Home_9 - Homework 9 Stephen Taylor Pages 74 75 5 Give the...

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Unformatted text preview: Homework 9 Stephen Taylor May 20, 2005 Pages 74- 75 : 5. Give the power series expansion of log z about z = i and find its radius of convergence. Letting f ( z ) = log z we find that its n-th derivative is given by f ( n ) ( z ) = (- 1) n- 1 ( n- 1)! z- n Using the standard Taylor series formula, we find f ( x ) has representation f ( z ) = log( i ) + ∞ X n =1 (- 1) n- 1 i n +1 n ( z- i ) n We note the radius of convergence of this series is 1 due to the singularity at the origin. This fact may also be verified by using the ratio test. 7. Evaluate the following (a) Z γ e iz z 2 dz, γ ( t ) = e it , ≤ t ≤ 2 π By Corollary (2.13) this integral has the value 2 πif (0) where f ( z ) = e iz . So we find the value of the integral to be- 2 π . (b) Z γ dz z- a , γ ( t ) = a + re it , ≤ t ≤ 2 π By Proposition (2.6) we find the value of this integral to be 2 πif ( a ) where f ≡ 1. So the value of the integral is 2 πi ....
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Home_9 - Homework 9 Stephen Taylor Pages 74 75 5 Give the...

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