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Orthogonal Property of Legendre Polynomials

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#ordinary-differential-equations #recurrence-relations #orthogonality #legendre-polynomials
237
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Prove that the Legendre polynomial holds true using the generating function and a binomial expansion.

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#generating-functions #legendre-polynomials
614
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Trying to prove that $\pi$ is irrational using Legendre Polynomials.

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#number-theory #irrational-numbers #legendre-polynomials
39
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Proof for a Legendre polynomial identity on a 2-sphere that appears in cosmology

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#legendre-polynomials #spherical-harmonics
791
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$L^2$ norm of the first derivative of Legendre polynomials

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#generating-functions #orthogonal-polynomials #legendre-polynomials
140
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Expressing spherical functions with zonal harmonics

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#legendre-polynomials #spherical-harmonics
235
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Recurrence relations and power series solution

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#recurrence-relations #power-series #legendre-polynomials
78
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Numerically approximating functions on the sphere by rotating zonal harmonics.

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#harmonic-analysis #legendre-polynomials #spherical-harmonics
229
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Integral Expression of Legendre Polynomials

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#analysis #legendre-polynomials
86
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Differing Forms of Legendre Polynomials

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#polynomials #legendre-polynomials
140
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How to convert a hypergeom function to the Legendre function?

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#hypergeometric-function #legendre-polynomials #legendre-functions
963
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Integral with Legendre Polynomial $\int x^2(P_l(x))^2dx$

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#definite-integrals #legendre-polynomials
489
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Rodrigues' formula proof

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#legendre-polynomials
569
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Evaluate $\int_0^1 x^{a-1}(1-x)^{b-1}\operatorname{Li}_3(x) \, dx$

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#integration #definite-integrals #hypergeometric-function #legendre-polynomials #polylogarithm
134
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$\left|\frac{d^n}{dx^n}(x^2-1)^n\right|\le\sqrt\frac2\pi\cdot\frac{2^nn!}{\sqrt n}\cdot\frac1{\sqrt[4]{1-x^2}}$, a bound for Legendre Polynomial

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#calculus #derivatives #inequality #special-functions #legendre-polynomials

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