All Questions
Tagged with legendre-polynomials orthogonality
25
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Deriving quadrature weights from discrete orthogonality of exponentials
In the proof of Lemma 2 of Driscoll and Healy, it says
\begin{align}
\sqrt{2}\delta_{k,0} &= \frac{1}{\sqrt{2}}\int_0^\pi P_k(\cos\theta)\sin\theta d\theta\\\\
&= \frac{1}{2\sqrt{2}}\int_{-\pi}...
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Is it possible to prove orthogonal form of integral of legendre polynomial solely from legendre's differential equation without using anything?
The differential equation for the Legendre polynomials
$P_n(x)$ is given by:
$(1 - x^2) \frac{d^2P_n}{dx^2} - 2x \frac{dP_n}{dx} + n(n + 1)P_n = 0$. Now I want to prove that $\begin{equation} \int_{-...
- 11
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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
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Alternative orthogonality relations between associated Legendre polynomials
The usual orthogonality relations quoted for associated Legendre polynomials is:
$$
\int_{-1}^{1}P_{n}^{m}(x)P_{n'}^{m}(x)dx=\frac{2(n+m)!}{(2n+1)(n-m)!}\delta_{n,n'}
$$
However, I have come across ...
- 469
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Integral of a product of Legendre polynomials
I would like to show that
$$
\int_{-1}^{1}P_{n}^{1}(x)P_{n'}^{0}(x)\frac{x}{\sqrt{1-x^{2}}}\,\mathrm dx
=
\begin{cases}
-\frac{2n}{2n+1},&n=n'>0\\
-2,&n>n'\text{ and } n-n' \text{ even}\\...
- 469
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Orthogonality of Legendre Polynomials support
I've recently been introduced to orthogonality of Legendre polynomials, I think I understand the idea behind it; that the integral of the products of two polynomials in the range $(1,-1)$ will equal $...
- 253
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Deriving normalization for the shifted associated Legendre function
Where can I find a solution for this integral:
$ \int_{a}^{b} P^m_l(c_1x + c_2b)P^{m'}_l(c_1x + c_2b)\,d(c_1x + c_2b)$, most solutions only solves for the interval [-1,1]. Of course I am looking for ...
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Orthogonality of Legendre polynomials using specific properties
I'm having significant issues with a problem and would appreciate any help at all with it. It is regarding proving the orthogonality of Legendre polynomials using a specific recursion formula and ...
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1
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What non-zero function is w-orthogonal to all the polynomials of degree less than or equal to $n$?
Background
It is understood that a function $q$ is $w$ orthogonal to a function $p$ over $[a,b]$ if there holds:
$$ \int_{a}^{b} q(x)w(x)p(x)dx = 0$$
For example, for $w(x):=1$, $[a,b]=[-1,1]$, the ...
- 79
2
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2
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972
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Using Orthogonality of Legendre Polynomials to determine a Recurrence Relation
This is something that I've been struggling on for a few hours now and would appreciate any help:
Rodrigue's formula:
$P_n(x)$ = $ \frac{1}{2^nn!} \frac{d^n}{dx^n}(x^2-1)^n $
The Legendre ...
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1
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Orthogonal Property of Legendre Polynomials
How can I get $$nu_{n} + (n-1)u_{n-1},$$ where
$$u_{n} = \int_{-1}^{1} x^{-1}P_{n-1}(x) P_{n}(x)\, \mathrm {d}x\; ? $$
I did many search, and also I did try by myself. But without a success.
Is ...
- 7
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698
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Orthogonality of the First Four Legendre Polynomials
Using the recurrence relation $$(n+1)P_{n+1}=x(2n+1)P_n(x)-nP_{n-1}(x) \ \ n\geq 1,$$ I've calculated the first four Legendre Polynomials as
\begin{align}
P_0(x)&=1 \\
P_1(x)&=x \\
P_2(x)&...
user557493
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Orthogonality of Legendre polynomials with logarithmic functions
I have to find the value of this integral:
$\int_{-1}^1 \ln(1-x)*P_3(x)\,dx$
where $P_3(x)$ is the Legendre polynomial.
I thought I can write $\ln(1-x)$ as a summation of Legendre polynomials and ...
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1
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654
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Showing that the first two Legendre polynomials, $P_0(x) = 1$ and $P_1(x) = x$ are orthogonal for $x \in [−1,1]$
I would like assistance with the following problem:
Show that the first two Legendre polynomials, $P_0(x) = 1$ and $P_1(x) = x$ are orthogonal for $x \in [−1,1]$.
Determine constants $\alpha$ and $\...
- 4,322
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388
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Weighted orthogonality integrals of associated Legendre polynomials
Question
What does the following integral evaluate to?
$$
\int_{-1}^1 \mathrm dx\,\sqrt{1-x^2}P_m^n(x)P_l^n(x)?
$$
Context
I have developed a condition that may be written as
$$
\sum_{m=0}^\infty \...
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