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. Lexaila

    Quadratic Residue and Quadratic Reciprocity Law QRL

    (p-6/p)=(-1/p)(2/p)(3/p) Make a table, so at the head row you have p(mod24), (-1/p), (2/p), QRL+-, (p/3) and finally (p-6/p), with in the head column below p (mod 24): 1,5,7,11
  2. G

    Apply the Legendre Transformation to the Entropy S as a function of E

    Hi, Unfortunately I am not getting anywhere with task three, I don't know exactly what to show Shall I now show that from ##S(T,V,N)## using Legendre I then get ##S(E,V,N)## and thus obtain the Sackur-Tetrode equation?
  3. M

    Without evaluating the Legendre symbols, prove the following....

    Since ## p=73 ## in this problem, how should I prove that ## \sum_{r=1}^{73-1}r(r|73)=0 ##? Given that ## 73=1\pmod {4} ##.
  4. M

    Evaluate the Legendre symbol ## (999|823) ##

    Consider ## (999|823) ##. Then ## 999\equiv 176\pmod {823} ##. This implies ## (999|823)=(176|823)=(16|823)(11|823)=(4^{2}|823)(11|823) ##. Since ## (a^{2}|p)=1 ##, it follows that ## (4^{2}|823)=1 ##. Thus ## (999|823)=(11|823) ##. Applying the Quadratic reciprocity law, we have that ##...
  5. SaintRodriguez

    I Legendre Transformation & Hamilton-Jacobi Formalism: A Relationship?

    Hey I have a question about the relation between Legendre transformation and Hamilton-Jacobi formalism. Is there some relation? Cause Hamilton-Jacobi is the expression of Hamiltonian with a transformation.
  6. 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...
  7. 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...
  8. 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...
  9. TheGreatDeadOne

    I Using recurrence formula to solve Legendre polynomial integral

    I am trying to prove the following expression below: $$ \int _{0}^{1}p_{l}(x)dx=\frac{p_{l-1}(0)}{l+1} \quad \text{for }l \geq 1 $$ The first thing I did was use the following relation: $$lp_l(x)+p'_{l-1}-xp_l(x)=0$$ Substituting in integral I get: $$\frac{1}{l}\left[ \int_0^1 xp'_l(x)dx...
  10. 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...
  11. 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.
  12. J

    Legendre polynomial - recurrence relations

    Note: $P_n (x)$ is legendre polynomial $$P_{n+1}(x) = (2n+1)P_n(x) + P'_{n-1}(x) $$ $$\implies P_{n+1}(x) = (2n+1)P_n(x) + \sum_{k=0}^{\lfloor\frac{n}{2}\rfloor} (2(n-1-2k)+1)P_{n-1-2k}(x))$$ How can I continue to use induction to prove this? Help appreciated.
  13. 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: $$...
  14. bryanso

    Associated Legendre Function with Angles

    In Wikipedia https://en.m.wikipedia.org/wiki/Associated_Legendre_polynomials, Section Reparameterization in terms of angles, I see this argument: Let ## x = cos\,\theta ## ## \sqrt{1 - x^2} = sin\,\theta ## This is also in Griffiths' Introduction to Quantum Mechanics. Why is this a valid...
  15. L

    Minimizing as a function of variables

    As promised, here is the original question, with the integral written in a more legible form.
  16. 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 =...
  17. S

    A Derive / verify Legendre P (cos x)

    hello, I am trying (and failing) to verify / derive the result of the Legendre polynomial P11 (cos x) = sin x Griffiths Quantum chapter 4 Table 4.2 I figured it would not be too bad. I have attempted this 3 or 4 times trying to be careful. I keep getting sin(x) times some additional trig...
  18. redtree

    I Legendre Transform: Momentum & Velocity

    I apologize for the simplicity of the question. I have been reading a paper on the Legendre transform (https://arxiv.org/pdf/0806.1147.pdf), and I am not understanding a particular step in the discussion. In the paper, Equation 16, where ##\mathcal{H} = \sqrt{\vec{p}^2 + m^2} ##...
  19. 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...
  20. 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...
  21. A

    MHB How to integral legendre polynomial

    Question \int_{-1}^{1} cos(x) P_{n}(x)\,dx ____________________________________________________________________________________________ my think (maybe incorrect) \int_{-1}^{1} cos(x) P_{n}(x)\,dx \frac{1}{2^nn!}\int_{-1}^{1} cos(x) \frac{d^n}{dx^n}(x^2-1)^n\,dx This is rodrigues formula by...
  22. 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...
  23. 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 ?
  24. 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...
  25. C

    I Is there a geometric interpretation of orthogonal functions?

