# What is Legendre polynomials: Definition and 89 Discussions

In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications.
Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions.

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1. ### I Expansion of 1/|x-x'| into Legendre Polynomials

we know that we can expand the following function in Legendre polynomials in the following way in the script given yo us by my professor, ##\frac 1 {|\vec x -\vec x'|}## is expanded using geometric series in the following way: However, I don't understand how ##\frac 1 {|\vec x -\vec...
2. ### Potential of a charged ring in terms of Legendre polynomials

hi guys I am trying to calculate the the potential at any point P due to a charged ring with a radius = a, but my answer didn't match the one on the textbook, I tried by using $$V = \int\frac{\lambda ad\phi}{|\vec{r}-\vec{r'}|}$$ by evaluating the integral and expanding denominator in terms of...

11. ### I Legendre polynomials in boosted temperature approximation

Hi all, In S. Weinberg's book "Cosmology", there is a derivation of the slightly modified temperature of the cosmic microwave background as seen from the Earth moving w.r.t. a frame at rest in the CMB. On Page 131 (1st printing), an approximation (Formula 2.4.7) is given in terms of Legendre...
12. ### Legendre polynomials, Hypergeometric function

Homework Statement _2F_1(a,b;c;x)=\sum^{\infty}_{n=0}\frac{(a)_n(b)_n}{(c)_nn!}x^n Show that Legendre polynomial of degree ##n## is defined by P_n(x)=\,_2F_1(-n,n+1;1;\frac{1-x}{2}) Homework Equations Definition of Pochamer symbol[/B] (a)_n=\frac{\Gamma(a+n)}{\Gamma(a)} The Attempt at a...
13. ### MHB Problem about Rodrigues' formula and Legendre polynomials

using Rodrigues' formula show that \int_{-1}^{1} \,{P}_{n}(x){P}_{n}(x)dx = \frac{2}{2n+1} {P}_{n}(x) = \frac{1}{2^nn!}\frac{d^n}{dx^n}(x^2-1)^n my thoughts \int_{-1}^{1} \,{P}_{n}(x){P}_{n}(x)dx = \frac{1}{2^{2n}(n!)^2}\int_{-1}^{1} \,\frac{d^n}{dx^n}(x^2-1)^n\frac{d^n}{dx^n}(x^2-1)^ndx...
14. ### MHB What is Associated Legendre polynomials

hey i have doubt about Legendre polynomials and Associated Legendre polynomials what is Associated Legendre polynomials ? It different with Legendre polynomials ?
15. ### I Defining Legendre polynomials in (1,2)

Hello everyone. The Legendre polynomials are defined between (-1 and 1) as 1, x, ½*(3x2-1), ½*(5x3-3x)... My question is how can I switch the domain to (1, 2) and how can I calculate the new polynomials. I need them to construct an estimation of a random uniform variable by chaos polynomials...
16. ### Understanding the Legendre Recurrence Relation for Generating Functions

Homework Statement I am having a slight issue with generating function of legendre polynomials and shifting the sum of the genertaing function. So here is an example: I need to derive the recurence relation ##lP_l(x)=(2l-1)xP_{l-1}(x)-(l-1)P_{l-2}## so I start with the following equation...
17. ### A Problems with identities involving Legendre polynomials

I am studying the linear oscillation of the spherical droplet of water with azimuthal symmetry. I have written the surface of the droplet as F=r-R-f(t,\theta)\equiv 0. I have boiled the problem down to a Laplace equation for the perturbed pressure, p_{1}(t,r,\theta). I have also reasoned that...

24. ### I Legendre polynomials and Rodrigues' formula

Source: http://www.nbi.dk/~polesen/borel/node4.html#1 Differentiating this equation we get the second order differential eq. for fn, (1-x^2)f''_n-2(n-1)xf'_n+2nf_n=0 ....(22) But when I differentiate to 2nd order, I get this instead, (1-x^2)f''_n+2(n-1)xf'_n+2nf_n=0Applying General Leibniz...

