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Class_17new

# Class_17new - PHYS809 Class 17 Notes Eigenfunction...

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1 PHYS809 Class 17 Notes Eigenfunction expansions for Green functions When we considered the Green function for the interior of a rectangle, we found that there was an asymmetry in the dependences on the coordinates. Here we briefly discuss how to develop Green functions that put the coordinates on an equal footing. Note that this can be done only at the expense of an additional sum over indices. Consider the equation 2 0, λ ∇ Φ + Φ = (17.1) subject to boundary conditions 0 Φ = on x = 0, x = a , y = 0 and y = b . By the method of separation of variables we find that non-trivial solutions are possible only if λ takes discrete eigenvalues given by 2 2 , mn m n a b π = + (17.2) where m and n are positive integers. The solutions are then the orthonormal eigenfunctions 2 sin sin . mn m x n y a b ab Φ = (17.3) The complete solution is a linear combination of terms of this form. The Green function is a solution of the Poisson equation

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Class_17new - PHYS809 Class 17 Notes Eigenfunction...

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