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Given a curve c in a domain d c and a function f dc

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Unformatted text preview: e upper half plane to the lower half plane (c) Maps the second quadrant onto the right-half plane (d) What happens to the strip {x+iy | 0 ≤ y ≤ 1, x ≥ 0} under the map f(z) = ie-z? 2. A Möbius transformation is a complex function of the form az + b f(z) = . cz + d (a) Find a Möbius transformation f with the property that f(1) = 1, f(0) = i, and f(-1) = -1. (b) Prove that your function is the only possible Möbius transformation with this property. (It is suggested you do some research in the Section 12.9 of the textbook.) 4. Contour Integrals & the Cauchy-Goursat Theorem (§18.1–18.4 in the text) A curve C in the complex plane C is a pair of piecewise continuous functions x = x(t), y I = y(t) for a ≤ t ≤ b. (This is just a piecewise continuous curve in 2-dimensional space). Given a curve C in a domain D ¯ C and a function f: DÆC , we can define the I I corresponding contour integral, Û Ù f(z) dz ı C as the limit of a Riemann sum of the form £f(zi*)∆zi associated with a partition a = t0 < t1 ≤ ... ≤ tn = b, where the limit is taken as max {|∆zi|} Æ 0. If we write f(z) as u(x, y) + iv(x, y) and d...
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