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. D

    Why Does the Integral of Legendre Polynomials Yield a Kronecker Delta?

    I am doing a Laplace's equation in spherical coordinates and have come to a part of the problem that has the integral... ∫ P(sub L)*(x) * P(sub L')*(x) dx (-1<x<1) The answer to this integral is given by a Kronecker delta function (δ)... = 0 if L...
  2. fluidistic

    Legendre polynomials, Jackson's book problem, potential

    Homework Statement I'm stuck in evaluating an integral in a problem. The problem can be found in Jackson's book page 135 problem 3.1 in the third edition. As I'm not sure I didn't make a mistake either, I'm asking help here. Two concentric spheres have radii a,b (b>a) and each is divided into...
  3. L

    A problem while verifying the generating function of Legendre Polynomials.

    Our professor gave us an a problem to solve, she asked us to prove or verify the following identity: http://img818.imageshack.us/img818/5082/6254.png Where \Phi is the Generating function of Legendre polynomials given by: \Phi(x,h)= (1 - 2hx + h2)-1/2 2. This Identity is from...
  4. A

    An integral over three Legendre polynomials

    Homework Statement I encountered the following integral in my research, and I've yet to find an analytic solution: I(n_1,n_2,n_3) = \int_{-1}^{1} d(\cos\theta_1) \int_{-1}^{1} d(\cos\theta_2) P_{n_1}(\cos\theta_1) P_{n_2}[\cos(\theta_1-\theta_2)] P_{n_3}(\cos\theta_2) where P_n(x) is the nth...
  5. J

    What is the Proof for the Relation Between Legendre Polynomials and Sums?

    Following relation seems to hold: \int^{1}_{-1}\left(\sum \frac{b_{j}}{\sqrt{1-μ^{2}}} \frac{∂P_{j}(μ)}{∂μ}\right)^{2} dμ = 2\sum \frac{j(j+1)}{2j+1} b^{2}_{j} the sums are for j=0 to N and P_{j}(μ) is a Legendre polynomial. I have tested this empirically and it seems correct. Anyway, I...
  6. E

    Integrating Legendre Polynomials Pl & Pm

    Homework Statement Integrate the expression Pl and Pm are Legendre polynomials Homework Equations The Attempt at a Solution Suppose that solution is equal to zero.
  7. M

    Orthogonality Relationship for Legendre Polynomials in Cylindrical Coordinates

    Hello everyone, Sorry if this is in the wrong sub-forum, I wasn't sure exactly where to place it. I was wondering if there is an orthogonality relationship for the Legendre polynomials P^{0}_{n}(x) that have been converted to cylindrical coordinates from spherical coordinates, similar to...
  8. C

    Proof that the legendre polynomials are orthogonal polynomials

    I'm now studying the application of legendre polynomials to numerical integration in the so called gaussian quadrature. There one exploits the fact that an orthogonal polynomial of degree n is orthogonal to all other polynomials of degree less than n with respect to some weight function. For...
  9. Demon117

    Orthogonality of Legendre Polynomials from Jackson

    Hello all! I am trying to work through and understand the derivation of the Legendre Polynomials from Jackson's Classical electrodynamics. I have reached a part that I cannot get through however. Jackson starts with the following orthogonality statement and jumps (as it seems) in his proof...
  10. V

    MATLAB Integration of a product of legendre polynomials in matlab

    I am trying to find a way to integrate the following expression Integral {Ylm(theta, phi) Conjugate (Yl'm'(theta, phi) LegendrePolynomial(n, Cos[theta])} dtheta dphi for definite values of l,m,n,l',m' . You normally do this in Mathematica very easily. But it happens that I need to use this...
  11. P

    Legendre Polynomials and Complex Analysis

    Hi all, I am currently a 2nd year mathematics and physics student. I am working, for the first time, on my own research and just sort of getting my feet wet. I got in touch with a professor that studies Special Functions and he led me to the Legendre functions and associated Legendre...
  12. W

    Orthogonality in Legendre polynomials

    Homework Statement There is a recursion relation between the Legendre polynomial. To see this, show that the polynomial x p_k is orthogonal to all the polynomials of degree less than or equal k-2. Homework Equations <p,q>=0 if and only if p and q are orthogonal. The Attempt at a...
  13. O

    Completeness of Legendre Polynomials

    I've recently been working with Legendre polynomials, particularly in the context of Spherical Harmonics. For the moment, it's enough to consider the regular L. polynomials which solve the differential equation [(1-x^2) P_n']'+\lambda P=0 However, I've run into a problem. Why in the...
  14. R

