Download Introductory Complex and Analysis Applications by William R. Derrick PDF

By William R. Derrick

ISBN-10: 0122099508

ISBN-13: 9780122099502

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Evaluate J | , | = i | ^ + 1 | \dz\. 2 THE CAUCHY-GOURSAT THEOREM In Exercise 3(a) and Example I of the last section we found that the line integral of a polynomial along a pwd closed curve vanishes, but that |z| = l ζ However, the function l/z is not analytic at the origin. Might it be true that the line integral of a function along a pwd Jordan curve vanishes when the function is analytic on and inside the curve? Surprisingly, the answer is in the affirmative. T h e first step in obtaining this result is the following theorem, due to Cauchy and Goursat: 2 34 Theorem rectangle C O M P LX E I N T E G R A TN I O Let the function f(z) be analytic on a domain containing the given by the inequalities a < χ

Without computing the integral, show that the circle dz An < —. = 2. z ^ + l ~ 3 |z| = 7. If y is the semicircle | z | = R, |arg z | < π/2, R> \, show that Γ Log ζ dz 4(Log/^^-^), and hence that the value of the integral tends to zero as 8. Evaluate e"^ dz, where y is: (a) (b) (c) -> o o . the straight line path joining 1 to /, the path is the first quadrant along the circle | z | = 1 joining 1 to /, the path along the coordinate axes joining 1 to /. 9. If y is the ellipse z(t) = α cos í + ib sin í, O < í < 2n, that dz = - b^ = l, show ±2n.

24 1 A N A L Y C T I F U N C T I O S N e-{0} [e-{0}]" Figure 1 . 1 4 Since e'^ = cos + / sin y and e cos y = • 2 = cosy — i sin y it follows that ' sin y = e'' — e 2i We extend these definitions to the complex plane: Definition cos ζ = e'' + e2 sm ζ = 2i These functions are entire as they are sums of entire functions and satisfy (cos z)' = (sinz) = = —sm z, 2i - - = = cosz. The other four trigonometric functions, defined in terms of the sine and cosine function by the usual relations tan ζ = sec ζ = sm ζ cos z ' 1 cos z ' cot ζ = COS ζ - , sm ζ 1 CSC ζ = — sm ζ are analytic except where their denominators vanish, and satisfy the usual rules of differentiation (tan z)' = sec^ z, (sec z)' = sec ζ tan z, (cot zY = - c s c ^ z, (esc z)' = -CSC ζ cot z.

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