Functions of a Complex Variable: Mid 1 Exam Solutions, Fall 2008, 18.112, MIT, Exams of Calculus

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Problems for 18.112 Mid 1 (Open Book)

Oct. 20, 200

1. (10’) Find all solutions z to equation z^3 = − 8 i.
2. (15’) Evaluate the integral dz (^). |z− 1 |= 21 (1^ −^ z)^3
3. (20’) Evaluate the integral γ e zz −+ 2 z (^) dz in the two cases: 1) γ = {z : |z| = 1}; 2) γ = {z : |z| = 3}.
4. (30’) (^) singularityLet f (z) be at analytic ∞. Show in thethat whole f (z) planeis a polynomial. and assume it has a nonessential
5. (Hint: (25’) Look (^) someSuppose at n f and (fm ()z(0) |)z is| for> analytic 100.large Show m in.) the that whole f is planea polynomial. and suppose |f (z)| < |z|n^ for

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