What is Legendre: Definition and 224 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. A

    Derivation of the thermodynamic potentials using Legendre transformations

    Hello guys, I'm studying Thermodynamics and I don't totally see how you introduce the potencials using Legendre transformations. I have seen a non formal explanation showing how you can interpret them, but not a rigorous demonstration of how you get them via the Legendre transformations...
  2. 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...
  3. K

    Legendre Polynomial and Rodrigues' Formula

    I am reading Jackson's electrodynamics book. When I went through the Legendre polynomial, I have a question. In the book, it stated that from the Rodrigues' formula we have Consider only the odd terms...
  4. S

    Gauss Legendre numerical intergration

    Gauss Legendre numerical intergration The attachment file contain solved example i don't know how he subsitute and why a2=2 done disappear in the answer please expalin in details
  5. P

    Legendre Expansion solution to sphere of potential V with charge q outside

    Hi, It seems that there is no much examples of this particular case. OK, we all know how to write the general solution to Laplace equation in spherical coordinates in terms of Legendre polynomials (when there is azimuthal symmetry). There are a lot of cases here but I would like to know...
  6. 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...
  7. 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...
  8. D

    Sums of Legendre Symbols Question

    Proposition: \sum_{i=0}^{p-1} (\frac{i^2+a}{p})=-1 for any odd prime p and any integer a. (I am referring to the Legendre Symbol). I was reading a paper where they claimed it was true for the a=1 case and referred to a source that I don't have immediate access to. So I was wondering if...
  9. A

    Questions about Legendre Equation

    1. I understand that the x in Legendre Equation (1-x^2)y''-2xy'+l(l+1)y=0 is often related to θ in spherical coordinates. We want the latter equation to have a solution at θ=0 and θ=pi. Therefore, we require that Legendre Equation has a solution at x=±1 And it is claimed that "we require the...
  10. A

    Recurrence relation for the Legendre functions

    My book wants to find solutions to Legendre's equation: (1-x2)y'' - 2xy' 0 l(l+1)y = 0 (1) By assuming a solution of the form: y = Ʃanxn , the sum going from 0->∞ (2) Now by plugging (2) into (1) one finds: Ʃ[n(n-1)anxn-2-n(n-1)anxn - 2nanxn + l(l+1)anxn = 0...
  11. 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...
  12. J

    Solution to Legendre equation in trig form

    hey guys, I've been trying to solve this question, http://img515.imageshack.us/img515/2583/asfj.jpg so the general solution would be y(cos(theta)) = C Pn(cos(theta)) + D Qn(cos(theta)) right? and since n = 2 in this case y(cos(theta)) = C P_2 (cos(theta)) + D Q_2...
  13. L

    Checking Qn(x) is a solution to legendre eq. (n=3)

    Hey, I have a question which ends by asking to verify that Q3(x) is a solution to the legendre equation, I took the first and second derivatives of it and before I continue with this messy verification I wanted to know if there was a simpler way to check. Q3(x) = (1/4)x(5x^2 -...
  14. P

    Legendre poly, generating function

    hey guys, my lecturer skipped the proof to show that \frac{1}{\sqrt{1+u^2 -2xu}} is a generating function of the polynomials, he told us that we should do it as an exercise by first finding the binomial series of \frac{1}{\sqrt{1-s}} then insert s = -u2 + 2xu he then said to expand...
  15. L

    Binomial expansion comparison with legendre polynomial expansion

    Hi, I've been working on this question which asks to show that {{P}_{n}}(x)=\frac{1}{{{2}^{n}}n!}\frac{{{d}^{n}}}{d{{x}^{n}}}{{\left( {{x}^{2}}-1 \right)}^{n}} So first taking the n derivatives of the binomial expansions of (x2-1)n...
  16. J

    Trigonometric form of Legendre equation

    hey, (1-{{x}^{2}}){{y}^{''}}-2x{{y}^{'}}+n(n+1)y=0,\,\,\,\,\,-1\le x\le 1 to convert the legendre equation y(x) into trig form y(cos\theta) is it simply, set x=cos\theta then (1-{{\cos }^{2}}\theta ){{y}^{''}}-2{{y}^{'}}\cos \theta +n(n+1)y=0 for -\pi \le x\le \pi {{\sin }^{2}}\theta...
  17. S

