This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 3 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 ) ....
View Full Document
- Fall '07