This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
 Fall '07
 Lim
 Math, dx, 1 g, dz, AugustinLouis Cauchy, 2Eb

Click to edit the document details