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. M

    Reduction of Order - Legendre Eqn

    Legendre's eq of order n>=0 is (1-x^2)y'' -2xy' +n(n+1)y = 0. You are given the soln y = P_n(x) for n=0,1,2,3 to be P_0(x)=1 ; P_1(x)=x ; P_2(x)=(3x^2-1)/2 ; P_3(x)=(5x^3 -3x)/2. Use reduction of order to find the second independent soln's Q_n(x) OK I've found Q_1(x) = ln(1-x)(1+x)...
  2. 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.
  3. 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...
  4. C

    Very simple (Dis)proof of Riemann hypothesis, Goldbach, Polignac, Legendre conjecture

    (Dis)proof of Riemann hypothesis,Goldbach,Polignac,Legendre conjecture I'm just an amateur and not goot at english ^^;
  5. A

    Factoring Legendre function?

    Consider the multipole expansion of Newtonian potential 1/R in 2D in terms of Legendre functions. \begin{align} \mathbf{r_1} &= (r , \theta_1) \\ \mathbf{r} &= (r , \theta) \\ \phi(\mathbf{r}-\mathbf{r_1}) &= \frac{1}{\left|\mathbf{r}-\mathbf{r_1}\right|} \\...
  6. Y

    Please help in integration of Associate Legendre function

    I don't understand why I solve the integration in two different ways and get two different answers! To find: \int_0^{\pi} P_1^1(cos \theta) sin \theta d \theta 1) Solve in \theta P_1(cos \theta) = cos \theta \;\Rightarrow \; P_1^1(cos \theta)= -sin \theta \int_0^{\pi}...
  7. Y

    Question on orthogonal Legendre series expansion.

    This start out as homework but my question is not about helping me solving the problem but instead I get conflicting answers depend on what way I approach the problem and no way to resolve. I know the answer. I am not going to even present the original question, instead just the part that I have...
  8. pellman

    Hamiltonian as Legendre transformation?

    The definition of a Legendre transformation given on the Wikipedia page http://en.wikipedia.org/wiki/Legendre_transformation is: given a function f(x), the Legendre transform f*(p) is f^*(p)=\max_x\left(xp-f(x)\right) Two questions: what does \max_x mean here? And why is it not...
  9. 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...
  10. 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...
  11. nicksauce

    Proving Bessel to Legendre in Dodelson's Cosmology Book

    In Dodelson's cosmology book it is claimed that "For large x, J_0(x\theta)\rightarrow P_{x}(cos\theta)". Does anyone have any insight on how to begin proving this?
  12. P

    Writing a polynomial in terms of other polynomials (Hermite, Legendre, Laguerre)

    Homework Statement The first 3 parts of this 4 part problem were to derive the first 5 Hermite polynomials (thanks vela), The first 5 Legendre polynomials, and the first 5 Laguerre polynomials. Here is the last part: Write the polynomial 2x^4-x^3+3x^2+5x+2 in terms of each of the sets of...
  13. V

    Potential from a Quadrupole using Legendre polynml's

    1. Problem Statement: There are charges of q placed at distance +a and -a from the origin on the z-axis. There is a charge at the origin of -2q. Express the potential of this point-like linear quadrupole in Legendre polynomials The distance between origin and point is r, the distance between...
  14. K

    Computing the Legendre Symbol (2/p) for Odd Primes: A Proof and Explanation

    "Let p be an odd prime, then we proved that the Legendre symbol Note that this can be easily computed if p is reduced modulo 8. For example, if p=59, then p≡3 (mod 8) and (-1)^{(p^2-1)/8} = (-1)^{(3^2-1)/8}" (quote from my textbook) ==================================== Now I don't...
  15. G

    How Does the Legendre Transformation Apply to Thermodynamics with Tension Force?

    Homework Statement At an elastic bar we give work because of hydrostatic pressure P and applied tension force F at the axis length that has length l. Homework Equations 1) Give an expression for dU. 2) With the help of legendre transformations find the thermodynamic equations and the...
  16. P

    Why do you need a convex/concave function to do a Legendre transform?

    I have been trying to figure this out for a couple weeks now. Why does the Legendre transform require that the function be convex? Is it because g(x) has to be solved to get x(g) and finding this inverse means g(x) should be bijective? (And if g is bijective then dg/dx will always be positive...
  17. Y

    Help regarding Legendre Rodrigue's formula problem.

