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Unformatted text preview: 1 2 O A tan x B C sin x x Figure 1: Trigonometric circle. 5. Consider a periodic function f C r per ([ , ]). Let a n = 1 2 i - e-inx f ( x ) dx. Prove that there exists a constant C > 0 such that | a n | C | n | r , n n = 0 . Prove that if f C 2 per ([ , ]), then its Fourier series converges uni-formly to f . 6. Dene the Gamma function in the following way: ( x ) = i t x-1 e-t dt, < x < . (a) Prove that ( x ) is well dened for 0 < x < . 3 (b) Provet that ( x + 1) = x ( x ), and conclude that ( n + 1) = n !. (c) Prove that log is convex on (0 , ). 7. Compute the following limits: (a) lim x + x x . (b) lim x + x x x ....
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This note was uploaded on 12/27/2011 for the course MATH 118c taught by Professor Garcia during the Fall '09 term at UCSB.
- Fall '09