hw3 - 1 2 O A tan x B C sin x x Figure 1: Trigonometric...

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Homework 3 – Math 118C, Spring 2010 Due on Tuesday, April 20th, 2010 1. Let K N be the F´ ejer kernel, defned in the previous assignment. I± f ∈ R and f ( x +), f ( x ) exist ±or some x , prove that lim N →∞ σ N ( f ; x ) = f ( x +) + f ( x ) 2 . 2. Defne f ( x ) = x 3 sin 2 x tan x, (1) g ( x ) = 2 x 2 sin 2 x x tan x. (2) Find out, ±or each o± these two ±unctions, whether it is positive o± negative ±or all x (0 , π 2 ), or whether it changes sign. Prove your answer. 3. Defne the sine and cosine ±unctions by sin( x ) = s n =0 ( 1) n x 2 n +1 (2 n + 1)! , and cos( x ) = s n =0 ( 1) n x 2 n (2 n )! . Show that sin(2 x ) = 2 sin( x ) cos( x ) directly by multiplying power se- ries. 4. In Geometry, the sine and cosine ±unctions are defned using a circle o± radius one, usually called Trigonometric circle, as depicted in fgure 1. Using this defnition o± the sine, and elementary geometry, prove that lim x 0 sin x x = 1 .
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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.

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hw3 - 1 2 O A tan x B C sin x x Figure 1: Trigonometric...

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