MATH300 Midterm: MATH 300 UofA Exam Solution 1
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Let f (x) = cos a x, 0 x < a f (x) =( cos x. |x| > . (a) find the fourier integral of f. (b) for which values of x does the integral converge to f (x)? (c) evaluate the integral for < x < . Let fc denote the fourier cosine transform and fs denote the fourier sine transform. Assume that f (x) and xf (x) are both integrable. (a) show that (b) show that. Chebyshev"s di erential equation reads (1 x2)y xy + y = 0, y(1) = 1, 1 < x < 1 (a) divide by 1 x2 and bring the di erential equation into sturm-liouville form. Decide if the resulting (b) for n 0, the chebyshev polynomials are de ned as follows: Tn(x) = cos(n arc cos x), 1 x 1. Show that tn(x) is an eigenfunction of this sturm-liouville problem and for each n 0 nd the corresponding eigenvalue.
