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ch13

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Part D. COMPLEX ANALYSIS Major Changes In the previous edition, conformal mapping was distributed over several sections in the first chapter on complex analysis. It has now been given greater emphasis by consolidation of that material in a separate chapter (Chap. 17), which can be used independently of a CAS (just as any other chapter) or in part supported by the graphic capabilities of a CAS. Thus in this respect one has complete freedom. Recent teaching experience has shown that the present arrangement seems to be preferable over that of the 8th edition. CHAPTER 13 Complex Numbers and Functions SECTION 13.1. Complex Numbers. Complex Plane, page 602 Purpose. To discuss the algebraic operations for complex numbers and the representation of complex numbers as points in the plane. Main Content, Important Concepts Complex number, real part, imaginary part, imaginary unit The four algebraic operations in complex Complex plane, real axis, imaginary axis Complex conjugates Two Suggestions on Content 1. Of course, at the expense of a small conceptual concession, one can also start immediately from (4), (5), z x iy , i 2 1 and go on from there. 2. If students have some knowledge of complex numbers, the practical division rule (7) and perhaps (8) and (9) may be the only items to be recalled in this section. (But I personally would do more in any case.) SOLUTIONS TO PROBLEM SET 13.1, page 606 2. Note that z 2 2 i and iz 2 2 i lie on the bisecting lines of the first and second quadrants. 4. z 1 z 2 0 if and only if Re ( z 1 z 2 ) x 2 x 1 y 2 y 1 0 and Im ( z 1 z 2 ) y 2 x 1 x 2 y 1 0. Let z 2 0, so that x 2 2 y 2 2 0. Now x 2 2 y 2 2 is the coefficient determinant of our homogeneous system of equations in the “unknowns” x 1 and y 1 , so that this system has only the trivial solution; hence z 1 0. 8. 23 2 i 10. 9, 16 244 im13.qxd 9/21/05 12:35 PM Page 244

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12. z 1 / z 2 7/41 (22/41) i , z 1 / z 2 ( z 1 / z 2 ) 7/41 (22/41) i 14. 5/13 (12/13) i , 5/13 (12/13) i 16. 3 x 2 y y 3 , y 3 18. Im [ (1 i ) 8 z 2 ] Im [ (2 i ) 4 z 2 ] Im [ 2 4 z 2 ] 32 xy SECTION 13.2. Polar Form of Complex Numbers. Powers and Roots, page 607 Purpose. To give the student a firm grasp of the polar form, including the principal value Arg z , and its application in multiplication and division. Main Content, Important Concepts Absolute value z , argument , principal value Arg Triangle inequality (6) Multiplication and division in polar form n th root, n th roots of unity (16) SOLUTIONS TO PROBLEM SET 13.2, page 611 2. 2(cos 1 _ 2 i sin 1 _ 2 ), 2(cos ( 1 _ 2 ) i sin ( 1 _ 2 )) 4. 1 _ 4 _ 1 16 2 (cos arctan 1 _ 2 i sin arctan 1 _ 2 ) 6. Simplification shows that the quotient equals 3. Answer: 3(cos i sin ). 8. Division shows that the given quotient equals _ 22 41 _ 7 41 i . Hence the polar form is _ 1 41 22 2 7 2 (cos arctan _ 7 22 i sin arctan _ 7 22 ). 10. 3.09163, 3.09163. Of course, the problem should be a reminder that the principal value of the argument is discontinuous along the negative real axis, where it jumps by 2 .
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ch13 - im13.qxd 12:35 PM Page 244 Part D COMPLEX ANALYSIS...

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