All Questions
Tagged with legendre-polynomials ordinary-differential-equations
89
questions
1
vote
0
answers
33
views
Derivation of the associated Legendre Polynomials
I have been struggling to find a proper derivation of the associated Legendre Polynomials and a derivation of
$$P_l^{-m}(\mu)=(-1)^m\frac{(l-m)!}{(l+m)!}P_l^m(\mu)$$
Can someone point to a proper ...
0
votes
1
answer
67
views
Could someone explain the reason behind using Legendre Polynomials?
Where $x = \cos(\phi)$ and therefore (and this is very important) $x=[-1,1]$, we have the (special such that m = 0) associated Legendre Equation $(1-x^2)\frac{d^2 y}{dx^2} - 2x \frac{dy}{dx}+ky = 0$, ...
- 583
0
votes
1
answer
68
views
Is the Gram-Schmidt Orthogonalization process for functions the same as it is for vectors? [closed]
I was not able to find resources online and was wondering so since it would greatly help me with my work if I can directly apply what I have learned from linear algebra.
- 17
0
votes
1
answer
47
views
Finding $l$ such that the Legendre differential equation has a polynomial solution
I was given this problem for practice and was wondering if my approach was correct:
$$
(1-x^2)y'' - 2xy' + 3ly = 0.
$$
At first I thought of just using $l = 2$ since the Legendre DE is defined in ...
- 17
2
votes
0
answers
67
views
Fourier-Legendre series for $x^n$
I'm struggling to find the Legendre expansion for $x^n$ (exercise 15.1.17 from Mathematical Methods for Physicists).
I'm trying to evaluate the following integral:
$$a_m = \frac{2m+1}{2} \int_{-1}^{1} ...
- 29
0
votes
0
answers
25
views
Legendre Differential equation, n(n-1) or n(n+1)
I am confused regarding the Legendre Differential Equations' coefficients.
In some books its,
$(1-x^2)y''-2xy'+n(n-1)y=0$
and somewhere it is,
$(1-x^2)y''-2xy'+n(n+1)y=0$
what is its correct form?
- 25
1
vote
1
answer
133
views
Legendre's Polynomial and spherical harmonics
The differential equation that is satisfied by the Legendre's polynomials is:
$$(1-x^2)y'' - 2xy' + \lambda y = 0 (*)$$
I have also been told that the Legendre's polynomial with the parameter $x = \...
- 376
0
votes
0
answers
19
views
Spheroidal eigenvalues with shifted boundary conditions
I was studying the spheroidal differential equation in relation to calculating solutions for fields in a general Kerr background metric and, as far as I can tell, the eigenvalues $\lambda$ that enter ...
- 175
2
votes
1
answer
354
views
How to obtain the Legendre Polynomials from the power series solution (of the Legendre's equation)?
Solve the differential equation $$(1-x^2)\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+n(n+1)y=0.$$ Show that a polynomial, say $P_n(x)$ is a solution of the above equation, when $n$ is an integer.
I tried ...
- 3,073
3
votes
1
answer
414
views
Proof of Bonnet's Recursion Formula for Legendre Functions of the Second Kind?
I'm doing some self-study on Legendre's Equation. I have seen and understand the proof of Bonnet's Recursion Formula for the Legendre Polynomials, $P_n(x)$.
$$(n+1)P_{n+1}(x) = (1+2n)xP_n(x) - nP_{n-...
2
votes
1
answer
230
views
Prove that polynomial $p(x)$ od degree $n$ is equal to $cP_{n}(x)$, where $P_{n}(x)$ is Legendre polynomial.
This is the problem $5$ from chapter $45$ on properties of Legendre polynomials from Simmons book "Differential Equations with Applications and Historical Notes".
If $p(x)$ is a polynomial ...
- 31
0
votes
1
answer
23
views
How do we determine if an operator over real functions is normal?
We have the operator $T(f) = (pf')'$, where $p(x) = x^2 - 1$. The inner product is $\displaystyle (f,g) = \int_{-1}^1 f(x)g(x) dx$. How do we infer whether eigenfunctions corresponding to different ...
0
votes
1
answer
165
views
What will be the explicit formula for Shifted Legendre's polynomial in interval $x\in[a,b]$
If I defined shifted Legendre polynomial $\tilde P_{n}(x)=P_{n}(\frac{2x-b-a}{b-a}) for\;all x\in[a,b]$ Then what will be the explicit formula $P_{n}(x)=\sum_{k=0}^{n} (-1)^{n+k} \frac{(n+k)!}{(n-k)! ...
1
vote
0
answers
167
views
Problem about the solution of Legendre's equation
I am reading a textbook on differential equations. This book is not written in English and surely you have not heard of it. The chapter I am studying now is about solving differential equations using ...
- 411
1
vote
0
answers
58
views
Differentiating Legendre polynomials
I am a physics undergraduate. I am consciously and specifically asking this question in math stack exchange and not physics stack exchange. I am working on solving for electric potential in charge ...
- 557