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. deuteron

    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. patric44

    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...
  3. R

    Expanding potential in Legendre polynomials (or 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 polynomials (or even spherical harmonic ) like this: $$ \begin{aligned} &\frac{1}{\left.\mid...
  4. L

    Prove eigenvalues of the derivatives of Legendre polynomials >= 0

    The problem has a hint about finding a relationship between ##\int_{-1}^1 (P^{(k+1)}(x))^2 f(x) dx## and ##\int_{-1}^1 (P^{(k)}(x))^2 g(x) dx## for suitable ##f, g##. It looks they're the weighting functions in the Sturm-Liouville theory and we may be able to make use of Parseval's identity...
  5. P

    Legendre Polynomials as an Orthogonal Basis

    If you had legendre polynomials defined in ##L^2([-1,1])##, with ##||Pn_2||^2=\frac{2}{2n+1}##, show that for any polynomial with p a set of ##L^2([-1,1])##, with degree less than n, we have the inner product of ##P_n## and p = 0. Find the polynomials ##P_0,... P_4## Tried to use the integral...
  6. L

    Separation of Variables: Find the potential b/w concentric hemispheres

    I'm having troubles setting up this problem. I know we are to use boundary conditions to determine An and Bn since in this case (a<r<b) neither can be set to 0. I don't know how the given potentials translate into boundary conditions, especially the V3 disk.
  7. LCSphysicist

    Orthogonality Relationship for Legendre Polynomials

    Suppose p = a + bx + cx². I am trying to orthogonalize the basis {1,x,x²} I finished finding {1,x,x²-(1/3)}, but this seems different from the second legendre polynomial. What is the problem here? I thought could be the a problem about orthonormalization, but check and is not.
  8. Vick

    A Associated Legendre Function of Second Kind

    The associated Legendre Function of Second kind is related to the Legendre Function of Second kind as such: $$ Q_{n}^m(z)= (-1)^m (1-z^2)^{m/2} \frac{d^m}{dz^m}(Q_{n}(z)) $$ The recurrence relations for the former are the same as those of the first kind, for which one of the relations is: $$...
  9. CrosisBH

    I Solving an ODE with Legendre Polynomials

    From Griffiths E&M 4th edition. He went over solving a PDE using separation of variables. It got to this ODE \frac{d}{d\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right)= -l(l+1)\sin \theta \Theta Griffths states that this ODE has the solution \Theta = P_l(\cos\theta) Where $$P_l =...
  10. migueldbg

    Potential from point charge at distance ##l## from conducting sphere

    After looking around a bit, I found that, considering the polar axis to be along the direction of the point charge as suggested by the exercise, the following Legendre polynomial expansion is true: $$\begin{equation}\frac{1}{|\mathbf{r} - \mathbf{r'}|} = \sum_{n=0}^\infty...
  11. J

    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. L

    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. A

    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. A

    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. C

    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. T

    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. H

    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...
  18. Alan Sammarone

    I Integration of Legendre Polynomials with different arguments

    Hi everybody, I'm trying to calculate this: $$\sum_{l=0}^{\infty} \int_{\Omega} d\theta' d\phi' \cos{\theta'} \sin{\theta'} P_l (\cos{\gamma})$$ where ##P_{l}## are the Legendre polynomials, ##\Omega## is the surface of a sphere of radius ##R##, and $$ \cos{\gamma} = \cos{\theta'}...
  19. E

    I How do you Calculate the Points in Gaussian Quadrature?

    How do you calculate the necessary points in a function to numerically integrate it using the Gaussian Quadrature? If I were to evaluate a function using two points, the Gaussian Quadrature needs the value of the function at ##\displaystyle{\pm \sqrt{\frac{1}{3}}}## with weights of unity. How...
  20. R

    A Legendre Polynomials -- Jackson Derivation

    Hello all, I'm reading through Jackson's Classical Electrodynamics book and am working through the derivation of the Legendre polynomials. He uses this ##\alpha## term that seems to complicate the derivation more and is throwing me for a bit of a loop. Jackson assumes the solution is of the...
  21. sitkican

    Multipole Expansion of a Thin rod

    Homework Statement Consider a very thin rod lying on the z axis from z = −L/2 to z = L/2. It carries a uniform charge density λ. Show that away from the rod, at the point r (r >>L), the potential can be written as V (r, θ) = (2Lλ/4πε0)(1/L)[ 1 + 1/3(L/2r)2P2(cos θ) + 1/3(L/2r)4 P4(cos θ) + · ·...
  22. dykuma

    Legendre Polynomials & the Generating function

    Homework Statement Homework Equations and in chapter 1 I believe that wanted me to note that The Attempt at a Solution For the first part of this question, as a general statement, I know that P[2 n + 1](0) = 0 will be true as 2n+1 is an odd number, meaning that L is odd, and so the Legendre...
  23. T

    I Legendre Differential Equation

    I just started learning Legendre Differential Equation. From what I learn the solutions to it is the Legendre polynomial. For the legendre DE, what is the l in it? Is it like a variable like y and x, just a different variable instead? Legendre Differential Equation: $$(1-x^2) \frac{d^2y}{dx^2}...
  24. T

    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...
  25. avikarto

    I Associated Legendre polynomials: complex vs real argument

    I am having trouble understanding the relationship between complex- and real-argument associated Legendre polynomials. According to Abramowitz & Stegun, EQ 8.6.6, $$P^\mu_\nu(z)=(z^2-1)^{\mu/2}\cdot\frac{d^\mu P_\nu(z)}{dz^\mu}$$ $$P^\mu_\nu(x)=(-1)^\mu(1-x^2)^{\mu/2}\cdot\frac{d^\mu...
  26. W

    Legendre polynomial of zero

    Homework Statement Using the Generating function for Legendre polynomials, show that: ##P_n(0)=\begin{cases}0 & n \ is \ odd\\\frac{(-1)^n (2n)!}{2^{2n} (n!)^2} & n \ is \ even\end{cases}## Homework Equations Generating function: ##(1-2xt+t^2)^{-1/2}=\displaystyle\sum\limits_{n=0}^\infty...
  27. H

    Proof of the Rodrigues formula for the Legendre polynomials

    How is the below expression for ##a_{n-2k}## motivated? I verified that the expression for ##a_{n-2k}## satisfies the recurrence relation by using ##j=n-2k## and ##j+2=n-2(k-1)## (and hence a similar expression for ##a_{n-2(k-1)}##), but I don't understand how it is being motivated. Source...
  28. H

    Verify the Rodrigues formula of the Legendre polynomials

    How does (6.79) satisfy (6.70)? After substitution, I get $$(1-w^2)\frac{d^{l+2}}{dw^{l+2}}(w^2-1)^l-2w\frac{d^{l+1}}{dw^{l+1}}(w^2-1)^l+l(l+1)\frac{d^{l}}{dw^{l}}(w^2-1)^l$$ Using product rule in reverse on the first two terms...
  29. ognik

    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. H

    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. ognik

    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. Coffee_

    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. B

    Associated Legendre polynomials

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

    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. T

    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. M

    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. R

    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. schrodingerscat11

    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. O

    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. K

    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. Y

    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. retro10x

    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. D

    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. R

    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. R

    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. M

    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. R

    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. H

    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. L

    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. C

    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...