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Unformatted text preview: Exercise Solutions for Complex Analysis Stephen Taylor April 1, 2005 2 Chapter 1 The Complex Number System 1.1 Pages 23 1. Find the real and imaginary parts for each of the following: In the following let the complex variable z = x + iy for { x,y } R (i) 1 z = 1 x + iy = 1 x + iy ( x iy ) ( x iy ) = x iy x 2 + y 2 Re( z ) = x x 2 + y 2 and Im( z ) = y x 2 + y 2 (ii) Let a R z a z + a = x + iy a x + iy + a ( x iy + a ) ( x iy + a ) = x 2 + y 2 a 2 + 2 ayi x 2 + y 2 + 2 ax + a 2 Re( z ) = x 2 + y 2 a 2 x 2 + y 2 + 2 ax + a 2 and Im( z ) = 2 ay x 2 + y 2 + 2 ax + a 2 (iii) z 3 = ( x + iy ) 3 = x 3 3 xy 3 + i 3 x 2 y iy 3 Re( z ) = x 3 3 xy 3 and Im( z ) = 3 x 2 y y 3 (iv) 3 4 CHAPTER 1. THE COMPLEX NUMBER SYSTEM z = 3 + 5 i 1 + 7 i = 3 + 5 i 1 + 7 i 1 7 i 1 7 i = 19 8 i 25 Re( z ) = 19 / 25 and Im( z ) = 8 / 25 (v) z = 1 + i 3 2 3 = 1 Re( z ) = 1 and Im( z ) = 0 (vi) z = 1 i 3 2 6 = 1 Re( z ) = 1 and Im( z ) = 0 (vii) i n for 2 n 8 n = 2 : Re( z ) = 1 and Im( z ) = 0 n = 3 : Re( z ) = 0 and Im( z ) = 1 n = 4 : Re( z ) = 1 and Im( z ) = 0 n = 5 : Re( z ) = 0 and Im( z ) = 1 n = 6 : Re( z ) = 1 and Im( z ) = 0 n = 7 : Re( z ) = 0 and Im( z ) = 1 n = 8 : Re( z ) = 1 and Im( z ) = 0 1.1. PAGES 23 5 (viii) 1+ i 2 n for 2 n 8 n = 2 : Re( z ) = 0 and Im( z ) = 1 n = 3 : Re( z ) = 2 / 2 and Im( z ) = 2 / 2 n = 4 : Re( z ) = 1 and Im( z ) = 0 n = 5 : Re( z ) = 2 / 2 and Im( z ) = 2 / 2 n = 6 : Re( z ) = 0 and Im( z ) = 1 n = 7 : Re( z ) = 2 / 2 and Im( z ) = 2 / 2 n = 8 : Re( z ) = 1 and Im( z ) = 0 2. Find the absolute value and the conjugate of each of the following: (i) z 1 = 2 + i  z 1  = 5 and z 1 = 2 i (ii) z 2 = 3  z 2  = 3 and z 1 = 3 (iii) z 3 = (2 + i )(4 + 3 i ) (2 + i )(4 + 3 i ) = 5 + 10 i  z 3  = 125 = 5 5 and z 1 = 5 10 i (iv) z 4 = 3 i 2+3 i 3 i 2 + 3 i = 3 11 ( 2 1) i 11 ( 2 + 9) 6 CHAPTER 1. THE COMPLEX NUMBER SYSTEM  z 4  = 110 11 and z 1 = 3 11 ( 2 1) + i 11 ( 2 + 9) (v) z 5 = i i +3 i i + 3 = 1 + 3 i 10  z 5  = 10 10 and z 1 = 1 3 i 10 (vi) z 6 = (1 + i ) 6 (1 + i ) 6 = 8 i  z 6  = 8 and z 1 = 8 (vii) z 7 = i 17  z 7  = 1 and z 1 = i 17 = i 3. Show that z is a real number iff z = z Proof: ( z is a real number) Let z = x + iy for { x,t } R . By definition of the conjugate, we find z = x iy . But since z is real, y = 0. Therefore z = x = z . ( z = z ) By hypothesis we find z = x + iy = x iy = z So 2 iy = 0 which gives y = 0. Therefore z = x is a real number. 4. If z and w are complex numbers, prove the following equations: Let z = x 1 + y 1 and w = x 2 + y 2 for { x 1 ,x 2 ,y 1 ,y 2 } R (i)  z + w  2 =  z  2 + 2Re[ z w ] +  w  2  z + w  2 =  ( x 1 + iy 1 ) + ( x 2 + iy 2 )  2 =  ( x 1 + x 2 ) + i ( y 1 + y 2 )  2 = ( x 1 + x 2 ) 2 + ( y 1 + y 2 ) 2 = ( x 2 1 + y...
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