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Unformatted text preview: But anything that is 0 should not appear in the final expression. Fancy ways of writing 0 will be penalized. Solution. Because the function is even, only cosine terms will be present. Thus f ( x ) = 1 2 a + ∞ ∑ n =1 a n cos nπx, where a n = ∫ 1 − 1 x 2 cos nπx dx. (b) (5 points) At what points x of the interval [ − 1 , 1] is f ( x ) equal to the sum of its Fourier series? Solution. Because f is continuous at all points of the interval, and because f (1 − ) = f (1+), the series converges to f ( x ) at ALL points of [ − 1 , 1]. (c) (5 points) To what values, if any, does the Fourier series converge at the points x = 3 . 2 , 4 . 9, and − 6? Solution. The function that is the sum of the Fourier series is periodic of period 2 and coincides with x 2 for  x  ≤ 1. Thus: • Limit or sum at 3 . 2 = same as at − . 8, so to ( − , 8) 2 = 0 . 64. • Limit or sum at 4 . 9 = same as at 0 . 9, so to 0 . 9 2 = 0 . 81. • Limit or sum at − 6 = same as at 0, so to 0 2 = 0. 2. Consider the following problem: (1 + x 2 ) y ′′ ( x ) + 2 xy ′ + λ (1 + x 2 ) y = , < x < 1 , y ′ (0) = y ′ (1) = 0 . 2 (a) (10 points) Write it out in the form of a regular Sturm–Liouville problem. What areWrite it out in the form of a regular Sturm–Liouville problem....
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 Spring '13
 Schonbek
 Sin, Boundary value problem, Partial differential equation, Sturm–Liouville theory

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