MIT 18.112 Complex Variable Final Exam: Problems & Solutions, Fall 2008, Exams of Calculus

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18.112 Functions of a Complex Variable Fall 2008

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Problems for 18.112 Final Examination

(Open Book)

Dec. 20, 200

1. (20’) Let a, b, c be complex numbers satisfying

b − a a − c =. c − a b − c

Considering the triangle with vertices a, b, c. Prove

|b − a| = |c − a| = |b − c|.

2. (15’) Find where the series �∞^ n z

n=1^ 1 +^ z^2 n

converges and determine where the sum f (z) is holomorphic. Give reasons for your answer.

3. (15’) Evaluate |z|ez dz γ z^2

where γ is the circle with radius 2 and center 0.

4. (15’) Prove that if f (z) has a pole of order h at z 0 , then

1 dh−^1 Resz=z 0 f (z) = (z − z 0 )hf (z). (h − 1)! dzh−^1 z=z 0