# 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...

10. ### 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. ### 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. ### 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.

17. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...
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### 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. ### 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. ### 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. ### 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. ### 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...
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### 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...

40. ### 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. ### 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. ### 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...

48. ### 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. ### Writing $x^3$ 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. ### 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)...