Questions tagged [eigenfunctions]

For questions on eigenfunctions, each of a set of independent functions that are the solutions to a given differential equation.

498 questions
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Proof that Legendre Polynomials are Complete

Can somebody either point me to, or show me a proof, that the Legendre polynomials, or any set of eigenfunctions, are complete?
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Finding eigenvalues and eigenfunctions for a BVP

Find the eigenvalues and eigenfunctions for $$y'' + \lambda y = 0, y(0) = 0, y'(\pi/2) = 0$$ According to my book we must check 3 cases: $\lambda < 0$, $\lambda = 0$, $\lambda > 0$. I started ...
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Understanding orthogonality in two-scale asymptotic expansion (cf. G. Allaire)

This question is about equation (2.16) of Lecture 2 on Homogenization in Porous Media_ by Allaire page 28. There are two spacial scales: $x$ being macroscopic and $y=\dfrac{x}{\varepsilon}$ being ...
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Show that the function is an eigenfunction of the equation

I'm not sure how to use the bbcode so I've taken a screenshot instead: Came up on a past exam paper that I'm working towards and I'm not sure how to answer it. I assumed that EQN . EIGENFUNCTION = ...