    Homework Statement Question: Use Rodrigues' formula and integral by parts to show: \int^1 _{-1}f(x)P_n (x)dx=\frac{(-1)^n}{2^n n!}\int^1_{-1}f^{(n)}(x)(x^2 -1)^n dx (As a convention f^{(0)}(x)=f(x) Homework Equations Rodrigues' Formula: P_n(x)=\frac{1}{2^n...
  18. 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 =...
  19. Y

    Please help with Legendre problem.

    Homework Statement Determine the IVP has bounded solution: Legendre equation: (1-x^2)y''-2xy'+6y=0 ; y(0)=0, y'(0)=1 Homework Equations P_2 (x)=\frac{1}{2}[3x^2 -1] Q_2 (x)=P_2 (x)\int \frac{dx}{[P_2 (x)]^2 (1-x^2)} y'(x)=c_1 P'_2 (x) + c_2 {P'_2(x)\int \frac{dx}{[P_2 (x)]^2...
  20. Y

    Legendre Differential equation question.

    Legendre equation: (1-x^2)y''-2xy'+n(n+1)y=0 Where -1< x < 1 General solution is y(x)=c_1 P_n (x)+c_2 Q_n (x) Where P_n (x) is bounded and Q_n (x) is unbounded on (-1,1). Q_n (x)=P_n (x)\int \frac{dx}{[P_n (x)]^2 (1-x^2)} Question: Why is Q_n (x) unbounded on (-1,1)? I tried...
  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. N

    Finding Legendre Transform of f(x) = (|x|+1)2

    Hi guys I am looking at f(x) = (|x|+1)2. I write this as f(x) = \left\{ {\begin{array}{*{20}c} {x^2 + 1 + 2x\,\,\,\,\,\,\,\,\,\,\,\,\,for\,\,\,\,x > 0} \\ {x^2 + 1 - 2x\,\,\,\,\,\,\,\,\,\,\,\,\,for\,\,\,\,x < 0} \\...
  23. S

    Integral of Legendre polynomial

    Homework Statement In solving a question I got a problem of solving the following integral. Your comments are appreciated. Homework Equations \int_{-1}^{1}xP_l'(x)dx=? The Attempt at a Solution I tried to solve by integration by parts, i.e...
  24. B

    Deriving Relations Between Generating Functions via Legendre Transformations

    Homework Statement Problem 9.7(a) of Goldstein, 3rd edition: If each of the four types of generating functions exists for a given canonical transformation, use the Legendre transformations to derive the relations between them. Homework Equations F = F1(q,Q,t) p = partial(F1)/partial(q) P =...
  25. 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...
  26. M

    Proof of orthogonality of associated Legendre polynomial

    I want to prove orthogonality of associated Legendre polynomial. In my textbook or many posts, \int^{1}_{-1} P^{m}_{l}(x)P^{m}_{l'}(x)dx = 0 (l \neq l') is already proved. But, for upper index m, \int^{1}_{-1} P^{m}_{n}(x)P^{k}_{n}(x)\frac{dx}{ ( 1-x^{2} ) } = 0 (m \neq k) is not...
  27. N

    How to Derive Legendre Polynomials Using Orthogonalization?

    Homework Statement The Legendre polynomials P_l(x) are a set of real polynomials orthogonal in the interval -1< x <1 , l\neq l' \int dx P_l(x)P_l'(x)=0, -1<x<1 The polynomial P_l(x) is of order l , that is, the highest power of x is x^l. It is normalized to P_l(x)=1 Starting with the set...
  28. 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...
  29. B

    Exploring Associated Legendre Functions: Applications and Uses

    Hello: What are Associated Legendre functions? What are they good for in terms of applications?
  30. D

    Legendre and Riemann: A Conjecture Comparison

    I recall reading somewhere that Legendre's conjecture implies the Riemann Hypothesis. But the Wiki article suggests that Legendre imposes lighter bounds on the density of primes than does RH, so I would think the other way around, if anything. Thanks for any enlightenment.
  31. Z

    Help with Legendre Differential equation

    I have never seen the Legendre Function and the Legendre function of the second kind multiplied together for a solution. Can someone point me in the right direction to learn more about solving these equations with solutions like this? Thanks very much
  32. P

    Solving a differential equation similar to Legendre

    I am trying to solve the following differential equation: (\frac{L^2}{6k^2}+\frac{w\sqrt{3}}{2}\sin^2\theta\ sin 2\phi)\psi=E\psi where is the angular momentum given by: L^2 = \frac{1}{\sin\theta}\frac{\partial}{\partial\theta...
  33. P

    Differential equation similar to Legendre

    I am trying to solve the following differential equation: (\frac{L^2}{6k^2}+\frac{w\sqrt{3}}{2}\sin^2\theta\cos 2\phi)\psi=E\psi where L^2 is the angular momentum given by...
  34. M

    Why use a Legendre transform instead of a simple change of variable?