    Normalization constant for Legendre Polynomials

    Homework Statement I am following a derivation of Legendre Polynomials normalization constant. Homework Equations I_l = \int_{-1}^{1}(1-x^2)^l dx = \int_{-1}^{1}(1-x^2)(1-x^2)^{l-1}dx = I_{l-1} - \int_{-1}^{1}x^2(1-x^2)^{l-1}dx The author then gives that we get the following...
  15. B

    Expanding 6x^2 in Terms of Legendre Polynomials

    Given the Legendre polynomials P0(x) = 1, P1(x) = x and P2(x) = (3x2 − 1)/2, expand the polynomial 6(x squared) in terms of P l (x). does anyone know what this question is asking me? what is P l (x)? thanks in advance
  16. P

    Legendre Polynomials - expansion of an isotropic function on a sphere

    Hello. I don't know what to do with one integral. I am sure it is something very simple, but I just don't see it... For some reason I am not able to post the equations, so I am attaching them as a separatre file. Many thanks for help.
  17. X

    Orthogonality of Legendre Polynomials

    Homework Statement For spherical coordinates, we will need to use Legendre Polynomials, a.Sketch graphs of the first 3 – P0(x), P1(x), and P2(x). b.Evaluate the orthogonality relationship (eq 3.68) to show these 3 functions are orthogonal to each other. (3 integrals). c.Show that the...
  18. T

    Electrostatic potential in Legendre polynomials

    Homework Statement Two spherical shells of radius ‘a’ and ‘b’ (b>a) are centered about the origin of the axes, and are grounded. A point charge ‘q’ is placed between them at distance R from the origin (a<R<b). Expand the electrostatic potential in Legendre polynomials and find the Green...
  19. N

    Generating function for Legendre polynomials

    Homework Statement Using binomial expansion, prove that \frac{1}{\sqrt{1 - 2 x u + u^2}} = \sum_{k} P_k(x) u^k. Homework Equations \frac{1}{\sqrt{1 + v}} = \sum_{k} (-1)^k \frac{(2k)!}{2^{2k} (k!)^2} v^k The Attempt at a Solution I simply inserted v = u^2 - 2 x u, then...
  20. Y

    Question on Rodrigues' equation in Legendre polynomials.

    I have problem understand in one step of deriving the Legendre polymonial formula. We start with: P_n (x)=\frac{1}{2^n } \sum ^M_{m=0} (-1)^m \frac{2n-2m)}{m!(n-m)(n-2m)}x^n-2m Where M=n/2 for n=even and M=(n-1)/2 for n=odd. For 0<=m<=M \Rightarrow \frac{d^n}{dx^n}x^2n-2m =...
  21. F

    Integrating legendre polynomials with weighting function

    Homework Statement I recently came across this integral while doing a problem in electromagnetism (I'm not sure if there exists a nice analytic answer): \int_{0}^{\pi}P_m(\cos(t))P_n(\cos(t)) \sin^2(t) = \int_{-1}^{1}P_m(x) P_n(x) \sqrt{1-x^2}, Homework Equations P_m(x) is the m^th...
  22. M

    Taking legendre polynomials outside the integral in a multipole expansion

    Homework Statement A chare +Q is distributed uniformly along the z axis from z=-a to z=+a. Find the multipole expansion. Homework Equations Here rho has been changed to lambda, which is just Q/2a and d^3r to dz. The Attempt at a Solution I have solved the problem correctly...
  23. Somefantastik

    MATLAB Is There a MATLAB Routine for Simple Legendre Polynomials of a Specific Degree?

    I see in MATLAB that you can call legendre(n,X) and it returns the associated legendre polynomials. All I need is is the simple Legendre polynomial of degrees 0-299, which corresponds to the first element in the array that this function returns. I don't want to call this function and get this...
  24. B

    Determining Legendre polynomials (Boas)

    I am having trouble evaluating the Legendre Polynomials (LPs). I can do it by Rodrigues' formula but I am trying to understand how they come about. Basically I have been reading Mary L. Boas' Mathematical Methods in the Physical Sciences, 3E. Ch.12 §2 Legendre Polynomials pg566. In the...
  25. A

    Decompositoin of f(x) in Legendre polynomials

    Hi, In Wikipedia it's stated that "... Legendre polynomials are useful in expanding functions like \frac{1}{\sqrt{1 + \eta^{2} - 2\eta x}} = \sum_{k=0}^{\infty} \eta^{k} P_{k}(x) ..." Unfortunately, I am failing to see how this can be true. Is there a way of showing this...
  26. H

