problems can be effectively solved using series expansions, such as the utilization of Legendre polynomials. Consider the function $\left( 1+\epsilon^2 - 2\epsilon x \right)^\frac{1}{2}$....
- 395
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2
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Does this Fourier Legendre series inverting $\frac65\frac{x^x}{x+1}-\frac35$ diverge or it is a software issue past $\approx 25$ series terms?
$\def\P{\operatorname P}$
The goal of the Fourier Legendre series is to find an exact explicit series solution to $x^x=x+1$, not just $x=a+b+c+\dots$ with unknown series coefficients. Lagrange ...
- 10.6k
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26
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Establishing Recurrence Relation for Legendre Polynomials using Rodrigues' Formula [duplicate]
Hello Math Stack Exchange community,
I am currently studying Legendre Polynomials and I encountered a recurrence relation that I would like to derive from Rodrigues' Formula. The recurrence relation ...
- 139
1
vote
0
answers
50
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Asymptotic expansion of Legendre functions at $x=-1$
Let $P_{\nu}(z),\,-1<z\leq1,\,\nu\in\mathbb{C}$ Legendre functions of the first kind defined as $$P_{\nu}(z)=\,_{2}F_{1}\left(-\nu,\nu+1;1;\frac{1-z}{2}\right).$$ I found this formula on Wolfram ...
- 11
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80
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Integration of a square of Conical (Mehler) function
I want to evaluate
$$\int_{\cos\theta}^1 \left( P_{-1/2+i\tau}(x) \right)^2 dx,$$
where $P$ is the Legendre function of the first kind, $i$ is the imaginary unit, and $\tau$ is a real number.
Are ...
- 23
2
votes
1
answer
316
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Derivation of an integral containing the complete elliptic integral of the first kind
This is a repost of mathoverflow to draw broader attentions.
https://mathoverflow.net/questions/439770/derivation-of-an-integral-containing-the-complete-elliptic-integral-of-the-first
I found the ...
- 23
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0
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188
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Confusion about spherical harmonics, Legendre polynomials
I'm quite new to the ideas behind spherical harmonics and Legendre polynomials. I have a couple of questions about them.
Spherical harmonics, as I understand them, are functions that can be used to ...
- 87
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0
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67
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How to integrate products of Legendre functions over the interval [0,1]
The associated Legendre polynomials are known to be orthogonal in the sense that
$$
\int_{-1}^{1}P_{k}^{m}(x)P_{l}^{m}(x)dx=\frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{k,l}
$$
This is intricately linked to ...
- 425
2
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1
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276
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Proof of Bonnet's Recursion Formula for Legendre Functions of the Second Kind?
I'm doing some self-study on Legendre's Equation. I have seen and understand the proof of Bonnet's Recursion Formula for the Legendre Polynomials, $P_n(x)$.
$$(n+1)P_{n+1}(x) = (1+2n)xP_n(x) - nP_{n-...
2
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42
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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 ...
- 457
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0
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60
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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
$$\int_{0}^{\infty} x^{\mu+...
- 85
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0
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54
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Finding a closed form for $\int_0^\pi (z+\sqrt{z^2-1} \cos x)^q \cos(nx)\ dx$
I wish to find a closed form expression for the integral:
$$\int_0^\pi (z+\sqrt{z^2-1} \cos x)^q \cos(nx)\ dx$$
where $z > 1$ and $q,n \in \mathbb{R}$. Thankfully, there is a closed form result if $...
- 194
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82
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a little problem about Rolle theorem
As Rolle theorem goes,if
$f(x)$ is continuous and well-defined in $[a,b]$,
derivable in $(a,b)$, and $f'(x)$ is bounded,
$f(a)=f(b)$,
then there exists $c$ ($a<c<b$), which satisfies $f'(c)=0$....
- 13
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66
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Linear Elliptic PDE Variable Coefficients Non-Separating Variables
I am trying to obtain Analytical solutions for the following Linear Elliptic PDE in the dependent variable U(x,y) having variable coefficients. 'x' is a (pseudo) radial coordinate, and 'y' is an ...