29. ### MHB Proving Self-Adjoint ODE for Legendre Polynomials

(I haven't encountered these before, also not in the book prior to this problem or in the near future ...) Show that the 1st derivatives of the legendre polynomials satisfy a self-adjoint ODE with eigenvalue $\lambda = n(n+1)-2$ Wiki shows a table of poly's , I don't think this is what the...
30. ### Solving Laplace's equation in spherical coordinates

The angular equation: ##\frac{d}{d\theta}(\sin\theta\,\frac{d\Theta}{d\theta})=-l(l+1)\sin\theta\,\Theta## Right now, ##l## can be any number. The solutions are Legendre polynomials in the variable ##\cos\theta##: ##\Theta(\theta)=P_l(\cos\theta)##, where ##l## is a non-negative integer...
31. ### Legendre polynomials in the reverse direction

I have just written a program to calculate Legendre Polynomials, finding for Pl+1 using the recursion (l+1)Pl+1 + lPl-1 - (2l+1).x.Pl=0 That is working fine. The next section of the problem is to investigate the recursive polynomial in the reverse direction. I would solve this for Pl-1 in...
32. ### Visualizing legendre polynomials in the hydrogen atom.

1. The way we solved this problem was proposing that the wave function has to form of ##\Psi=\Theta\Phi R## where the three latter variables represent the anlge and radius function which are independent. The legendre polynomials were the solution to the ##\Theta## part. I am having some trouble...
33. ### Associated Legendre polynomials

m=1 and l=1 x = cos(θ) What would be the solution to this? Thanks.
34. ### Hermite and Legendre polynomials

Hi, I am just curious, are Hermite and Legendre polynomials related to one another? From what I have learned so far, I understand that they are both set examples of orthogonal polynomials...so I am curious if Hermite and Legendre are related to one another, not simply as sets of orthogonal...
35. ### How to derive Legendre Polynomials?

Homework Statement Could someone explain how Legendre polynomials are derived, particularly first three ones? I was only given the table in the class, not steps to solving them...so I am curious. Homework Equations P0(x) = 1 P1(x) = x P2(x) = 1/2 (3x2 - 1) The Attempt at a Solution ...
36. ### Details regarding Legendre Polynomials

I just had a few questions not directly addressed in my textbook, and they're a little odd so I thought I would ask, if you don't mind. :) -Firstly, I was just wondering, why is it that Legendre polynomials are only evaluated on a domain of {-1. 1]? In realistic applications, is this a limiting...
37. ### Legendre Polynomials - how to find P0(u) and P2(u)?

Pl(u) is normalized such that Pl(1) = 1. Find P0(u) and P2(u) note: l, 0 and 2 are subscript recursion relation an+2 = [n(n+1) - l (l+1) / (n+2)(n+1)] an n is subscript substituted λ = l(l+1) and put n=0 for P0(u) and n=2 for P2(u), didnt get very far please could someone...
38. ### Derivation: Normalization condition of Legendre polynomials

Greetings! :biggrin: Homework Statement Starting from the Rodrigues formula, derive the orthonormality condition for the Legendre polynomials: \int^{+1}_{-1} P_l(x)P_{l'}(x)dx=(\frac{2}{2l + 1}) δ_{ll'} Hint: Use integration by parts Homework Equations P_l=...
39. ### Electric field and Legendre Polynomials

Homework Statement I want to varify that the components of a homogenous electric field in spherical coordinates \vec{E} = E_r \vec{e}_r + E_{\theta} \vec{e}_{\theta} + E_{\varphi} \vec{e}_{\varphi} are given via: E_r = - \sum\limits_{l=0}^\infty (l+1) [a_{l+1}r^l P_{l+1}(cos \theta) - b_l...
40. ### First derivative of the legendre polynomials

show that the first derivative of the legendre polynomials satisfy a self-adjoint differential equation with eigenvalue λ=n(n+1)-2 The attempt at a solution: (1-x^2 ) P_n^''-2xP_n^'=λP_n λ = n(n + 1) - 2 and (1-x^2 ) P_n^''-2xP_n^'=nP_(n-1)^'-nP_n-nxP_n^' ∴nP_(n-1)^'-nP_n-nxP_n^'=(...
41. ### MHB Understanding Legendre Polynomials: A Guide for Students

