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ps3sol - ACM 95a/100a Problem Set 3 Solutions Prepared by...

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ACM 95a/100a Problem Set 3 Solutions Prepared by: Zhiyi Li October 18, 2007 Total: 102 points Include grading section: 2 points Problem 1 (15 points total) Consider two functions f ( z ) and g ( z ) analytic at z 0 with f ( z 0 ) = g ( z 0 ) = 0 and g ( z 0 ) = 0. a) (12 points) Use the definition of the complex derivative to justify L’Hopital’s rule: lim z z 0 f ( z ) g ( z ) = f ( z 0 ) g ( z 0 ) Hint: f ( z ) /g ( z ) can be rewritten in a useful form noting f ( z 0 ) = g ( z 0 ) = 0. b) (3 points) Evaluate the limit lim z πi sinh z e z + 1 Solution 1 a) Since f ( z 0 ) = g ( z 0 ) = 0, we have lim z z 0 f ( z ) g ( z ) = lim z z 0 f ( z ) - f ( z 0 ) g ( z ) - g ( z 0 ) = lim z z 0 f ( z ) - f ( z 0 ) z - z 0 g ( z ) - g ( z 0 ) z - z 0 (1) Now, by definition lim z z 0 f ( z ) - f ( z 0 ) z - z 0 = f ( z 0 ) (2) and lim z z 0 g ( z ) - g ( z 0 ) z - z 0 = g ( z 0 ) = 0 (3) Therefore, lim z z 0 f ( z ) g ( z ) = lim z z 0 f ( z ) - f ( z 0 ) z - z 0 lim z z 0 g ( z ) - g ( z 0 ) z - z 0 = f ( z 0 ) g ( z 0 ) (4) 1
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b) We may use L’Hopital’s rule to evaluate the limit: lim z πi sinh z e z + 1 = lim z πi cosh( z ) e z = e + e - 2 e = 1 (5) Problem 2 (10 points) If z = x + iy , determine where 3 x 2 y 2 - 6 ix 2 y 2 is differentiable and where it is analytic. Solution 2 For 3 x 2 y 2 - i 6 x 2 y 2 , u = 3 x 2 y 2 and v = - 6 x 2 y 2 . We check the Cauchy-Riemann equations: u x = v y = 6 xy 2 = - 12 x 2 y = 6 xy ( y + 2 x ) = 0 (6) u y = - v x = 6 x 2 y = 12 xy 2 = 6 xy ( x - 2 y ) = 0 (7) These both hold for either x = 0 or y = 0, i.e., on the real and imaginary axes. In addition, the partials u x , u y , v x , v y are continuous there. Therefore, 3 x 2 y 2 - i 6 x 2 y 2 is differentiable on the real and imaginary axes . However, there is no -neighborhood of a point on the axes which contains only points where the function is differentiable. Therefore, it is nowhere analytic .
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