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Complex Analysis: Midterm Examination 10:20 AM - 12:00 PM, April 14, 2009. [1] (10 %) Find all values of ± - 8 - i 8 3 ² 1 / 4 in the form of a + ib . [2] (10 %) Prove f ( z )= e y e ix is nowhere analytic, where z = x + iy . [3] (15 %) True or false (If it is false, explain brie±y why it isn’t true) (a) (5 %) If f ( z ) is analytic on a closed contour C , then ³ C ± f ( z ) dz =0. (b) (5 %) If f ( z ) is diﬀerentiable at a point z 0 and at every point in some neighborhood of z 0 , then f ( z ) is an entire function. (c) (5 %) S = { z | Re( z ) ² =3 } is a domain ( open connected set ). [4] (15 %) Verify that u ( x, y )= e x ( x cos y - y sin y ) is harmonic. Find v ( x, y ), the conjugate harmonic function of u ( x, y ). [5] (10 %) Evaluate ³ C 1 z dz in the form of a + ib , where C is the arc of the circle z =4 e it with - π/ 2 t π/ 2. [6] (10 %) Evaluate ³ C ± ´ 3 z +2 - 1 z - 2 i µ dz , where C is the circle | z | =5. [7] (10 %) Expand f ( z )= 1+
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Unformatted text preview: z 1-z in the Taylor series centered at z = i , and give the radius of convergence of this series. [8] (10 %) Let f ( z ) = u ( x, y ) + iv ( x, y ) where the ²rst partial derivatives of u ( x, y ) and v ( x, y ) are continuous. Prove that f ( z ) is analytic at z if and only if ∂u ∂x = ∂v ∂y and ∂u ∂y =-∂v ∂x [9] (10 %) Assume f ( z ) is analytic in a domain D , and C is a closed contour lying entirely in D . Use the fact that f ± ( z ) = 1 2 πi ³ C ± f ( z ) ( z-z ) 2 dz , with z within C , to prove f ±± ( z ) = 2! 2 πi ³ C ± f ( z ) ( z-z ) 3 dz . Hint: f ±± ( z ) = lim Δ z → f ± ( z + Δ z )-f ± ( z ) Δ z . 1...
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