math185f08-hw3

math185f08-hw3 - MATH 185 COMPLEX ANALYSIS FALL 2008/09...

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Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 3 Ω ⊆ C will always denote a region unless specified otherwise. For f : Ω → C and c ∈ C a constant, we write f ≡ c to mean that f ( z ) = c for all z ∈ Ω. 1. Let f : Ω → C with f ( x + iy ) = u ( x,y )+ iv ( x,y ). Let z ∈ Ω and suppose there exists a function ϕ : C → C such that lim h → f ( z + h )- f ( z )- ϕ ( h ) h = 0 . Recall from Problem Set 2 , Problem 2 that f is real differentiable if ϕ is real linear and f is complex differentiable if ϕ is complex linear. Recall from Problem Set 2 , Problem 1 that a real linear ϕ satisfies ϕ ( x + iy ) = ( ax + by ) + i ( cx + dy ) (1.1) for some a b c d ∈ R 2 × 2 and a complex linear ϕ satisfies ϕ ( 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 ∈ Ω, 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 ) ....
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math185f08-hw3 - MATH 185 COMPLEX ANALYSIS FALL 2008/09...

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