    For the following properties, show that either f(a) = 1 for all a, or f(a) = Legendre

    Let p be an odd prime. Let f(a) be a function defined for a prime to p satisfying the following properties: (i) f(a) only takes the values ±1. (ii) If a=b (mod p), then f(a)=f(b). (iii) f(ab) = f(a)f(b) for all a and b. Show that either f(a) = 1 for all a or that f(a) = (\frac{a}{b})
  18. 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...
  19. 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...
  20. T

    Legendre Polynomial (anti)symmetry proof

    Homework Statement Let P_{n}(x) denote the Legendre polynomial of degree n, n = 0, 1, 2, ... . Using the formula for the generating function for the sequence of Legendre polynomials, show that: P_{n}(-x) = (-1)^{n}P_{n}(x) for any x \in [-1, 1], n = 0, 1, 2, ... . Homework Equations...
  21. conquest

    Finding Hamiltonian as Legendre transform on SO(3)

    Homework Statement We need to find the Hamiltonian that corresponds to a given Lagrangian by finding the Legendre transform. The system is a rigid body pinned down in some point. This means the motion is described essentialy by SO(3). So the Lagrangian is given in terms of these matrices and...
  22. M

    Legendre Transforms: U=U(S,V) vs U(V,P)

    When people do Legendre transforms they suppose that U=U(S,V). But you can see in some books that heat is defined by: dQ=(\frac{\partial U}{\partial P})_{V}dP+[(\frac{\partial U}{\partial V})_P+P]dV So they supposed obviously that U=U(V,P). In some books you can that internal energy is...
  23. 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...
  24. 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.
  25. S

    Thermodynamics, Helmholtz free energy, Legendre transformation

    Homework Statement The Helmholtz free energy of a certain system is given by F(T,V) = -\frac{VT^2}{3}. Calculate the energy U(S,V) with a Legendre transformation. Homework Equations F = U - TS S = -\left(\frac{\partial F}{\partial T}\right)_V The Attempt at a Solution We...
  26. C

    Legendre differential equation and reduction of order

    Homework Statement Question is to find a general solution, using reduction of order to: (1-x^2)y" - 2xy' +2y = 0 (Legendre's differential equation for n=1) Information is given that the Legendre polynomials for the relevant n are solutions, and for n=1 this means 'x' is a solution...
  27. 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...
  28. V

    Prove infinitude of primes of form 4k+1 using properties of Legendre symbol (-1/p)

    Homework Statement Show that there are infinitely many primes 4k+1 using the properties of \left(\frac{-1}{p}\right). Homework Equations \left(\frac{-1}{p}\right) = \begin{cases} 1, & \text{if }p\equiv 1\ (mod\ 4), \\ -1, & \text{if }p\equiv 3\ (mod\ 4). \end{cases}...
  29. 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...
  30. 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...
  31. S

    Legendre Differential Equation Transformation by Substitution

    Homework Statement Show that the differential equation: sin(theta)y'' + cos(theta)y' + n(n+1)(sin(theta))y = 0 can be transformed into Legendre's equation by means of the substitution x = cos(theta). Homework Equations Legendre's Equation: (1 - x^2)y'' - 2xy' + n(n+1)y = 0 The Attempt at a...
  32. R

    Integral of associated Legendre Function

    I need to evaluate the following integral: [tex]\int_0^{\pi} \lleft(\frac{P_n^1}{\sin\theta} \frac{d P_l^1}{d\theta}\right)\, \sin\theta\, d\theta [tex] This integral, I think, has a closed form expression. Itarises in elastic wave scattrering. I am an engineer and do not have suficient...
  33. 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...
  34. K

    Lagrangian, Hamiltonian and Legendre transform of Dirac field.

