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For questions about Legendre polynomials, which are solutions to a particular differential equation that frequently arises in physics.
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Simplification of an integral involving a linear combination of tensor products of Legendre polynomials
I have this expression: $$W(x,y)=\sum_{i=0}^{I}\sum_{j=0}^{J}W_{ij}P_i(x)P_j(y)$$ Where $I, J, W_{ij}$ are constants and $P_i , P_j$ are Legendre polynomials. I want to compute the following equation: ... integration laplacian legendre-polynomials- 11
How can I prove differential recurrence relation of Legendre polynomials without generating function?
What I am trying to prove is $P'_{n+1}(x)−P'_{n−1}(x)=(2n+1)P_n(x)$ What I can use are here: $P_0(x)=1$, $P_1(x)=x$ $\int_{-1}^1 P_m(x)P_n(x) \;dx = 0 \;(n\neq m)$ (the orthogonality of Legendre ... recurrence-relations legendre-polynomials- 11
Verifying Legendre's equation
Use the following results: $$(l+1)P_{l+1}(x)-x(2l+1)P_l(x)+lP_{l-1}(x)=0,$$ $$P_l(x)+2xP'_l(x)=P'_{l+1}(x)+P'_{l-1}.$$ in order to show the following recurrence relations: $$(2l+1)P_l(x)=P'_{l+1}(x)-P'... ordinary-differential-equations mathematical-physics legendre-polynomials- 121
Calculate the integral where $P_{n}$ and $P_{m}$ are Legendre Polynomials
Calculate the folowing integral: $$I_{k,m}=\int_{-1}^{1} x(1-x^2)P'_{n}(x)P'_{m} dx $$ So, my attempt to solve this consisted in: First, I thought of manipulating the folowing relations so i could get ... calculus integration mathematical-physics legendre-polynomials legendre-transformation- 29
$I(x) = -\int_0^1 \frac{1}{z}\ln\left(\frac{1-x z + \sqrt{1-2 x z+ z^2}}{2}\right)\,dz$
Is there a closed form integral for $$I(x) = -\int_0^1 \frac{1}{z}\ln\left(\frac{1-x z + \sqrt{1-2 x z+ z^2}}{2}\right)\,dz$$ for $-1 < x < 1$? This integral is related to Legendre polynomials ... integration asymptotics generating-functions closed-form legendre-polynomials- 1,166
Orthogonality of the Legendre polynomials with respect to $L_2$ (Integration by parts)
I want to show, that the Legendre polynonmials are orthogonal with respect to scalar product in $L_2[-1,1]$. The Legendre polynomials are defined as follows: $$P_n(x) = \left( \frac{2n+1}{2}\right)^{\... integration functional-analysis orthogonality legendre-polynomials gram-schmidt- 119
Partial wave decomposition of a function with Gamma functions
I have the following function $$f(x,s) = \frac{\Gamma(1-\frac{s}{2}(x-1))}{\Gamma(1+\frac{s}{2}(x-1))}$$ where $s>0$, $x \in [-1,1] $ and $\Gamma(z)$ are Gamma functions. I would like to decompose ... integration gamma-function legendre-polynomials- 532
Derive Jackson Equation 3.26
I want to derive equation 3.26 from jackson's book, classical electrodynamics. $(2l+1)\int_{0}^{1}P_l(x)dx=(-\frac{1}{2})^{(l-1)/2}\dfrac{(2l+1)(l-2)!!}{2(\dfrac{l+1}{2})!}$ where l is odd, using the ... legendre-polynomials- 1
Legendre polynomial property
How do i show from the Legendre's Polynomial equation $$P_n(x)=\sum_{k=0}^N \dfrac{(-1)^k(2n-2k)!}{2^nk!(n-k)!(n-2k)!}x^{n-2k}$$ where $N=n/2$ for even $n$ and $ N=(n-1)/2$ for odd $n$. Using just ... legendre-polynomials- 2,567
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 ... integration orthogonal-polynomials legendre-polynomials- 1
Creating a spanning set out of Legendre polynomials
In the book "An Introduction to Partial Differential Equations with MATLAB" by Matthew P Coleman (2nd edition), exercise 2 in chapter 11.3 states Show that the functions $$\phi_n(x) = P_n(x)... legendre-polynomials- 143
recurrence relation associated Legendre functions
I need a little help to find the recurrence relation $$\sqrt{1-x^2}P_l^m(x) = \frac{1}{2l+1} (P_{l-1}^{m+1}-p_{l+1}^{m+1})$$ Using the identity $$(2l+1)P_l(x) = \frac{d}{dx}(P_{l+1}(x)-P_{l-1}(x))$$ I ... recurrence-relations legendre-polynomials legendre-functions- 13
Evaluating an integral with derivatives of Associated Legendre polynomials
I came across the following integral $$\int_{-1}^{+1} (1-x^{2}) \frac{\partial P_{lm}(x)}{\partial x} \frac{\partial P_{km}(x)}{\partial x} dx$$ where $P_{lm}(x)$ is an associated Legendre polynomial, ... integration special-functions legendre-polynomials- 361
How to evaluate the following sum: $\sum_{n=1}^\infty \frac{P_n(x) - P_{n-1}(x)}{ n } \cos(nt)$
I am trying to find a closed form expression for the following sum, $$ F(x,t)= \frac{1}{\log\left(\frac{1+x}{2}\right)}\sum_{n=1}^\infty \frac{P_n(x) - P_{n-1}(x)}{ n } \cos(nt) ~, $$ where $P_n(x)$ ... summation legendre-polynomials- 109
Prove that the Legendre polynomial recurrence relationship satisfies the defining differential equation
I am trying to show that from this recurrent relationship $$ (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'' - ... polynomials approximation computational-mathematics orthogonal-polynomials legendre-polynomials- 477
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