smid_95-100aSOLUTION

smid_95-100aSOLUTION - ACM95a/100a Last updated November 4,...

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Unformatted text preview: ACM95a/100a Last updated November 4, 2009 Mid-Term Exam Solutions Prepared by Faisal Amlani Problem 1. (30 pts.) (a) (10 pts.) Express the following complex numbers in the form a + ib or re iθ : (i) (5 pts.) (1 + 2 i ) i , (ii) (5 pts.) log ( e π 4 i ) . (b) (10 pts.) Show that for real θ the relation | 1 − e iθ | = 2 | sin( θ/ 2) | holds. (c) (10 pts.) Show that for 2 π < θ < 4 π we have arg(1 − e iθ ) = θ 2 + (2 n + 1 2 ) π, n = 0 , ± 1 , ± 2 , . . . SOLUTION : (a) (i) (1 + 2 i ) i = exp( i log(1 + 2 i )) = exp( i ( Log( √ 5) + i Arctan (2 / 1) + 2 πin )) = exp( − Arctan (2) − 2 πn + i Log(5) / 2) = exp( − Arctan (2) − 2 πn ) exp( i Log(5) / 2) . (ii) log ( e iπ/ 4 ) = i π 4 + i 2 πn = iπ parenleftbigg 2 n + 1 4 parenrightbigg (b) vextendsingle vextendsingle 1 − e iθ vextendsingle vextendsingle = radicalbig (1 − e iθ )(1 − e- iθ ) = radicalBig ( e- iθ/ 2 − e iθ/ 2 )( e iθ/ 2 − e- iθ/ 2 ) = 2 radicalBigg e- iθ/ 2 − e iθ/ 2 − i 2 e iθ/ 2 − e- iθ/ 2 i 2 = 2 radicalBig sin( θ/ 2)sin( θ/ 2) = 2 | sin( θ/ 2) | . One can also use trig identities to prove this. (c) arg ( 1 − e iθ ) = arg ( e iθ/ 2 ( e- iθ/ 2 − e iθ/ 2 )) = θ 2 + arg ( − 2 i sin( θ/ 2)) For 2 π < θ < 4 π , sin( θ/ 2) is a negative real number. Thus − 2 i sin( θ/ 2) is a positive imaginary number with argument π/ 2 + 2 πn , i.e. arg ( 1 − e iθ ) = θ 2 + parenleftbigg 2 n + 1 2 parenrightbigg π, n ∈ Z . One can also use trig identities to prove this. Problem 2. (20 pts.) (a) (5 pts.) Find all the finite branch points and the number of branches for the function f ( z ) = parenleftbigg z − i z + i parenrightbigg 1 / 2 . (b) (5 pts.) Using cuts which do not contain either z = − 1 nor z = 1, construct a single-valued branch of the function in (a) satisfying f (1) = √ 2(1 − i ) / 2. What is f ( − 1) for this branch? (c) (5 pts.) Consider the function w 1 ( z ) = log bracketleftBigg parenleftbigg z − i z + i parenrightbigg 1 2 bracketrightBigg . Is a cut along the imaginary axis from z = − i to z = i sufficient to define a single valued branch of w 1 ? Justify your answer....
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This note was uploaded on 01/14/2010 for the course ACM 1 taught by Professor Prof during the Fall '09 term at Caltech.

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smid_95-100aSOLUTION - ACM95a/100a Last updated November 4,...

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