math185f08-hw3sol - MATH 185: COMPLEX ANALYSIS FALL 2008/09...

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Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 3 SOLUTIONS C will always denote a region unless specified otherwise. For f : C and c C a constant, we write f c to mean that f ( z ) = c for all z . 1. Let f : C with f ( x + iy ) = u ( x,y )+ iv ( x,y ). Let z and suppose there exists a function : C C such that lim h f ( z + h )- f ( z )- ( h ) h = 0 . Recall from Problem Set 2 , Problem 2 that f is real differentiable if is real linear and f is complex differentiable if is complex linear. Recall from Problem Set 2 , Problem 1 that a real linear satisfies ( x + iy ) = ( ax + by ) + i ( cx + dy ) (1.1) for some a b c d R 2 2 and a complex linear satisfies ( x + iy ) = ( ax- cy ) + i ( cx + ay ) (1.2) for some [ a- c c a ] R 2 2 . (a) Show that if f is real differentiable at z = x + iy , then the matrix a b c d is given by a b c d = u x ( x ,y ) u y ( x ,y ) v x ( x ,y ) v y ( x ,y ) . Solution. Taking limits along 0, R , we get 0 = lim f ( z + )- f ( z )- ( ) = lim f ( z + )- f ( z )- b- ic = lim u ( x + ,y )- u ( x ,y )- a + i lim v ( x + ,y )- v ( x ,y )- c = lim u ( x + ,y )- u ( x ,y ) - a + i lim v ( x + ,y )- v ( x ,y ) - c = [ u x ( x ,y )- a ] + i [ v x ( x ,y )- c ] . Hence u x ( x ,y ) = a and v x ( x ,y ) = c. Date : October 24, 2008 (Version 1.0). 1 Taking limits along i 0, i i R , we get 0 = lim f ( z + i )- f ( z )- ( i ) i = lim f ( z + i )- f ( z )- b- id i =- i lim u ( x ,y + )- u ( x ,y )- b + lim v ( x ,y + )- v ( x ,y )- d =- i lim u ( x ,y + )- u ( x ,y ) - b + lim v ( x ,y + )- v ( x ,y ) - d =- i [ u y ( x ,y )- b ] + [ v y ( x ,y )- d ] . Hence u y ( x ,y ) = b and v y ( x ,y ) = d. (b) Show that if f is complex differentiable at z = x + iy , then the matrix [ a- c c a ] is given by a- c c a = u x ( x ,y )- u y ( x ,y ) u y ( x ,y ) u x ( x ,y ) . Solution. By Problem Set 2 , Problem 2 (a), we know that complex differentiability at z implies real differentiability at z . So part (a) holds in this case, ie. there exists a having the form in (1.1) with a b c d = u x ( x ,y ) u y ( x ,y ) v x ( x ,y ) v y ( x ,y ) ....
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This note was uploaded on 06/15/2009 for the course MATH 185 taught by Professor Lim during the Fall '07 term at University of California, Berkeley.

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math185f08-hw3sol - MATH 185: COMPLEX ANALYSIS FALL 2008/09...

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