Transcription of Math 520a - Homework 2
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Math 520a - Homework 2
math 520a - Homework 21. Use Cauchy s integral formula (for an analytic function or its derivatives)to evaluate(a) For the contour (t) =eit, 0 t 2 , the integral eizz2dz(b) For the contour (t) = 1 +12eit, 0 t 2 , the integral ln(z)(z 1)ndzSolution:I ll just give answers for this one.(a) 2 (b) Integral is 0 forn= 1. Forn >1 it is ( 1)n2 i/(n 1).2. Letf(z) be an entire function such that there are constantsC, Dwith|f(z)| C+D|z|n, zProve thatfis a polynomial of degree at :Sincefis entire it has a power series about the origin whichconverges for (z) = k=0akzkThe coefficients are given byak=f(k)(0)/k!. We will show thatf(k)(0) = 0fork > n.
Math 520a - Homework 2 1. Use Cauchy’s integral formula (for an analytic function or its derivatives) to evaluate (a) For the contour γ(t) = eit, 0 ≤ t ≤ 2π, the integral Z
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