Legendre Definition and 202 Threads

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...

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...

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...

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...

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 -...

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...

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...

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})

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...

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...

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...

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...

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.

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...

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...

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...

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...

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...

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...

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...

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...

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...

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) \,.

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...

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...

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...

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 <...

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...

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...

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...

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...

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...

[FONT="Palatino Linotype"]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

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)...

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...

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)...

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.

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...

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

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}...

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...

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...

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...

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...

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?

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...

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...

"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[FONT=Times New Roman]≡3 (mod 8) and (-1)^{(p^2-1)/8} = (-1)^{(3^2-1)/8}" (quote from my textbook)...

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...

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...