    In most of the physical systems, if we have a Lagrangian L(q,\dot{q}), we can define conjugate momentum p=\frac{\partial L}{\partial{\dot{q}}}, then we can obtain the Hamiltonian via Legendre transform H(p,q)=p\dot{q}-L. A important point is to write \dot{q} as a function of p. However, for the...
  35. Simfish

    Legendre Transformation of the Hamiltonian

    It's given as this H\left(q_i,p_j,t\right) = \sum_m \dot{q}_m p_m - L(q_i,\dot q_j(q_h, p_k),t) \,. But if it's a Legendre transformation, then couldn't you also do this? H\left(q_i,p_j,t\right) = \sum_m \dot{p}_m q_m - L(p_i,\dot p_j(p_h, q_k),t) \,.
  36. 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...
  37. 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...
  38. Y

    QFT. Effective action and the skeleton expansion, how the legendre transform works

    QFT. Effective action and the skeleton expansion, how the legendre transform works! Homework Statement I've written a presentation on the effective action and have been posed a few questions to look out for. I think I know the answer to the first but am stumped by the second. "you've...
  39. V

    Derivatives of Legendre Series Expansions

    I am working on an advanced fundamental engineering theory. For that I need to solve a system of differential equations in R2 by expanding my variables as Legendre series expansion. Thus: u(x,y)=\sum\sumAmnPm(x)Pn(y) The equations contain of each variable derivatives up to the fourth...
  40. S

    Associated Legendre functions and orthogonality

    According to the orthogonality property of the associated Legendre function P_l^{|m|}(cos\theta) we have that: \int_{0}^{\pi}P_{l}^{|m|}(cos\theta){\cdot}P_{l'}^{|m'|}(cos\theta)sin{\theta}d\theta=\frac{2(l+m)!}{(2l+1)(l-m)!}{\delta}_{ll'} What I am looking for is an orthogonality...
  41. R

    Conjectures by Legendre and Brocard made stronger.

    If this series https://www.physicsforums.com/showthread.php?t=485665 is proved to be infinite, then proofs of these two conjectures can be done as simple corollaries. Legendre's Conjecture states that for every $n\ge 1,$ there is always at least one prime \textit{p} such that $n^2 < p <...
  42. 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...
  43. L

    Determining Legendre derivitives

    Determining Legendre derivitives Homework Statement if i need to find the derivative of the first Legendre polynomial, P1(cos\Theta) can i sub in cos\Theta for x in P1(x) = x? Homework Equations The Attempt at a Solution if that's the case the derivitive is just -sin(\Theta), which...
  44. G

    Legendre Funcs: Homework & Equations Explained

    Homework Statement Hi everyone, I am having issues understanding how Legendre functions work especially the recursion and what the subscripts mean in general. I am attempting to make a program to compute the legendre functions Pnm(cos(theta)) and the normalized version and then verifying it by...
  45. D

    Legendre transformation of the CR3BP equations

    I'm not quite sure where to post this but I suppose it should go here given it's about classical mechanics... Anyhoo. I'm currently on the long road to implementing a symplectic integrator to simulate the closed restricted 3 body problem and I'm in the process of deriving the Hamiltonian...
  46. 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...
  47. 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
  48. J

    Legendre differential equation- power series

    Homework Statement http://mathworld.wolfram.com/LegendreDifferentialEquation.html I have a question about how the website above moves from one equation to another etc. 1./ Equations (4), (5) and (6) When differentiating (4) to (5) shouldn't the the limit be from n=1, which means (5)...
  49. I

    Exploring Lagrangians that Cannot be Transformed with Legendre

    Hi! Our TA told us, that it may be not always possible to change lagrangian into hamiltonian using Legendre transformation. As far as I'm concerned the only such possibility is that we can not substitute velocity (dx/dt) with momenta and location(s). And so, we've been tryging to come up with an...
  50. O

    Extended version of associated Legendre eq. Can I use the Frobenius method?

    Hello, Let us assume we have a differential equation ( \frac{d^2}{dx^2}+ \frac{2}{1+1/n}\Theta_n^{n-1} - \nu^2 ) y = 0 where \Theta_n = (1-\mu^2)^{1/(n+1)} is a function of \mu which is a function of x : \frac{d\mu}{dx} = \Theta_n^n = (1-\mu^2)^{n/(n+1)} In terms of \mu...
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