TEAM PROJECT. Integration. (a) Comparison. Write a short

Chapter 14, Problem 14.1

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TEAM PROJECT. Integration. (a) Comparison. Write a short repOit comparing the essential points of the two integration methods. (b) Comparison. Evaluate I/(:::) d::: by Theorem l and check the result by Theorem 2, where: (i) f(::.) = ::4 and C is the semicircle Izi = 2 from - 2i to 2i in the right half-plane, (ii) f(:.:) = e2z and C is the shonest path from 0 to 1 + 2i. (e) Continuous deformation of path. Experiment with a family of paths with common endpoints, say, z(t) = 1 + ia sin t, 0 ~ t ~ 71. with real parameter a. Integrate nonanalytic functions (Re:c, Re (:.:2), etc.) and explore how the result depends on a. Then take analytic functions of your choice. (Show the details of your work.) Compare and comment. (d) Continuuus deformation of path. Choose another family. for example. semi-ellipses z(t) = a cos 1 + i sin I, -7T/2 ~ t ~ 71'/2, and experiment as in (c).