math185-hw2 - MATH 185 COMPLEX ANALYSIS FALL 2007\/08 PROBLEM SET 2 Throughout the problem set i =-1 and whenever we write x yi it is implicit that x y R

math185-hw2 - MATH 185 COMPLEX ANALYSIS FALL 2007/08...

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MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 2 Throughout the problem set, i = - 1; and whenever we write x + yi , it is implicit that x, y R . The real and imaginary parts of z C are denoted by Re( z ) and Im( z ) respectively, ie. if z = x + yi , then Re( z ) = x and Im( z ) = y . If Ω C , we let Ω R := { ( x, y ) R 2 | x + iy Ω } . 1. Suppose f ( z ) = n =0 a n z n has a radius of convergence R > 0. Let d be a fixed positive integer. Let g ( z ) = X n =0 n d a n z n and h ( z ) = X n =0 a n n ! z n . Find the radii of convergence of g ( z ) and h ( z ). Show that for any r with 0 < r < R , there is a constant M > 0 such that | h ( z ) | ≤ Me | z | /r . 2. Suppose f ( z ) = n =0 a n z n has a radius of convergence R = . Show that if Im[ f ( x )] = 0 and Re[ f ( iy )] = 0 for all x, y R , then f ( - z ) = - f ( z ) for all z C . 3. Let the functions f, g : C C be defined by f ( z ) = x 3 y 2 + ix 2 y 3 and g ( z ) = e x cos y + ie x sin y for z = x + yi . Determine the set of points z C at which f is differentiable and do the same for g . 4. Let Ω C be a region. Let f = u + iv be analytic on Ω. (a) Show that | f 0 | 2 = det u x u y v x v y = u 2 x + v 2 x = u 2 y + v 2 y on Ω.
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Unformatted text preview: (b) If u,v ∈ C 2 (Ω R ), ie. all second order partial derivatives of u,v exist and are continuous on Ω R , then u xx + u yy = 0 and v xx + v yy = 0 on Ω R (in this case u and v are called harmonic functions ). 5. Let Ω ⊆ C be a region. Let f = u + iv be analytic on Ω. (a) Determine all f for which g = u 2 + iv 2 (ie. g ( z ) := [ u ( x,y )] 2 + i [ v ( x,y )] 2 ) is also analytic on Ω. (b) Show that if there exist α,β ∈ C × such that αu + βv is constant on Ω, then f must be constant on Ω. (c) Show that if for all z ∈ Ω, either f ( z ) = 0 or f ( z ) = 0, then f must be constant on Ω. Date : September 19, 2007 (Version 1.1); posted: September 17, 2007; due: September 24, 2007. 1...
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