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. Bassa

    Change of Variable With Legendre Equation

    Homework Statement [/B] Change the independent variable from x to θ by x=cosθ and show that the Legendre equation (1-x^2)(d^2y/dx^2)-2x(dy/dx)+2y=0 becomes (d^2/dθ^2)+cotθ(dy/dθ)+2y=0 2. Homework Equations The Attempt at a Solution [/B] I did get the exact form of what the equation...
  2. 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...
  3. J

    Legendre Second Kind: $Q_n(x)$ Functions

    Legendre functions $Q_n(x)$ of the second kind \begin{equation*} Q_n(x)=P_n(x) \int \frac{1}{(1-x^2)\cdot P_n^2(x)}\, \mathrm{d}x \end{equation*} what to do after this step? how can I complete ? I need to reach this formula \begin{equation*} Q_n(x)=\frac{1}{2} P_n(x)\ln\left( \frac{...
  4. duc

    Coefficient of a polynomial defined by Legendre polynomial

    Homework Statement The polynomial of order ##(l-1)## denoted ## W_{l-1}(x) ## is defined by ## W_{l-1}(x) = \sum_{m=1}^{l} \frac{1}{m} P_{m-1}(x) P_{l-m}(x) ## where ## P_m(x) ## is the Legendre polynomial of first kind. In addition, one can also write ## W_{l-1}(x) = \sum_{n=0}^{l-1} a_n \cdot...
  5. H

    Raabe's test says Legendre functions always converge?

    The Legendre functions are the solutions to the Legendre differential equation. They are given as a power series by the recursive formula (link [1] given below): ##\begin{align}y(x)=\sum_{n=0}^\infty a_n x^n\end{align}## ##\begin{align}a_{n+2}=-\frac{(l+n+1)(l-n)}{(n+1)(n+2)}a_n\end{align}##...
  6. S

    Helmholtz Free Energy Legendre Transformation

    Homework Statement Show how a Legendre transformation is used to obtain the Helmholtz free energy A(T,V) from the internal energy and derive the general expression for the differential of A. Homework Equations Internal Energy is a function of Entropy and Volume. U Ξ (S, V) A Ξ (T,V) A = U...
  7. Fantini

    MHB Legendre Polynomial and Legendre Equation

    Given $f(x) = (x^2-1)^l$ we know it satisfies the ordinary differential equation $$(x^2-1)f'(x) -2lx f(x) = 0.$$ The book defines the Legendre polynomial $P_l(x)$ on $\mathbb{R}$ by Rodrigues's formula $$P_l(x) = \frac{1}{2^l l!} \left( \frac{d}{dx} \right)^l (x^2-1)^l.$$ I'm asked to prove by...
  8. DOTDO

    Legendre Transformation of Lagrangian density ?

    Hi I began to study the basics of QED. Now I am studying Lagrangian and Hamiltonian densities of Dirac Equation. I'll call them L density and H density for convenience :)Anyway, the derivation of the H density from L density using Legendre transformation confuses me :( I thought because...
  9. Coffee_

    Generating functions and Legendre transforms

    I am confused about the ''4 basic types'' of generating functions. I have searched for this a bit on google but haven't found anything that truly made the click for me on this concept so I'll try here: What I do understand and need no elaboration on: 1) When considering the Hamiltonian and...
  10. C

    How many values of k can be determined, such that

    Homework Statement How many values of k can be determined in general, such that (k/p) = ((k+1) /p) = 1, where 1 =< k <=p-1? Note: (k/p) and ((k+1)/p) are legendre symbols Question is more clearer on the image attached.Homework Equations On image. The Attempt at a Solution I've tried...
  11. 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...
  12. evinda

