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line. These are just some basic notes on Bessel functions and their application to finding the eigenfunctions of the Laplacian. 1 Where Bessel Functions Arise Consider the problem of finding the eigenvalues/vectors for the Laplacian in 2 dimensions: We wish to find a pair (u,λ) which solves ∆u = −λu, (1) (since the eigenvalues of ∆ will turn out to be non-positive, we express them as −λ where λ will be non-negative). Let’s expressu = u(r,θ) as a function in polar coordinates and recall that in polar coordinates the Laplacian can be expressed a

Year: 2012
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