AEM_3e_Chapter_17

AEM_3e_Chapter_17 - Part V Complex Analysis 17 Functions of...

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Part V Complex Analysis 17 17 Functions of a Complex Variable EXERCISES 17.1 Complex Numbers 3. i 8 =( i 2 ) 4 1) 4 =1 6. 3 9 i 9. 11 10 i 12. 2 2 i 15. 2 4 i 3+5 i · 3 5 i 3 5 i = 14 22 i 34 = 7 17 11 17 i 18. 3 i 11 2 i · 11 + 2 i 11 + 2 i = 35 5 i 125 = 7 25 1 25 i 21. (1 + i )(10 + 10 i ) = 10(1 + i ) 2 =20 i 24. (2+3 i )( i ) 2 = 2 3 i 27. x x 2 + y 2 30. 0 33. 2 x +2 yi = 9+2 i implies 2 x = 9 and 2 y = 2. Hence z = 9 2 + i . 36. x 2 y 2 4 x +( 2 xy 4 y ) i =0+0 i implies x 2 y 2 4 x = 0 and y ( 2 x 4) = 0. If y = 0 then x ( x 4) = 0 and so z = 0 and z =4 .If 2 x 4=0or x = 2 then 12 y 2 =0or y = ± 2 3. This gives z = 2+2 3 i and z = 2 2 3 i . 39. | z 1 z 2 | = | ( x 1 x 2 )+ i ( y 1 y 2 ) | = p ( x 1 x 2 ) 2 y 1 y 2 ) 2 which is the distance formula in the plane. 276
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17.2 Powers and Roots EXERCISES 17.2 Powers and Roots 3. 3 µ cos 3 π 2 + i sin 3 π 2 6. 5 2 µ cos 7 π 4 + i sin 7 π 4 9. 3 2 2 µ cos 5 π 4 + i sin 5 π 4 12. z = 8+8 i 15. z 1 z 2 =8 · cos µ π 8 + 3 π 8 + i sin µ π 8 + 3 π 8 ¶¸ i ; z 1 z 2 = 1 2 · cos µ π 8 3 π 8 + i sin µ π 8 3 π 8 ¶¸ = 2 4 2 4 i 18. h 4 2 ³ cos π 4 + i sin π 4 ´i · 2 µ cos 3 π 4 + i sin 3 π 4 ¶¸ · cos µ π 4 + 3 π 4 + i sin µ π 4 + 3 π 4 ¶¸ = 8 21. 2 9 · cos 9 π 3 + i sin 9 π 3 ¸ = 512 24. (2 2) 4 · cos 8 π 3 + i sin 8 π 3 ¸ = 32+32 3 i 27. 8 1 / 3 =2 · cos 2 3 + i sin 2 3 ¸ , k =0 ,1 ,2 w 0 = 2[cos0 + i sin0] = 2; w 1 · cos 2 π 3 + i sin 2 π 3 ¸ = 1+ 3 i w 2 · cos 4 π 3 + i sin 4 π 3 ¸ = 1 3 i 30. ( i ) 1 / 3 1 / 6 · cos µ π 4 + 2 3 + i sin µ π 4 + 2 3 ¶¸ , k w 0 1 / 6 h cos π 4 + i sin π 4 i = 1 3 2 + 1 3 2 i . 7937 + 0 . 7937 i w 1 1 / 6 · cos 11 π 12 + i sin 11 π 12 ¸ = 1 . 0842 + 0 . 2905 i w 2 1 / 6 · cos 19 π 12 + i sin 19 π 12 ¸ . 2905 1 . 0842 i 33. The solutions are the four fourth roots of 1; w k = cos π +2 4 + i sin π 4 ,k , 1 , 2 , 3 . We have w 1 = cos π 4 + i sin π 4 = 2 2 + 2 2 i w 2 = cos 3 π 4 + i sin 3 π 4 = 2 2 + 2 2 i w 3 = cos 5 π 4 + i sin 5 π 4 = 2 2 2 2 i w 4 = cos 7 π 4 + i sin 7 π 4 = 2 2 2 2 i. 277
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17.2 Powers and Roots 36. · 8 µ cos 3 π 8 + i sin 3 π 8 ¶¸ 3 h 2 ³ cos π 16 + i sin π 16 ´i 10 = 2 9 2 10 · cos µ 9 π 8 10 π 16 + i µ 9 π 8 10 π 16 ¶¸ = 1 2 ³ cos π 2 + i sin π 2 ´ = 1 2 i 39. (a) Arg( z 1 )= π , Arg( z 2 π 2 , Arg( z 1 z 2 π 2 , Arg( z 1 ) + Arg( z 2 3 π 2 6 = Arg( z 1 z 2 ) (b) Arg( z 1 /z 2 π 2 , Arg( z 1 ) Arg( z 2 π π 2 = π 2 6 = Arg( z 1 /z 2 ) EXERCISES 17.3 Sets in the Complex Plane 3. 6. 9. 12. 15. 18. 21. 24. | Re( z ) | = | x | is the same as x 2 and | z | = p x 2 + y 2 . Since y 2 0 the inequality x 2 p x 2 + y 2 is true for all complex numbers. 278
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17.4 Functions of a Complex Variable EXERCISES 17.4 Functions of a Complex Variable 3. x = 0 gives u = y 2 , v = 0. Since y 2 0 for all real values of y , the image is the origin and the negative u -axis.
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This note was uploaded on 01/03/2011 for the course BIS 511 at Yale.

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AEM_3e_Chapter_17 - Part V Complex Analysis 17 Functions of...

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