BÄ°LKENT COMPLEX CALCULUS

BÄ°LKENT COMPLEX CALCULUS - Date: March...

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Time: 13:00-15:00 ¨ Math 206 Complex Calculus – Midterm Exam I – Solutions Q-1) Find all the fourth roots of 3 i - 1. Write the resulting numbers in rectangular form. Answer: 3 i - 1 = 2exp[ i ( 2 π 3 + 2 )], n Z . The fourth roots are c k = 4 2exp[ i ( π 6 + 2 )] for k = 0 , 1 , 2 , 3 . , which gives: c 0 = 4 2 ˆ 3 2 + i 1 2 ! , c 1 = 4 2 ˆ - 1 2 + i 3 2 ! , c 2 = 4 2 ˆ - 3 2 - i 1 2 ! , c 3 = 4 2 ˆ 1 2 - i 3 2 ! . Note for the curious: c 0 = - . 5946035575 + 1 . 029883572 i . Q-2) Find all the values of ( - 1 - i ) 3+4 i . Write the principal value in rectangular form. Answer: ( - 1 - i ) 3+4 i = exp((3 + 4 i )log( - 1 - i )) , = exp ± (3 + 4 i )(ln 2 + i [ - 3 π 4 + 2 ]) , n Z , = exp ± ( 3 2 ln2 + (3 - 8 n ) π ) + i (2ln2 - 9 π 4 + 6 ) . The principal value is obtained when n = 0 and it is given by 2 2 e 3 π cos(2ln 2 - π 4 ) + i sin(2ln2 - π 4 ) · . Note here that
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This note was uploaded on 11/08/2011 for the course MATH 332 taught by Professor Feritöztürk during the Spring '05 term at Bilkent University.

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BÄ°LKENT COMPLEX CALCULUS - Date: March...

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