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hw5sol

# hw5sol - Complex Analysis Spring 2001 Homework V Solutions...

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Complex Analysis Spring 2001 Homework V Solutions 1. Conway, chapter 4, section 5, problem 7. Let γ ( t ) = 1 + e it for 0 t 2 π. Find γ ( z z - 1 ) n dz for all positive integers n. By Corollary 5.8, this is 2 πi ( n - 1)! times the n - 1rst derivative of f ( z ) = z n evaluated at z = 1 . The n - 1rst derivative of z n is n ! z and so the result is 2 nπi. 2. Conway, chapter 4, section 5, problem 9. Show that if f : C C is continuous and analytic off the interval [ - 1 , 1] then f is entire. This is an application of Morera’s theorem. One has to show that the integral of f over the boundary of any rectangle is equal to zero. For rectangles which are disjoint from the interval, this follows from Cauchy’s theorem. Rectangles which meet the interval can be broken into a union of subrectangles which lie on and above the x-axis and rectangles which lie on or below the axis. For any such rectangle which intersects the interval, show that the integral is given as the limit as δ 0 of the integrals over rectangles where the side lying on the horizontal axis is moved distance δ away from the axis. This is where continuity is used, since if

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