Question

Question of 9 Laurent Series and the Residue Theorem - 9.4 Argument Principle.

I want #2 to be answered.

Exercises 9.59. 1. If f(2) is analytic inside and on the simple closed contour C, and f(z) on C, show that the number of times f(z) C is given by assumes the value a inside f'(2)dz. 1 2πί Jσ f(2)- simple closed contour C except for finite number of poles inside C. Denote the zeros by z1,. . , Zn (none C) and the poles by w1, ... ,Wm. If g(z) is analytic 2. Let f(z) be analytic inside and on a a of which lies on inside and on C, prove that f'(z) g(2) f(z) m 1 de Σ3)-Σw) . 2Ti j=1 j=1 where each zero and pole occurs as its multiplicity. 3. If P(z) = a+ a12+ often in the sum as is required by +anz", evaluate zP'(2) dz

Answer 1

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