Q1 (20 points) Consider the Helmholtz equation vệu + 4 = 0, on the periodic cut disk domain Do = {r > 0; 0 Sose" }, defi

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Q1 (20 points) Consider the Helmholtz equation vệu + 4 = 0, on the periodic cut disk domain Do = {r > 0; 0 Sose" }, defined in terms of polar coordinates (r, 0) and with 0 < 0 3 27. The domain is periodic in the direction with period (*) (a) Using a separation of variables technique, show that the most general (*-periodic, bounded, separable solution to (*) is ulr, 6) = 4yelv, /, ('); 2пл 0 (1) e-00 for some coefficients An, where , is a Bessel function of the first kind of index v,. (You may quote without proof the general solution of Bessel's equation with either integer or non-integer index.) (b) Now suppose (* = 21. Show that u = ely, where y is the usual Cartesian coordinate, satisfies (*) and can thus be written in the form of (*). In this case, show that the coefficients An satisfy 25 AJ\(v) = ." elitr sin 0-10) de. (c) By evaluating this expression at a particular value of r, show that Ao = 1. By considering the limiting behaviour of the n'h derivative of J.(r) as r → 0), determine A, for all n > 0. Explain clearly how you can also calculate the values of A, for all n < 0, and hence deduce that erle-r' )2 = "J,(").