    Hi all. So to start I'll say I'm just dealing with functions of a real variable. In my linear algebra courses one thing was drilled into my head: "Algebraic invariants are geometric objects" So with that in mind, is there any geometric connection between two orthoganal functions on some...
  26. M

    MATLAB Associated Legendre Polynomial of 1st and 2nd kind

    Hi PF! In MATLAB I'm trying to use associated Legendre polynomials of the 1st and second kind, widely regarded as ##P_i^j## and ##Q_i^j##, where ##j=0## reduces these to simply the Legendre polynomials of the 1st and second kind (not associated). Does anyone here know if MATLAB has a built in...
  27. C

    Coefficient Matching for different series

    Homework Statement Hello, I have a general question regarding to coefficient matching when spanning some function, say , f(x) as a linear combination of some other basis functions belonging to real Hilbert space. Homework Equations - Knowledge of power series, polynomials, Legenedre...
  28. J

    MHB Complex Variables - Legendre Polynomial

    We define the Legendre polynomial $P_n$ by $$P_n (z)=\frac{1}{2^nn!}\frac{d^n}{dz^n}(z^2-1)^n$$ Let $\omega$ be a smooth simple closed curve around z. Show that $$P_n (z)=\frac{1}{2i\pi}\frac{1}{2^n}\int_\omega\frac{(w^2-1)^n}{(w-z)^{n+1}}dw$$ What I have: We know $(w^2-1)^n$ is analytic on...
  29. 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...
  30. 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...
  31. M

    Legendre Polynomial Integration

    Homework Statement Simplify $$\int_{-1}^1\left( (1-x^2)P_i''-2xP'_i+2P_i\right)P_j\,dx$$ where ##P_i## is the ##i^{th}## Legendre Polynomial, a function of ##x##. Homework Equations The Attempt at a Solution Integration by parts is likely useful?? Also I know the Legendre Polynomials are...
  32. 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'}...
  33. Mayan Fung

    I Determining the coefficient of the legendre polynomial

    We know that the solution to the Legendre equation: $$ (1-x^2)\frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + n(n+1) = 0 $$ is the Legendre polynomial $$ y(x) = a_n P_n (x)$$ However, this is a second order differential equation. I am wondering why there is only one leading coefficient. We need two...
  34. mercenarycor

    Question about Legendre elliptic integrals

    Homework Statement [/B] J(a, b, c;y)=∫aydx/√((x-a)(x-b)(x-c)), let a<b<cHomework Equations f(θ, k)=∫0θdx/√(1-k2sin2(x)), k≤1 The Attempt at a Solution This is an example from my study material, and I don't understand the first step they do. Let x=a+(b-a)t, dx=(b-a)dt Wait...what? Why? How...
  35. 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...
  36. S

    A Chain rule - legendre transformation

    let df=∂f/∂x dx+∂f/∂y dy and ∂f/∂x=p,∂f/∂y=q So we get df=p dx+q dy d(f−qy)=p dx−y dqand now, define g. g=f−q y dg = p dx - y dq and then I faced problem. ∂g/∂x=p←←←←←←←←←←←←←←← book said like this because we can see g is a function of x and p so that chain rule makes ∂g/∂x=p but I wrote...
  37. redtree

    A Conjugate variables in the Fourier and Legendre transforms

    In quantum mechanics, position ##\textbf{r}## and momentum ##\textbf{p}## are conjugate variables given their relationship via the Fourier transform. In transforming via the Legendre transform between Lagrangian and Hamiltonian mechanics, where ##f^*(\textbf{x}^*)=\sup[\langle \textbf{x}...
  38. 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...
  39. 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}...
  40. 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...
  41. RJLiberator

    Legendre Transformation of f(x) = x^3

    Homework Statement [/B] Find the Legendre Transformation of f(x)=x^3 Homework Equations m(x) = f'(x) = 3x^2 x = {\sqrt{\frac{m(x)}{3}}} g = f(x)-xm The Attempt at a Solution I am reading a quick description of the Legendre Transformation in my required text and it has the example giving for...
  42. N

    I Why use the Legendre transform over back substitution

    In thermodynamics we use a variation of the Legandere Legendre transform to move from one description of the system to another ( depending on what is the control variable...), but I don't understand why choose to use the Legandere Legendre transform over writing x in terms of s=dy/dx and back...
  43. 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...
  44. 1

    How do I set up this Legendre Transform for Hamiltonian

    Homework Statement Im trying to understand the Legendre transform from Lagrange to Hamiltonian but I don't get it. This pdf was good but when compared to wolfram alphas example they're slightly different even when accounting for variables. I think one of them is wrong. I trust wolfram over the...
  45. W

    What Determines the Values of Legendre Polynomials at 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...
  46. 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...
  47. 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...
  48. P

    Kronecker Delta in Legendre Series

    Hello everyone, I'm working through some homework for a second year mathematical physics course. For the most part I am understanding everything however there is one step I do not understand regarding the steps taken to solve for the coefficients of the Legendre series. Starting with setting...
  49. Msilva

    Writing [itex]x^3[/itex] in Legendre base

    Hello friends. I need help to write the function x^3 as a somatory using the Legendre polinomials as base. Something like: f(x)=\sum^{\infty}_{n=0}c_{n}P_{n}(x) Basically is to find the terms c_{n}. But, the problem is that Legendre polinomials does't form a orthonormal base: \langle...
  50. W

    Associated Legendre polynomial (I think)

    Homework Statement I'm not 100% sure what this type of problem is called, we weren't really told, so I'm having trouble looking it up. I'd really appreciate any resources that show solved examples, or how to find some! Anyway. For the solution to the spherical wave equation φ(t, θ, Φ) i)...