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math185f08-hw3sol

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

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MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 3 SOLUTIONS 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 0 Ω and suppose there exists a function ϕ : C C such that lim h 0 f ( z 0 + h ) - f ( z 0 ) - 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 0 = x 0 + iy 0 Ω, then the matrix a b c d is given by a b c d = u x ( x 0 , y 0 ) u y ( x 0 , y 0 ) v x ( x 0 , y 0 ) v y ( x 0 , y 0 ) . Solution. Taking limits along ξ 0, ξ R , we get 0 = lim ξ 0 f ( z 0 + ξ ) - f ( z 0 ) - ϕ ( ξ ) ξ = lim ξ 0 f ( z 0 + ξ ) - f ( z 0 ) - - icξ ξ = lim ξ 0 u ( x 0 + ξ, y 0 ) - u ( x 0 , y 0 ) - ξ + i lim ξ 0 v ( x 0 + ξ, y 0 ) - v ( x 0 , y 0 ) - ξ = lim ξ 0 u ( x 0 + ξ, y 0 ) - u ( x 0 , y 0 ) ξ - a + i lim ξ 0 v ( x 0 + ξ, y 0 ) - v ( x 0 , y 0 ) ξ - c = [ u x ( x 0 , y 0 ) - a ] + i [ v x ( x 0 , y 0 ) - c ] . Hence u x ( x 0 , y 0 ) = a and v x ( x 0 , y 0 ) = c. Date : October 24, 2008 (Version 1.0). 1
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Taking limits along 0, i R , we get 0 = lim η 0 f ( z 0 + ) - f ( z 0 ) - ϕ ( ) = lim η 0 f ( z 0 + ) - f ( z 0 ) - - idη = - i lim η 0 u ( x 0 , y 0 + η ) - u ( x 0 , y 0 ) - η + lim η 0 v ( x 0 , y 0 + η ) - v ( x 0 , y 0 ) - η = - i lim η 0 u ( x 0 , y 0 + η ) - u ( x 0 , y 0 ) η - b + lim η 0 v ( x 0 , y 0 + η ) - v ( x 0 , y 0 ) η - d = - i [ u y ( x 0 , y 0 ) - b ] + [ v y ( x 0 , y 0 ) - d ] . Hence u y ( x 0 , y 0 ) = b and v y ( x 0 , y 0 ) = d. (b) Show that if f is complex differentiable at z 0 = x 0 + iy 0 Ω, then the matrix [ a - c c a ] is given by a - c c a = u x ( x 0 , y 0 ) - u y ( x 0 , y 0 ) u y ( x 0 , y 0 ) u x ( x 0 , y 0 ) . Solution. By Problem Set 2 , Problem 2 (a), we know that complex differentiability at z 0 implies real differentiability at z 0 . So part (a) holds in this case, ie. there exists a ϕ having the form in (1.1) with a b c d = u x ( x 0 , y 0 ) u y ( x 0 , y 0 ) v x ( x 0 , y 0 ) v y ( x 0 , y 0 ) . By the Cauchy-Riemann equations, we know that complex differentiability at z 0 implies a = u x ( x 0 , y 0 ) = v y ( x 0 , y 0 ) = d and b = u y ( x 0 , y 0 ) = - v x ( x 0 , y 0 ) = - c. (c) Suppose f is analytic on Ω (ie. complex differentiable at all z Ω) and that u ( x, y ) = ϕ ( x ), v ( x, y ) = ψ ( y ), ie. f takes the form f ( x + iy ) = ϕ ( x ) + ( y ) .
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