ans14 - Math 185 Sample Answers to Problem Set#14 Page 296...

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Unformatted text preview: Math 185. Sample Answers to Problem Set #14 Page 296 1. The function F ( s ) has simple poles at s = ± √ 2 and ± √ 2 i . Therefore, we may take γ = 2. For all s ∈ {± √ 2 , ± √ 2 i } , we have Res s = s ( e st F ( s ) ) = 2 s 3 e s t 4 s 3 = e s t 2 . Therefore, f ( t ) = Res s = √ 2 e st F ( s ) + Res s =- √ 2 e st F ( s ) + Res s = √ 2 i e st F ( s ) + Res s =- √ 2 i e st F ( s ) = e √ 2 t + e- √ 2 t + e √ 2 it + e- √ 2 it 2 = cosh √ 2 t + cos √ 2 t , provided the italicized statement on page 289 is true. To check this, we have, for all R > 2 + √ 2, | F (2 + e iθ ) | = 2(2 + Re iθ ) 3 (2 + Re iθ ) 4- 4 ≤ 2(2 + R ) 3 ( R- 2) 4- 4 , which goes to 0 as R → ∞ . Thus the above expression for f ( t ) is true. 5. The function F ( s ) has poles at ± ai , so we may take γ = 1. To check the italicized statement on page 289, we have | F (1 + Re iθ ) | = 8 a 3 | 1 + Re iθ | 2 | (1 + Re iθ ) 2 + a 2 | 3 ≤ 8 a 3 (1 + R ) 2 (( R- 1) 2- a 2 ) 3 for all R > 1 + a . This bound goes to zero when R → ∞ , so the italicized statement is true....
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This note was uploaded on 09/01/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at Berkeley.

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ans14 - Math 185 Sample Answers to Problem Set#14 Page 296...

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