    MHB Legendre Equation: Analyzing Differential Equation at Interval $(-1,1)$

    Hello! (Wave) $$(1-x^2)y''-2xy'+p(p+1)y=0, p \in \mathbb{R} \text{ constant } \\ -1 < x<1$$ At the interval $(-1,1)$ the above differential equation can be written equivalently $$y''+p(x)y'+q(x)y=0, -1<x<1 \text{ where } \\p(x)=\frac{-2x}{1-x^2} \\ q(x)= \frac{p(p+1)}{1-x^2}$$ $p,q$ can be...
  13. L

    Legendre Transform Homework Statement and Solution Attempt

    Homework Statement The problem is in the attached .PNG file. Homework Equations none The Attempt at a Solution I believe I am supposed to do a Legendre transform but have no idea where to begin...
  14. 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...
  15. W

    The Legendre transform of ##f(x) = \exp(\lvert x\rvert )##

    Homework Statement Let the single variable real function f:\mathbb{R}\rightarrow\mathbb{R} be given by f(x)=e^{|x|}. Determine the Legendre transform of f. Homework Equations Let I\subseteq\mathbb{R}be an interval, and f:I\rightarrow\mathbb{R}a convex function. Then its Legendre transform...
  16. B

    Associated Legendre polynomials

    m=1 and l=1 x = cos(θ) What would be the solution to this? Thanks.
  17. 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...
  18. Soumalya

    TextBooks for Some Topics in Mathematics

    Hi, I need suggestions for picking up some standard textbooks for the following set of topics as given below: Ordinary and singular points of linear differential equations Series solutions of linear homogenous differential equations about ordinary and regular singular points...
  19. 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 ...
  20. 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...
  21. bearcharge

    Associated Legendre differential equation involved in solving spin function?

    Amazed by the closeness of equations for orbital angular momentum L and spin angular momentum S, I can't help asking is associated Legendre differential equation involved in solving spin function? I only heard that spin naturally comes from treatment of quantum mechanics with relativity theory...
  22. ch3cooh

    Polynomial approximation: Chebyshev and Legendre

    Chebyshev polynomials and Legendre polynomials are both orthogonal polynomials for determining the least square approximation of a function. Aren't they supposed to give the same result for a given function? I tried mathematica but the I didn't get the same answer :( Is this precision problem or...
  23. S

    Schlaefli integral satisfies Legendre equation

    The integral representation of Legendre functions is P_\nu(z) = \oint_{\Gamma} \frac{(w^2-1)^\nu}{(w-z)^{\nu+1}} dw. I'm trying to show that this satisfies Legendre's equation. When I take the derivatives and plug it into the equation, I just get a nasty expression with nasty integrals times...
  24. C

    Purpose of doing Legendre Transform

    Hi, I'm new here, I was just wondering if anyone could help clarify a subject I'm having difficulties teaching myself... In thermo we perform a "Legendre transform" on the internal energy with respect to entropy. The stated purpose of this is so that we don't have to work in the entropy...
  25. H

    Associated Legendre Polynomial Identity

    Does anyone know how to prove this identity? I don't quote understand why the associated Legendre function is allowed to have arguments where |x|>1. h_n(kr)P_n^m(\cos\theta)=\frac{(-i)^{n+1}}{\pi}\int_{-\infty}^\infty e^{ikzt}K_m(k\rho\gamma(t))P_n^m(t)\,dt where \gamma(t)=\begin{cases}...
  26. STEMucator

    Legendre Equation & Polynomials

    I came across the Legendre differential equation today and I'm curious about how to solve it. The equation has the form: $$(1 - x^2)y'' - 2xy' + \nu(\nu +1)y = 0, (1)$$ Where ##\nu## is a constant. The equation has singularities at ##x_1 = \pm 1## where both ##p## and ##q## are not analytic...
  27. 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...
  28. individ

    MHB Solutions of Diophantine equations of Legendre.

    The problem is that when I got the formula that led below - there was a question whether they really describe all the decisions? But has not yet found a counterexample, and it exists at all interested or not? Although the discussion of this issue in many other forums and did not work. Everyone...
  29. A

    How do I find the n+1 and n-1 order expansion of a Legendre series?

