Appendixes - INSTRUCTORS SOLUTIONS MANUAL APPENDIX I. (PAGE...

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INSTRUCTOR’S SOLUTIONS MANUAL APPENDIX I. (PAGE A-10) APPENDICES Appendix I. Complex Numbers (page A-10) 1. z =− 5 + 2 i , Re ( z ) 5 , Im ( z ) = 2 y x z 5 + 2 i z 6 z π i z = 4 i z -plane Fig. .1 2. z = 4 i , Re ( z ) = 4 , Im ( z ) 1 3. z π i , Re ( z ) = 0 , Im ( z ) π 4. z 6 , Re ( z ) 6 , Im ( z ) = 0 5. z 1 + i , | z |= 2 , Arg ( z ) = 3 π/ 4 z = 2 ( cos ( 3 4 ) + i sin ( 3 4 )) 6. z 2 , | z 2 , Arg ( z ) = π z = 2 ( cos π + i sin π) 7. z = 3 i , | z 3 , Arg ( z ) = 2 z = 3 ( cos (π/ 2 ) + i sin (π/ 2 )) 8. z 5 i , | z 5 , Arg ( z ) 2 z = 5 ( cos ( 2 ) + i sin ( 2 )) 9. z = 1 + 2 i , | z 5 = Arg ( z ) = tan 1 2 z = 5 ( cos θ + i sin θ) 10. z 2 + i , | z 5 = Arg ( z ) = π tan 1 ( 1 / 2 ) z = 5 ( cos θ + i sin 11. z 3 4 i , | z 5 = Arg ( z ) π + tan 1 ( 4 / 3 ) z = 5 ( cos θ + i sin 12. z = 3 4 i , | z 5 = Arg ( z ) tan 1 ( 4 / 3 ) z = 5 ( cos θ + i sin 13. z = 3 i , | z 2 , Arg ( z ) 6 z = 2 ( cos ( 6 ) + i sin ( 6 )) 14. z 3 3 i , | z 2 3 , Arg ( z ) 2 3 z = 2 3 ( cos ( 2 3 ) + i sin ( 2 3 )) 15. z = 3 cos 4 π 5 + 3 i sin 4 π 5 | z 3 , Arg ( z ) = 4 π 5 16. If Arg ( z ) = 3 π 4 and Arg (w) = π 2 , then arg ( z w) = 3 π 4 + π 2 = 5 π 4 ,so Arg ( z w) = 5 π 4 2 π = 3 π 4 . 17. If Arg ( z ) 5 π 6 and Arg (w) = π 4 , then arg ( z /w) 5 π 6 π 4 13 π 12 Arg ( z /w) 13 π 12 + 2 π = 11 π 12 . 18. | z 2 , arg ( z ) = π z = 2 ( cos π + i sin 2 19. | z 5 = arg ( z ) = π sin θ = 3 / 5 , cos θ = 4 / 5 z = 4 + 3 i 20. | z 1 , arg ( z ) = 3 π 4 z = ± cos 3 π 4 + i sin 3 π 4 ² z 1 2 + 1 2 i 21. | z π, arg ( z ) = π 6 z = π ³ cos π 6 + i sin π 6 ´ z = π 3 2 + π 2 i 22. | z 0 z = 0 for any value of arg ( z ) 23. | z 1 2 , arg ( z ) π 3 z = 1 2 ³ cos π 3 i sin π 3 ´ z = 1 4 3 4 i 24. 5 + 3 i = 5 3 i 25. 3 5 i 3 + 5 i 26. 4 i 4 i 27. 2 i = 2 + i 28. | z 2 represents all points on the circle of radius 2 centred at the origin. 29. | z |≤ 2 represents all points in the closed disk of radius 2 centred at the origin. 30. | z 2 i 3 represents all points in the closed disk of radius 3 centred at the point 2 i . 31. | z 3 + 4 i 5 represents all points in the closed disk of radius 5 centred at the point 3 4 i . 32. arg ( z ) = 3 represents all points on the ray from the origin in the first quadrant, making angle 60 with the positive direction of the real axis. 661
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APPENDIX I. (PAGE A-10) R. A. ADAMS: CALCULUS 33. π arg ( z ) 7 π/ 4 represents the closed wedge-shaped region in the third and fourth quadrants bounded by the ray from the origin to −∞ on the real axis and the ray from the origin making angle 45 with the positive direction of the real axis. 34. ( 2 + 5 i ) + ( 3 i ) = 5 + 4 i 35. i ( 3 2 i ) + ( 7 3 i ) =− 3 + 7 + i + 2 i 3 i = 4 36. ( 4 + i )( 4 i ) = 16 i 2 = 17 37. ( 1 + i )( 2 3 i ) = 2 + 2 i 3 i 3 i 2 = 5 i 38. ( a + bi )( 2 a bi ) = ( a + bi )( 2 a + bi ) = 2 a 2 b 2 + 3 abi 39. ( 2 + i ) 3 = 8 + 12 i + 6 i 2 + i 3 = 2 + 11 i 40. 2 i 2 + i = ( 2 i ) 2 4 i 2 = 3 4 i 5 41. 1 + 3 i 2 i = ( 1 + 3 i )( 2 + i ) 4 i 2 = 1 + 7 i 5 42. 1 + i i ( 2 + 3 i ) = 1 + i 3 + 2 i = ( 1 + i )( 3 2 i ) 9 + 4 = 1 5 i 13 43. ( 1 + 2 i )( 2 3 i ) ( 2 i )( 3 + 2 i ) = 8 + i 8 + i = 1 44. If z = x + yi and w = u + v i , where x , y , u , and v are real, then z + w = x + u + ( y + v) i = x + u ( y + i = x + u v i = z + w.
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Appendixes - INSTRUCTORS SOLUTIONS MANUAL APPENDIX I. (PAGE...

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