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B&Auml;&deg;LKENT COMPLEX CALCULUS

# B&Auml;&deg;LKENT COMPLEX CALCULUS - Date Saturday...

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Date: March 10, 2007, Saturday Time: 13:00-15:00 ¨ Ozg¨uler & Sert¨oz 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 = 2 exp[ i ( 2 π 3 + 2 )], n Z . The fourth roots are c k = 4 2 exp[ 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 ln 2 + (3 - 8 n ) π ) + i (2 ln 2 - 9 π 4 + 6 ) . The principal value is obtained when n = 0 and it is given by 2 2 e 3 π cos(2 ln 2 - π 4 ) + i sin(2 ln 2 - π 4 ) · . Note here that 9 π 4 = π 4 + 2 π so we can use π 4 . Again for the curious we note P . V . ( - 1 - i ) 3+4 i = 28909 . 33549 - 19815 . 99860 i.

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Q-3) Find a harmonic conjugate v ( x, y ) for the function u ( x, y ) = e y sin x + xy
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B&Auml;&deg;LKENT COMPLEX CALCULUS - Date Saturday...

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