Does anyone understand this project? I desperately need your help! Please let me know. Appreciate a lot!
42. ### Using Legendre Polynomials in Electro

Homework Statement A conducting spherical shell of radius R is cut in half and the two halves are infinitesimally separated (you can ignore the separation in the calculation). If the upper hemisphere is held at potential V0 and the lower half is grounded find the approximate potential for...
43. ### MHB Legendre Polynomials: Pattern Analysis & Integration

Consider $f(x) = \begin{cases} 1, & 0\leq x\leq 1\\ -1, & -1\leq x\leq 0 \end{cases}$ Then $c_n = \frac{2n + 1}{2}\int_{0}^1\mathcal{P}_n(x)dx - \frac{2n + 1}{2}\int_{-1}^0\mathcal{P}_n(x)dx$ where $$\mathcal{P}_n(x)$$ is the Legendre Polynomial of...
44. ### Legendre polynomials and Bessel function of the first kind

Homework Statement Prove that \sum_{n=0}^{\infty}{\frac{r^n}{n!}P_{n}(\cos{\theta})}=e^{r\cos{\theta}}J_{0}(r\sin{\theta}) where P_{n}(x) is the n-th legendre polynomial and J_{0}(x) is the first kind Bessel function of order zero. Homework Equations...
45. ### Relationship between Legendre polynomials and Hypergeometric functions

Homework Statement If we define \xi=\mu+\sqrt{\mu^2-1}, show that P_{n}(\mu)=\frac{\Gamma(n+\frac{1}{2})}{n!\Gamma(\frac{1}{2})}\xi^{n}\: _2F_1(\frac{1}{2},-n;\frac{1}{2}-n;\xi^{-2}) where P_n is the n-th Legendre polynomial, and _2F_1(a,b;c;x) is the ordinary hypergeometric function...
46. ### Expanding an f(x) in terms of Legendre Polynomials

Homework Statement Expand f(x) = 1 - x2 on -1 < x < +1 in terms of Legendre polynomials. Homework Equations The Attempt at a Solution Unfortunately, I missed the class where this was explained and I have other classes during my professor's office hours. I have no idea how to begin this...
47. ### Associated Legendre polynomials for negative order

Homework Statement I just need to deduce the expression for the associated Legendre polynomial P_{n}^{-m}(x) using the Rodrigues' formula Homework Equations Rodrigues formula reads P_{n}(x)=\frac{1}{2^{n}n!}\frac{d^n}{dx^n}(x^2-1)^n and knowing that...
48. ### Integral involving product of derivatives of Legendre polynomials

Anyone how to evaluate this integral? \int_{-1}^{1} (1-x^2) P_{n}^{'} P_m^{'} dx , where the primes represent derivative with respect to x ? I tried using different recurrence relations for derivatives of the Legendre polynomial, but it didn't get me anywhere...
49. ### Legendre polynomials and binomial series

Homework Statement Where P_n(x) is the nth legendre polynomial, find f(n) such that \int_{0}^{1} P_n(x)dx = f(n) {1/2 \choose k} + g(n)Homework Equations Legendre generating function: (1 - 2xh - h^2)^{-1/2} = \sum_{n = 0}^{\infty} P_n(x)h^n The Attempt at a Solution I'm not sure if that...
50. ### Recurrence relations for Associated Legendre Polynomials

Homework Statement I'm working on problem 6.11 in Bransden and Joachain's QM. I have to prove 4 different recurrence relations for the associate legendre polynomials. I have managed to do the first two, but can't get anywhere for the last 2 Homework Equations Generating Function: T(\omega...