All Questions
Tagged with legendre-polynomials power-series
26
questions
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
0
votes
0
answers
75
views
Coefficient of $x^n$ in Legendre series expansion
Suppose we are approximating a function $f$ with a Legendre series of order $N$, namely
$$
f(x) \approx \sum_{n=0}^N c_n P_n(x) \equiv f_N(x)
$$
where $P_n(x)$ is the $n^{th}$ Legendre polynomial and
$...
- 31
4
votes
1
answer
130
views
Series of product of legendre polynomials with shifted degree
I am working on some quantum mechanics and I would love to find a closed expression for the series
$$
S(x,y) = \sum_{l=0}^\infty P_l(x) P_{l+1}(y)
$$
$x,y \in \left[ -1, 1\right]$ and also for its ...
1
vote
1
answer
112
views
Showing that a series that involves Legendre polynomial converges.
I am currently studying Legendre polynomials and I am trying to prove its generating function specifically using the recurrence relation
$$ (n+1)L_{n+1}(x) = x(2n+1)L_n(x) - nL_{n-1}(x), \quad \forall ...
- 1,141
1
vote
0
answers
20
views
further simplification of a summation involving Legendre and Associated Legendre polynomials
Was calculating something from a physics problem and found myself dealing with the following summation:
$$ \sum\limits_{\text{odd} \text{ } l}^{\infty}P_{l+1}(0)P'_l (\cos{\theta})\sin{\theta}\ $$
By ...
- 183
0
votes
0
answers
46
views
Formula for the derivative of finite power series in reversed order of terms.
I wanted to solve the polar part in Schrödinger's wave equation for the H-atom by direct substitution of functions of form:-
$$
\Theta_{lm}(\theta) = a_{lm} \sin^{|m|}\theta \sum_{r≥0}^{r≤(l-|m|)/2}(-...
- 1,210
2
votes
1
answer
342
views
Coefficient of the Highest Degree in the Power Series Solution to Legendre's Differential Equation
The Legendre's differential equation
$$(1-x^2)y''-2xy'+n(n+1)y=0$$
Substitute $y=\sum_{m=0}^\infty a_mx^m$
$$
\sum_{m=2}^\infty m(m-1)a_mx^{m-2}-\sum_{m=2} ^\infty m(m-1)a_mx^m-\sum_{m=1}^\infty ...
- 7,674
2
votes
0
answers
119
views
Legendre differential operator
I'm trying tu calculate the eigenvalues of Legendre differential operator, that is:
$$-\frac{d}{dx}\left( (1-x^2) \frac{du}{dx}\right) = \lambda\, u \,\,\,\text{in}\,\,\,x\in[-1,1]$$
Using series of ...
- 1,774
0
votes
1
answer
148
views
Non-analytic smooth functions and Legendre polynomials
From Wikipedia: a function $f$ is real analytic on an open set $D$ in the real line if for any $x_0 \in D$ one can write
$$f(x) = \sum_{n = 0} ^ \infty a_n (x - x_0) ^ n$$
in which the coefficients $...
- 15
0
votes
1
answer
240
views
Maclaurin series solution to Legendre equation and general expression for coefficients
Consider a Maclaurin series solution
$$y = (1−x^2)y′′ −2xy′ +α(α+1)y=0, −1<x<1.$$
Show that $$a_2 = \frac{-α(α+1)}{6}a_0$$ $$a_3=\frac{−(α−1)(α+2)}{6}a_1$$
and, for all $n≥2$,
$$a_{n+2} = \...
- 23
1
vote
1
answer
820
views
How to solve Legendre's differential equation without power series assumption?
Legendre's differential equation $\,(1-x^2)\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+\ell(\ell+1)y=0\, $ is usually solved in most text-books either by assuming a power series solution or by Frobenius method....
- 344
1
vote
1
answer
238
views
Recurrence relations and power series solution
I am given the following initial value problem: $$(1-x^2)y''+7xy'-26y=0 \qquad , \qquad y(0)=0 \qquad , \qquad y'(0)=4$$
I have solved for the singular points, which are $x= 1, -1$
The question ...
- 89
0
votes
1
answer
763
views
How to tell if series terminates (Legendre ODE)
when solving for the coefficients for the Legendre ODE $(1-x^2)y’’-2xy’+l(l+1)y=0$, I understand how to obtain the recurrence relation
$$a_{k+2}=\frac{k(k+1)-l(l+1)}{(k+2)(k+1)}a_k.$$
What I do not ...
- 2,238
2
votes
1
answer
568
views
differentiating the generating function of the Legendre equation
I need to differentiate the generating function
$$G(x,t)=\sum a_n(x)t^n$$ w.r.t. x in order to solve
$$\tfrac{d}{dx}[(1-x^2)\tfrac{dG}{dx}]+\tfrac{d}{dt}[t^2\tfrac{dG}{dt}]$$.
But I don't ...
- 2,815
2
votes
1
answer
84
views
An ordinary differential equation equal to an infinite sum of legendre polynomials
what is the method for solving a differential equation when a summation is involved from the start ?
ex. what method is required to find a particular solution to
$$(1-x^2)y''-2xy'=\sum_{n=1}^{\...
- 2,815