    Eigenfunction expansion in Legendre polynomials

    Homework Statement How to use eigenfunction expansion in Legendre polynomials to find the bounded solution of (1-x^2)f'' - 2xf' + f = 6 - x - 15x^2 on -1<= x <= 1 Homework Equations eigenfunction expansion The Attempt at a Solution [r(x)y']' + [ q(x) + λ p(x) ]...
  27. MathematicalPhysicist

    Legendre Polynomials: Expansion and Series Generation

    I need to expand the next function in lengendre polynomial series: f(x)=1 x in (0,1] f(x)=0 x=0 f(x)=-1 x in [-1,0). Now here's what I did: the legendre series is given by the next generating function: g(x,t)=(1-2tx+t^2)^(-1/2)=\sum_{0}^{\infty}P_n(x)t^n where P_n are legendre...
  28. Repetit

    Product of Legendre Polynomials

    Hey! Could someone please help me find out how to express the product of two Legendre polynomials in terms of a sum of Legendre polynomials. I believe I have to use the recursion formula (l+1)P_{l+1}(x)-(2l+1) x P_l(x) + l P_{l-1}(x)=0 but I am not sure how to do this. What is basically...
  29. I

    How Do You Derive the Formula for P-n(0) Using Legendre Polynomials?

    There is a question where you should find a formula for P-n(0) using the Legendre polynomials: P-n(x)=1/(2^n*n!) d^n/dx^n(x^2-1)^n , n=0,1,2,3... I tried to derive seven times by only substituting the n until n=7,I did that because i wanted to find something that i can build my formula but i...
  30. A

    Legendre polynomials proof question.Help

    Hello everyone i had some questions about legendre polynomials. I have solved most of them but i had just two not answered question. I tried to solve this problem by rodriguez rule but it was really hard for me. Could anyone help me or give me some hints for this question...
  31. S

    Proving Orthogonality of Legendre Polynomials P3 and P1

    To show that two Legendre polynomials(Pn and Pm) are orthogonal wht is the test that i have to use? is it this? \int_{-1}^{1} P_{n}(x)P_{m}(x) dx = 0 in that case to prove that P3 and P1 are orthogonal i have to use the above formula??
  32. L

    Expand a function in terms of Legendre polynomials

    Problem: Suppose we wish to expand a function defined on the interval (a,b) in terms of Legendre polynomials. Show that the transformation u = (2x-a-b)/(b-a) maps the function onto the interval (-1,1). How do I even start working with this? I haven't got a clue...
  33. L

    Proving Orthogonality of Legendre Polynomials

    Problem: Show that \int_{-1}^{1} x P_n(x) P_m(x) dx = \frac{2(n+1)}{(2n+1)(2n+3)}\delta_{m,n+1} + \frac{2n}{(2n+1)(2n-1)}\delta_{m,n-1} I guess I should use orthogonality with the Legendre polynomials, but if I integrate by parts to get rid of the x my integral equals zero. Any tip on...
  34. G

    Solution of hydrogen atom : legendre polynomials

    I was messing around with the \theta equation of hydrogen atom. OK, the equation is a Legendre differential equation, which has solutions of Legendre polynomials. I haven't studied them before, so I decided to take closed look and began working on the most simple type of Legendre DE. And the...
  35. G

    Legendre Polynomials Orthogonality Relation

    ...and orthogonality relation. The book says \int_{-1}^{1} P_n(x) P_m(x) dx = \delta_{mn} \frac{2}{2n+1} So I sat and tried derieving it. First, I gather an inventory that might be useful: (1-x^2)P_n''(x) - 2xP_n'(x) + n(n+1) = 0 [(1-x^2)P_n'(x)]' = -n(n+1)P_n(x) P_n(-x) = (-1)^n P_n(x)...
  36. R

    Legendre polynomials application

    I need some help. I fitted a 7th order legendre polynomial and got the L0 to L7 coefficients for different ANOVA classes. How can I get a back transformation in order to plot each class using the estimated coefficients? Thanks to anybody. Roberto.
  37. W

    Back Transformation for Legendre Polynomials

    some body who can explain for me the Legndre polynomials:eek: :eek:
  38. T

    Find Legendre Polynomials of Order 15+

    Hey there, does anyone know where I could find a list of Legendre Polynomials? I need them of the order 15 and above, and I haven't been able to find them on the net. Thanks!
  39. T

    Legendre Polynomials: Beginner's Guide

    hi folks! I have been trying to figure out some plausible geometric intrepretation to legendre polynomials and what are they meant to do. I have come across the concept of orthogonal polynomials while working with some boundary value problems in solid mechanics and wasn't able to come to...
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