ans6 - Math 185 Sample Answers to Problem Set#6 Page 153 1...

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Unformatted text preview: Math 185. Sample Answers to Problem Set #6 Page 153 1. In each case it suffices to show that the given function is analytic on the closed unit disc | z | ≤ 1 . b . This function is entire, hence analytic on | z | ≤ 1 . c . This function is analytic everywhere except for z satisfying z 2 +2 z +2 = 0 . By the quadratic formula, this equation is satisfied if and only if z = (- 2 ± √ 4- 8) / 2 =- 1 ± i . But |- 1 ± i | = √ 2 , so- 1 ± i lies outside of the closed unit disc; therefore 1 / ( z 2 +2 z +2) is analytic on the closed disc | z | ≤ 1 . f . This function is analytic everywhere except for those z for which z + 2 is zero or a negative real number. Therefore it is analytic everywhere except for the real interval (-∞ ,- 2) . This interval does not intersect the closed unit disc, so Log( z +2) is analytic everywhere on that disc. 2c. The function z/ (1- e z ) is analytic if and only if e z 6 = 1 , which holds if and only if z is not an integer multiple of 2 πi . Thus it is analytic at z if and only if z / ∈ { 2 πin : n ∈ Z } . Now if n = 0 then 2 πin = 0 , and this is interior to C 2 , so it does not lie between the two contours. If n 6 = 0 , then | 2 πin | = 2 π | n | ≥ 2 π > 4 , so z is outside of C 1 . Therefore f is analytic everywhere between and on the two contours, so Corollary 2 in Section 46 applies, giving the desired equality of integrals. 4. a . The integral of e- z 2 along the lower part of the given rectangle is Z a- a e- x 2 dx = 2 Z a e- x 2 dx since e- x 2 is an even function. Using the contour z ( x ) =- x + ib ,- a ≤ x ≤ a , the integral along the upper part is Z a- a e- ( x 2- b 2 )+2 bxi (- 1) dx =- e b 2 Z a- a e- x 2 cos 2 bxdx- ie b 2 Z a- a e- x 2 sin2 bxdx =- 2 e b 2 Z a e- x 2 cos 2 bxdx since the first integrand is an even function and the second is odd.since the first integrand is an even function and the second is odd....
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ans6 - Math 185 Sample Answers to Problem Set#6 Page 153 1...

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