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Tagged with legendre-polynomials orthogonal-polynomials
86
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Integral of product of Legendre polynomial and exponential function
Kindly help me with the following integral :
$
I_l(a) = \int_{-1}^{+1} dx\, e^{a x} P_l(x) \quad
$
($a$ is real and positive).
I thought to use the following relation given in Gradshteiyn and also ...
- 65
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0
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37
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Product of d-dimensional Legendre polynomials
Let $P_n:\mathbb{R}\rightarrow\mathbb{R}$ be the $d$-dimensional Legendre polynomials, that is they are orthonormal polynomials w.r.t the probability measure $\mu_d$ on $[-1,1]$ given by $\mu_d= \...
- 993
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Understanding the Dual Emergence of Legendre Polynomials in Differential Equations and Orthogonalization
I am currently examining the mathematical properties of Legendre polynomials and have observed their emergence in two distinct areas: as solutions to a specific class of differential equations (...
- 139
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106
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How do I find the leading coefficient of a Legendre polynomial?
I'm trying to construct the Legendre polynomials from the differential equation. As is done in this set of lecture notes, I can get an expression for the coefficient $c_{l-2k}$ in terms of $c_l$:
$$c_{...
- 11
1
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1
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Show that $K(u)=\sum_{k=0}^r P_k(0) P_k(u) \mathbf{1}_{\{|u| \leq 1\}}$ is a kernel of order $r$
I have a question concerning the construction of kernels wit orthogonal polynomials. The instructor defined the orthogonal polynomials as
$$P_0(x)=\frac{1}{\sqrt{2}}, P_m(x)=\sqrt{\frac{2 m+1}{2}} \...
- 275
1
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0
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89
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Generating function of orthogonal polynomial basis
I'm studying the bases made up by orthogonal polynomial such as: Hermite, Legendre, Laguerre, Chebyshev. On my book there is a theoretical introduction that gives the difinition of generating function ...
- 323
2
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85
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How to integrate products of Legendre functions over the interval [0,1]
The associated Legendre polynomials are known to be orthogonal in the sense that
$$
\int_{-1}^{1}P_{k}^{m}(x)P_{l}^{m}(x)dx=\frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{k,l}
$$
This is intricately linked to ...
- 469
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123
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Inner product of 4 Legendre Polynomials
Is there a closed form for the quadruple inner product of Legendre Polynomials such as:
\begin{align}
\int_{-1}^{1}P_k(x)P_l(x)P_m(x)P_n(x)dx
\end{align}
I am aware of solutions for the triple inner ...
- 23
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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:
$$P_{2n}(0) = \frac{(-1)^...
- 29
2
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1
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How to prove Legendre Polynomials' recurrence relation without using explicit formula?
Assume there is an inner product on linear space $V = \{ \text{polynormials}\}$:
$$\langle f, g\rangle = \int_{-1}^1 w(t) f(t)g(t) dt$$ with $w(t) \ge 0$ and not identically zero.
Then we can ...
- 1,231
3
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1
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Reference(book or article) for an explicit formula of Legendre polynomials
The following explicit formula is stated for Legendre polynomials on Wikipedia.
\begin{equation}
P_n(x)=\sum_{k=0}^n {n\choose k}{n+k \choose k} \left(\dfrac{x-1}{2}\right)^2
\end{equation}
Do you ...
- 45
1
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1
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267
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Uniqueness of the nodes for Gauss-Legendre quadrature
Gauss-Legendre quadrature approximates $\int_{~1}^{1}f(x)dx$ by $\sum_{i=1}^nw_if(x_i)$.
Wikipedia says that
This choice of quadrature weights $w_i$ and quadrature nodes $x_i$ is the unique choice ...
user1095295
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64
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Calculation of the integral of the Legendre polynomial of the second kind
Please tell me the possible options for calculating the integral of the form $\int\limits_{-\infty}^a\frac{Q_n(x)}{(x+b)^{n+2}}dx$, where $a\in[-2,-\infty)$; $b\in(-a,-\infty)$; $Q_n(x)$ - Legendre ...
0
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1
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350
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Prove that the Legendre polynomial recurrence relationship satisfies the defining differential equation
I am trying to show that from this recurrent relationship
$$ (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
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1
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270
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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\...
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