    The Legendre transformation creates a new function which contains the same information as the old, but is of a different variable. This is used to obtain the Hamiltonian from the Lagrangian. My question is, why is there more advantageous than simply rearranging the q's for p's and plugging them...
  35. 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...
  36. S

    Legendre Polynomial Orthogonality Integral Limits

    Good afternoon I have a question regarding the limits on the orthogonality integral of Legendre Polynomicals: \int_{-1}^1 P_l(u)P_{l'}du = 2/(2l+1) I am in the middle of a question involving the solution of Laplace's equation inside a hemisphere, which means that for the usual...
  37. Q

    Can Legendre polynomials be evaluated using a recurrence relation in Fotran 90?

    Is it possible to evaluate a legendre polynomial p(n,x) using the recurrence relation p(n,x)=p(n-1,x)*x-p(n-2,x) in fotran 90 {there are some other terms which i left out for brevity]
  38. S

    Convergence of expansion of Legendre generating function.

    The Legendre functions may be defined in terms of a generating function: g(x,t) = \frac{1}{\sqrt{1-2xt+t^2}} Of course, \frac{1}{\sqrt{1+x}} =\sum^{\infty}_{n=0} (\stackrel{-.5}{n})x^n . However, this series doesn't converge for all x. It only converges if |x| < 1. In our case, |t^2 -...
  39. B

    Integral with legendre generating function

    Homework Statement Use the Legendre generating function to show that for A > 1, \int^{\pi}_{0} \frac{\left(Acos\theta + 1\right)sin\thetad\theta}{\left(A^{2}+2Acos\theta+1\right)^{1/2}} = \frac{4}{3A} Homework Equations The Legendre generating function \phi\left(-cos\theta,A\right) =...
  40. H

    How can I compute the Legendre polynomial integral over a specific range?

    Homework Statement As part of a larger problem, I need to compute the following integral (over -1<\theta<1): \int \sin \theta P_{l}(\cos \theta) d (\cos \theta) Homework Equations \int P_{l}(x) P_{l'}(x) dx= \frac{2}{2l+1} \delta_{l',l} Also, solutions are known to the following...
  41. C

    Potential for Electric Charge over Spherical Shell using Legendre Functions

    Homework Statement Electric Charge is distributed over a thin spherical shell with a density which varies in proportion to the value of a single function P_l(cos \theta) at any point on the shell. Show, by using the expansions (2.26) and (2.27) and the orthongonality relations for the...
  42. 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...
  43. 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) ]...
  44. T

    SoS problem in legendre and bessel functions

    hello every body ... I am a new member in this forums ..:smile: and i need ur help in telling me what's the perfect way to study legendre and bessel function for someone doesn't know anything about them and having a hard time in trying to understand ... i`ll be thankful if u...
  45. M

    Position and Momentum operator in Legendre base

    First of all, I don't really know if this problem corresponds to this section, but anyway I have this as a probem in my matemathical physics class. The problem is stated something like this: Find the matrix elements of the position and momentum operators in the legendre base (on the...
  46. 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...
  47. 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...
  48. J

    Newtons Law of Gravity Legendre Polynomial & Harmonic functions

    I'm reading section 17 of Mathematical Physics by Donald H. Menzel on Harmonic functions. They start with Newtons law of gravitation (although the following method can be aplied to any potential field with a 1/r dependence.) see...
  49. J

    Legendre Transformation: Find f(T,v)

    Homework Statement du = T ds - p dv Find a Legendre transformation giving f(T,v) The Attempt at a Solution Can anyone check if this is remotely correct? f(T,v) df = \partial f/\partial T dT + \partial f/\partial v dv du = Tds - p dv u = f - vp d(f-vp) = Tds + v dp - p dv - v dp df = Tds -...
  50. M

    Easy question about associated legendre functions

    i've trying for hours, can anyone help me (tell me if this is not the right place to post this question)
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