Ps3sol - ACM 95a/100a Problem Set 3 Solutions Prepared by Zhiyi Li Total 102 points Include grading section 2 points Problem 1(15 points total

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Unformatted text preview: 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 with f ( z ) = g ( z ) = 0 and g ( z ) = 0. a) (12 points) Use the definition of the complex derivative to justify L’Hopital’s rule: lim z → z f ( z ) g ( z ) = f ( z ) g ( z ) Hint: f ( z ) /g ( z ) can be rewritten in a useful form noting f ( z ) = g ( z ) = 0. b) (3 points) Evaluate the limit lim z → πi sinh z e z + 1 Solution 1 a) Since f ( z ) = g ( z ) = 0, we have lim z → z f ( z ) g ( z ) = lim z → z f ( z )- f ( z ) g ( z )- g ( z ) = lim z → z f ( z )- f ( z ) z- z g ( z )- g ( z ) z- z (1) Now, by definition lim z → z f ( z )- f ( z ) z- z = f ( z ) (2) and lim z → z g ( z )- g ( z ) z- z = g ( z ) = 0 (3) Therefore, lim z → z f ( z ) g ( z ) = lim z → z f ( z )- f ( z ) z- z lim z → z g ( z )- g ( z ) z- z = f ( z ) g ( z ) (4) 1 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 iπ + e- iπ 2 e iπ = 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 ....
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This note was uploaded on 02/22/2010 for the course ACM 95A taught by Professor Nilesa.pierce during the Fall '06 term at Caltech.

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Ps3sol - ACM 95a/100a Problem Set 3 Solutions Prepared by Zhiyi Li Total 102 points Include grading section 2 points Problem 1(15 points total

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