20090930101035131_13장

20090930101035131_13장 - Homework 1 ...

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Unformatted text preview: Homework 1 § 13 . 1 1. (a) Show that multiplication by i ( z 7-→ iz ) is geometrically a counterclockwise rotation through π 2 . (b) Show that multiplication by α = cos θ + i sin θ ( z 7-→ αz ) is geometrically a counterclockwise rotation through θ . Sol. (a) Let z = x + iy ( 6 = 0). Then iz =- y + ix . Consider the triangle OAP with vertices O = (0 , 0) , A = ( x, 0) , P = ( x, y ) and the triangle OA P with vertices O = (0 , 0) , A = (0 , x ) , P = (- y, x ). Clearly, OA = OA , AP = A P , ∠ OAP = ∠ OA P = 90 ◦ . Thus 4 OAP ≡ 4 OA P . Since ∠ P OP = ∠ P OA + ∠ A OP = ∠ POA + ∠ A OP = 90 ◦ , multiplication by i ( z 7-→ iz ) is geometrically a counterclockwise rotation through π 2 . (b) Let z = x + iy ( 6 = 0). Then z = x + iy = p x 2 + y 2 ( x p x 2 + y 2 + i y p x 2 + y 2 ) =: r (cos δ + i sin δ ) , and αz = r (cos( θ + δ ) + i sin( θ + δ )). Thus multiplication by α = cos θ + i sin θ ( z 7-→ αz ) is geometrically a counterclockwise rotation through θ . 2. Show that z = x + iy is pure imaginary if and only if z =- z. Sol. ( ⇒ ) Suppose that z is pure imaginary, that is, z = iy, y ∈ R . Then z =- iy =- z. ( ⇐ ) Suppose that z satisfies z =- z, x- iy =- x- iy . Then x = 0, and so z = iy , z is pure imaginary. 3. Let z = x + iy . Find (a) Im [(1 + i ) 8 z 2 ], (b) Re (1 / ¯ z 2 ). Sol. (a) Note (1+ i ) 8 z 2 = (2 i ) 4 ( x + iy ) 2 = 16( x 2- y 2 +2 xyi ). Im [(1+ i ) 8 z 2 ] = 32 xy. Computational Science & Engineering (CSE) C. K. Ko Homework 2 (b)Note 1 ¯ z 2 = 1 ( x- iy ) 2 = ( x + iy ) 2 ( x- iy ) 2 ( x + iy ) 2 = x 2- y 2 + 2 xyi ( x 2 + y 2 ) 2 . Thus Re ( 1 ¯ z 2 ) = x 2- y 2 ( x 2 + y 2 ) 2 . § 13 . 2 1. Determine the principal value of the argument. (a) 7- 7 i (b)- π 2 . Sol. (a) 7- 7 i = 7 √ 2(cos(- π 4 + 2 nπ ) + i sin(- π 4 + 2 nπ )), n = 0 , ± 1 , ± 2 , ··· principal value of the argument :- π 4 (b)- π 2 = π 2 (cos( π + 2 nπ ) + i sin( π + 2 nπ )), n = 0 , ± 1 , ± 2 , ··· principal value of the argument : π 2. Find and graph all roots of (3 + 4 i ) 1 / 3 in the complex plane. Sol. 3 + 4 i = 5(cos α + i sin α ) , cos α = 3 / 5 , sin α = 4 / 5....
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This note was uploaded on 11/08/2011 for the course EE 111 taught by Professor Kim during the Spring '11 term at Korea University.

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20090930101035131_13장 - Homework 1 ...

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