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1
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485
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Prove that the Dirac delta function can be presented in terms of the Legendre Polynomials
I tried everything I could, but I can't find a derivation for this equation. Can I get some hints on how to approach to prove the following result?
$$\delta(1-x)=\sum_{n=0}^{\infty}{\frac{(2n+1)P_n(x)}...
- 21
1
vote
1
answer
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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 ...
- 13
4
votes
1
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193
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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 ...
- 225
2
votes
1
answer
215
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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 ...
- 10.6k
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72
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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
$$Q_0(z)=\frac{1}{2}\ln\left(\frac{z+1}{z-1}\right)$...
- 506
3
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1
answer
133
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Help showing the classical Legendre equation has limit circle boundary points.
I am following this paper by Krall and Zettl. I am trying to use the results of Sturm-Louiville (SL) theory to study eigen functions of the classical Legendre equation:
$$
\tag1 \frac{d}{dx}\left((1-x^...
- 533
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42
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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. ...
- 9,126
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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 $\mu,\nu\in \mathbb{R}$ (...
- 9,126
3
votes
1
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307
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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 ...
- 526
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100
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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 ...
- 157
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832
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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:
$$
\int_{b-a}^{b+a}\sigma h_{n}^{(1)}(\sigma)P_{n}^{m}\left(\frac{\sigma^{2}-a^{2}+b^{2}}{2b\sigma}\right)P_{n'}...
- 425
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How to deduce this Fourier cosine transform of the product of modified Bessel function
I want to know how to deduce
$$
\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(...
- 299
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0
answers
75
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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
$$K(k) \equiv \int_0^1\dfrac{\operatorname{d}...
- 65
5
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2
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416
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Integral of a product of Legendre polynomials
I would like to show that
$$
\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}\\...
- 425
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1
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429
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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 $j_{l}$...
- 425
1
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1
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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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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 ...
- 500
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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,...
1
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2
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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
$$
\left(1-x^2\right)y'' -2xy' + \lambda(\lambda+1)y = 0 \...
- 6,757
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Prove $1+\sum_{k=1}^{p} \frac{(-1)^k.n(n-1)(n-2)\cdots(n-2k+1)}{2^k.k!.(2n-1)(2n-3)\cdots(2n-2k+1)}=\frac{2^n(n!)^2}{(2n)!}$
Prove that $$
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\\
\...
- 7,495
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0
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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 ...
2
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0
answers
55
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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
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0
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66
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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
$$x^2+2y^2+3z^2=\sum_{l=0}^\infty \sum_{m=-l}^l a_{lm}\rho^l P_l^m (cos(\theta))e^{im\phi} |...
- 463
2
votes
0
answers
77
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Convergence of a serie to obtain the eigenvalues of general Legendre operator
We have the Legendre operator,
$$
\mathcal{L}u = - \frac{d}{dx}\bigg[ (1-x^2)\frac{du}{dx}\bigg],
$$
and we want to find its eigenvalues and eigenfunctions, therefore we must solve the following ODE,
$...
2
votes
0
answers
361
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Legendre Polynomials proving a relation using Rodrigue's Formula
I have been trying to prove the relation$\left (P'_n(1)=\frac{1}{2}n(n+1)\right )$ and have proved it directly from the Legendre differential equation as $P_n(x)${the Legendre polynomial} is the ...
- 87
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vote
0
answers
93
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Derivation of the relation between Associated Legendre Function and Gauss Hypergeometric Function
I was trying to derive the Hobson's relation between the associated Legendre function and Gauss Hypergeometric function. This is given by,
$$P^m_n(z)=\frac{1}{\Gamma(1-m)}\left(\frac{z+1}{z-1}\right)^{...
- 171
5
votes
1
answer
196
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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
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1
answer
84
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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 ...
- 21
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votes
1
answer
1k
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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}...
- 83
2
votes
1
answer
541
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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 ...
- 35
0
votes
0
answers
48
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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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votes
1
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305
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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 ...
- 1
0
votes
1
answer
526
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)...
user570048
2
votes
1
answer
145
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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 $$
By changing the variable in the equation to x = cos $\phi$, derive the self adjoint form of the Legendre equation:
$$\...
- 45