Legendre Definition and 202 Threads

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 =...

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...

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...

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...

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} \\...

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...

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...

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...

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...

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...

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

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.

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

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...

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...

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...

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...

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]

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 -...

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) =...

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...

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...

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...

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) ]...

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...

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...

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...

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 -...

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

There is a question where you should find a formula for P-n(0) using the Legendre polynomials: P-n(x)=1/(2^n*n!) d^n/dx^n(x^2-1)^n , n=0,1,2,3... I tried to derive seven times by only substituting the n until n=7,I did that because i wanted to find something that i can build my formula but i...

Homework Statement int x^m*P_n(x) dx=0 where integration is from (-1) to (+1).Given m<n Homework Equations The Attempt at a Solution I took integrand F(x) and saw that F(-x)=(-1)^(m+n)*F(x) should that help anyway?

Homework Statement I am to prove that P_n(-x)=(-1)^n*P_n(x) And, P'_n(-x)=(-1)^(n+1)*P'_n(x) Homework Equations The Attempt at a Solution I know that whether a Legendre Polynomial is an even or odd function depends on its degree.It follows directly from the solution of...

Homework Statement P_n (z) and Q_n (z) are Legendre functions of the first and second kinds, respectively. The function f is a polynomial in z. Show that Q_n (z) = \frac{1}{2} P_n (z) \ln \left(\frac{z+1}{z-1} \right) + f_{n-1} (z) implies Q_n (z) = \frac{1}{2} \int \frac{P_n (t) \...

Hello everyone i had some questions about legendre polynomials. I have solved most of them but i had just two not answered question. I tried to solve this problem by rodriguez rule but it was really hard for me. Could anyone help me or give me some hints for this question...

Hello everyone i had some questions about legendre polynomials. I have solved most of them but i had just two not answered question. I tried to solve this problem by rodriguez rule but it was really hard for me. Could anyone help me or give me some hints for this question...

Hello everyone i had some questions about legendre polynomials. I have solved most of them but i had just two not answered question. I tried to solve this problem by rodriguez rule but it was really hard for me. Could anyone help me or give me some hints for this question...

To show that two Legendre polynomials(Pn and Pm) are orthogonal wht is the test that i have to use? is it this? \int_{-1}^{1} P_{n}(x)P_{m}(x) dx = 0 in that case to prove that P3 and P1 are orthogonal i have to use the above formula??

Problem: Suppose we wish to expand a function defined on the interval (a,b) in terms of Legendre polynomials. Show that the transformation u = (2x-a-b)/(b-a) maps the function onto the interval (-1,1). How do I even start working with this? I haven't got a clue...

Problem: Show that \int_{-1}^{1} x P_n(x) P_m(x) dx = \frac{2(n+1)}{(2n+1)(2n+3)}\delta_{m,n+1} + \frac{2n}{(2n+1)(2n-1)}\delta_{m,n-1} I guess I should use orthogonality with the Legendre polynomials, but if I integrate by parts to get rid of the x my integral equals zero. Any tip on...

I was messing around with the \theta equation of hydrogen atom. OK, the equation is a Legendre differential equation, which has solutions of Legendre polynomials. I haven't studied them before, so I decided to take closed look and began working on the most simple type of Legendre DE. And the...

...and orthogonality relation. The book says \int_{-1}^{1} P_n(x) P_m(x) dx = \delta_{mn} \frac{2}{2n+1} So I sat and tried derieving it. First, I gather an inventory that might be useful: (1-x^2)P_n''(x) - 2xP_n'(x) + n(n+1) = 0 [(1-x^2)P_n'(x)]' = -n(n+1)P_n(x) P_n(-x) = (-1)^n P_n(x)...

Legendre transform... If we define a function f(r) with r=x,y,z,... and its Legnedre transform g(p) with p=p_x ,p_y,p_z,... then we would have the equality: Df(r)=(Dg(p))^{-1} (1) where D is a differential operator..the problem is..what happens when g(p)=0?...(this problem is...

I need some help. I fitted a 7th order legendre polynomial and got the L0 to L7 coefficients for different ANOVA classes. How can I get a back transformation in order to plot each class using the estimated coefficients? Thanks to anybody. Roberto.

some body who can explain for me the Legndre polynomials:eek: :eek:

Hi, I'm trying to prove the orthogonality of associated Legendre polynomial which is called to "be easily proved": Let P_l^m(x) = (-1)^m(1-x^2)^{m/2} \frac{d^m} {dx^m} P_l(x) = \frac{(-1)^m}{2^l l!} (1-x^2)^{m/2} \frac {d^{l+m}} {dx^{l+m}} (x^2-1)^l And prove \int_{-1}^1...

is the legendre equation an example of a frobenius equation?

can someone explain step-by-step why the legendre polynomial came into being? I'm having one hard time understanding it...

Hey there, does anyone know where I could find a list of Legendre Polynomials? I need them of the order 15 and above, and I haven't been able to find them on the net. Thanks!

Hi, I have a problem where I am given the Legendre equation and have been told 1 solution is u(x). It asks me to obtain an expression for the second solution v(x) corresponding to the same value of l. I think it requires Sturm Liouville treatment but don't have a clue how to begin. Please HELP!

Hi all, I've been doing a math problem about the Legendre differential equation, and finding there are two linearly independent solutions. When I was taught about quantum mechanics the polynomial solutions were introduced to me as the basis for spherical harmonics and consequently the...