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.
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...
I'm trying to implement and extend the work of Emmanuel Prados (http://www-sop.inria.fr/odyssee/research/prados-faugeras:04b/thesis.htm). I'm trying to follow how Appendix A, "How to transform a convex Hamiltonian into a HJB Hamiltonian; Legendre Transform", works for the provided example. I've...
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.
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...
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...
I am working on some homework that I already handed in, but I can't get one of the problems. The fourth problem on the HW was to prove the forms of (-1/p), (2/p), (3/p), (-5/p), and (7/p).
I did this for -1 and 2 using the quadratic residues and generalizing a form for them. for 3 and 7 i...
hi folks!
I have been trying to figure out some plausible geometric intrepretation to legendre polynomials and what are they meant to do.
I have come across the concept of orthogonal polynomials while working with some boundary value problems in solid mechanics and wasn't able to come to...