    Homework Statement Find the n+1 and n-1 order expansion of \stackrel{df}{dy}Homework Equations (n+1)Pn+1 + nPn-1 = (2n+1)xPn ƒn = \sum CnPn(x) Cn = \int f(x)*Pn(u)The Attempt at a Solution I know you can use the recursion relation for Legendre Polynomials once you combine Cn with the...
  30. 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=...
  31. 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...
  32. 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^'=(...
  33. K

    Byron,Fuller)question-spherical harmonics & associated legendre funct.

    Hello, Question is about equation 5.71 ---to-----5.72 All that was done is algebra or trigonometry? Where do I find such more example or reading or exercises. I apologize for not printing the actual equations, there were too many symbols in them. Thanks.
  34. N

    How to find solutions to a Legendre equation?

    Homework Statement Find the first three positive values of \lambda for which the problem: (1-x^2)y^n-2xy'+\lambda y = 0, \ y(0)=0, \ y(x) & y'(x) bounded on [-1, 1] has nontrivial solutions. Homework Equations When n is even: y_1(x) = 1 - \frac{n(n+1)}{2!}x^2 +...
  35. 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!
  36. alyafey22

    MHB Prove the Legendre equation on Elliptic integrals

    K(k)E'(k)+K'(k)E(k)-K(k)K'(k)=\frac{\pi}{2} Complete Elliptic integral of first kind K(k)= \int^1_0 \frac{dx}{\sqrt{1-x^2}\sqrt{1-k^2x^2}} Complete Elliptic integral of second kind E(k)= \int^1_0 \frac{\sqrt{1-x^2}}{\sqrt{1-k^2x^2}}dx Complementary integral...
  37. D

    MHB Simplifying a Legendre polynomial

    Given the following expression \[ \mathcal{P}_{n}(0) = \left.\frac{1}{2^{n}n!}\frac{d^{n}}{dx^{n}} \sum_{k = 0}^{n}\binom{n}{k}(x^2)^k(-1)^{n - k}\right|_{x = 0}, \qquad (*)...
  38. 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...
  39. 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...
  40. F

    Solve Legendre Polynomial using Method of Frobenius

    Not sure how this can be done. can anyone help?
  41. S

    Existence of solution legendre equation

    Hi all, I have my exam in differential equations in one week so I will probably post a lot of question. I hope you won't get tired of me! Homework Statement This is Legendres differential equation of order n. Determine an interval [0 t_0] such that the basic existence theorem guarantees...
  42. C

    Why are some coefficients negative when expanding a function in Legendre?

    I get a series of Legendre expansion coefficients for a function. Then I compute the value of the function via expansion coefficients. As I want to whether the code is right or not, finally I expand the function in Legendre again. the result is almost same as before, but some coefficients is...
  43. 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...
  44. 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...
  45. 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...
  46. 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...
  47. ShayanJ

    Solving associated Legendre equation

    I'm trying to solve the associated Legendre differential equation: y''+\frac{2x}{x^2-1}y'+[ \lambda+\frac{m^2}{x^2-1}]y=0 By series expansion around one of its regular singularities.(e.g. x_0=1) This equation is of the form: y''+p(x)y'+q(x)y=0 Which is solved by the...
  48. K

    Maximizing Functions with Legendre Transforms

    Trying an example from a textbook but I don't understand Legendre transform at all. "construct Legendre transforms of the entropy that are natural functions of (1/T, V, n) and (1/T, V, μ/T)". I don't really understand where to start. An example prior to this exercise just states: A = E - TS =...
  49. 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...
  50. M

    Legendre equation , the Bessel equation and Sturm Liouville equation

    Show that the Legendre equation as well as the Bessel equation for n=integer are Sturm Liouville equations and thus their solutions are orthogonal. How I can proove that ..? :(
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