All Questions
Tagged with legendre-polynomials polynomials
33
questions
2
votes
0
answers
39
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Multidimensional Legendre polynomials?
Legendre polynomials can be given by several expressions, but perhaps the most compact way to represent them is by Rodrigues' formula as
$$P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n.$$
I ...
- 934
0
votes
0
answers
49
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Interpolation and general Gaussian quadrature
I just finished a course on numerical mathematics, and have become quite interested in interpolation and how it ties into numerical integration. What suprised me while studiyng quadrature was the fact ...
- 358
1
vote
1
answer
83
views
Does it make sense to think about formal power series where the coefficients belong to a Ring?
Normally, one defines formal power series as below:
Let $F$ be a field. A formal power series is an expression of the form
$$ a_0 + a_1x + a_2x^2 + \dots = \sum_{n \geqslant 0} a_nx^n,$$
where $\{a_n\}...
- 1,141
0
votes
1
answer
74
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Could someone help prove some properties of Legendre Polynomials?
I have already proved other properties of the Legendre polynomials, like:
$$P_n(-x) = (-1)^n \, P_n(x)$$
$$P_{2n+1}(0) = 0$$
$$P_n(\pm1)= (\pm1)^n$$
But I can't get this one:
$$P_{2n}(0) = \frac{(-1)^...
- 29
0
votes
1
answer
64
views
Comparing the different bases for representing a function
Suppose I have some function $f(x)$.
I know that this function can be represented in several different bases. For example, we can express this using Legendre polynomials, Hermite polynomials, Fourier ...
0
votes
1
answer
350
views
Prove that the Legendre polynomial recurrence relationship satisfies the defining differential equation
I am trying to show that from this recurrent relationship
$$ (n+1)P_{n+1}(x) = (2n+1)xP_n(x) - nP_{n-1}(x) $$
that the Legendre polynomial $P_n(x)$ satisfies the differential equation
$$ (1-x^2)P'' - ...
- 553
1
vote
1
answer
270
views
How $x^n$ is linearly represented by Legendre polynomials
I recently come across a problem with respect to Legendre polynomial as follows.
Let $L^2[-1,1]$ be the Hilbert space of real valued square integrable functions on $[-1,1]$ equipped with the norm $\|f\...
- 493
4
votes
1
answer
177
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The values of $P_n(x)$ at the zeros of $P'_n(x)$
I recently come across a problem with respect to Legendre polynomial as follow.
For any $n \in \mathbb{N}$, $P_n(x) := \frac{1}{2^n n!}\frac{{\rm d}^n (x^2-1)^n}{ {\rm d} x^n} $ is the classical ...
- 493
0
votes
1
answer
432
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Proof the Recurrence formula with Rodrigues' Formula
My University Professor gave the task to proof the recurrence formula without generating function of Legendre Polynomial , only with Rodrigues' Formula .
So far, I used :
$P_l\left(x\right)=\frac{1}{{...
- 13
1
vote
1
answer
257
views
Initial guess in Newton-Raphson method.
To find roots using the Newton-Raphson method, the initial guess is very important otherwise it may take several iterations to give the value of roots. For the given Legendre polynomial ($ P _ 8 $), ...
- 21
1
vote
0
answers
319
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Understanding the dimensionality of Legendre polynomials.
I have been looking at the Laplace equation $\nabla^2 f = 0$ in various dimensions. In 3 dimensions, the angular equation leads to the well-known spherical harmonics, defined up to normalisation as
\...
- 241
0
votes
1
answer
480
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Orthonormal polynomial basis of $L^2([0,1])$
I was wondering if, given a natural number $i\in \mathbb N$, there exists an orthonormal basis (w.r.t. the standard scalar product) $(p_n)_{n \in \mathbb N}$ of $L^2([0,1])$ such that $p_n$ is a ...
- 1
1
vote
0
answers
46
views
Derivative of $(X^2-1)^m$ , Legendre polynomial
Let :
$n \in \mathbb{N}$
$\Phi(P)=[ (X^2-1)P']'$ for $P \in \mathbb{R}_n[X]$
$m \leq n$
$W_m(X)=(X^2-1)^m$
$P_m(X)= k_m \frac{ d^m}{dX^m}[ W_m(X)]$ with $k_m$ such that $P_m(1)=1$
We want to express ...
- 877
-1
votes
1
answer
79
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A question about Legendre polynomials
If $P_n(x)$ is a Legendre polynomial of degree $n$. If a is such that $P_n(a)=0$ that is $a$ is a root of $P_n(x)=0$.
Then the $P_{n-1}(a)$ and $P_{n+1}(a)$ is:
equal ?
or not equal ?
Or are of ...
3
votes
1
answer
1k
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Orthogonality of Legendre polynomials from generating function
Given the the Legendre polynomials generating function:
$$G(x,t)=\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n$$
prove the relation:
$$\int_{-1}^{1} (P_n(x))^2 dx = \frac{2}{2n+1}$$
My ...
