MA 513 - MA 513 Complex Variables First Test Solutions...

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Unformatted text preview: MA 513 Complex Variables First Test Solutions February 2009 1. (20 points) (a) Find all solutions of the equation z 3 + 1- i = 0 . Answer: z = 2 1 / 6 2 (1+ i ); z = 2 1 / 6 e i ( 4 2 3 ) : z = 2 1 / 6 2 2 (- 1- 3+ i ( 3- 1)); z = 2 1 / 6 2 2 (- 1+ 3- i ( 3+1)) (b) Find all the numbers (- 1) using the functions exp and log . Answer: z = e i 2 (2 k +1) , k = 0 , 1 , 2 , ... 2. (20 points) Let z = x + iy and suppose f ( z ) = u + iv is an entire function. (a) Write the Cauchy-Riemann equations for u, v. u x = v y , u y =- v x . (b) Prove that if f ( z ) = u- iv is also an entire function, then f ( z ) is constant. Proof: Cauchy-Riemann equations for f ( z ) = u- iv : u x =- v y , u y = v x . Therefore, u x = u y = v x = v y = 0 , so u = const. and v = const. 3. (15 points) Let C be the arc of the circle | z | = 2 from z = 2 to z = 2 i in the first quadrant of the complex plane. Show that (without evaluating the integral) vextendsingle vextendsingle...
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