380_sample_exam_solutions[1]

380_sample_exam_solutions[1] - Math 380 Solutions to Sample...

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Math 380 Solutions to Sample Exam Problems 1. Find each of the following: (a) (2 + i )(3 + 4 i ) = 2 + 11 i (b) Arg (1 p 3 i ) = 3 (c) Im( 2+ i 3 i ) = 1 2 (d) cos i = ( e 1 + e 1 ) = 2 (or cosh 1 ) (e) the polar form of (1 p 3 i ) = 2 e 3 (f) i 1 = 3 = e i ( 2+2 ) ± 1 = 3 = e i ( 6+2 3) : The three distinct values are e 6 = ² p 3 + i ³ = 2 ; e i (5 6) = ² p 3 + i ³ = 2 ; and e i (9 6) = i: 2. Find all values of i 2 i : i 2 i = i 2 i i = ( 1) e i ( 2+2 )( i ) = e 2+2 ; k = 0 ; ± 1 ; ± 2 ; ::: 3. Let f ( z ) = 2 xy + iy 2 . Find all points where f 0 ( z ) exisits and determine f 0 ( z ) at those points. Is f analytic at any point? Explain. Solution We need to check the Cauchy-Riemann conditions. We have u = 2 xy and v = y 2 : Since the partial derivatives of these are everywhere continuous, f 0 ( z ) exists precisely where the C-R equation hold, namely where u x = v y and u y = v x : That is, where 2 y = 2 y , and
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This note was uploaded on 09/16/2011 for the course MATH 380 taught by Professor Staff during the Spring '11 term at S.F. State.

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380_sample_exam_solutions[1] - Math 380 Solutions to Sample...

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