FINAL EXAM REVIEW

Format of the exam:

The exam will be a self-scheduled take-home exam. You are bound strictly by the honor code not to discuss any aspect of the exam with anyone after you have taken it. You have three hours to complete the exam. You may not use any books, notes, calculators, neighbors, computers, etc. In fact,

you may use nothing but a pencil, paper, and your brain.

Definitions you should be able to state:

1) Domain in C

2) Differentiable function f : S--> C

3) Analytic function

4) Holomorphic function

5) Open and closed set in C

6) Path connected set

7) Topologists' definition of continuity (prop. 2.4)

8) log, Log, Log_a, arg, arg_a, argument

9) Winding number of a path around a point.

10) Zero of order m

11) Isolated singularity

12) Removable singularity, pole, isolated essential singularity

13) Stereographic projection

14) Riemann sphere/extended complex plane

15) Behavior at infinity of f(z) (p. 207)

16) Meromorphic function

Theorems you should be able to state and prove:

1) The Paving lemma

2) Differentiation of power series

3) Cauchy Riemann equations

4) Partial converse to Cauchy Riemann equations

5) If |f| is constant on a domain D and f is differentiable then f is constant

6) Fundamental Theorem of contour integration

7) The Estimation Lemma

8) Antiderivative Theorem (Theorem 6.11)

9) Cauchy's Theorem for a triangle

10) Antiderivative theorem for star-shaped domains

11) Cauchy's Theorem

12) Generalized Cauchy's Theorem

13) Cauchy's Integral formula

14) Taylor's theorem (10.2 AND 10.3)

15) Morera's theorem

16) Cauchy's Estimate

17 )Liouville's Theorem

18) Fundamental Theorem of Algebra

19) Identity Theorem (10.10 and 10.11)

20) Maximum Modulus Theorem

21) Minimum Modulus Theorem

22) Laurent's Theorem (including uniqueness)

23) Differentiation of negative power series lemma

24) Behavior at an isolated singularity (Lemma 11.2)

25) Behavior at a pole (prop. 11.4 and corollary 11.6)

26) Weierstrass-Casorati Theorem

27) STATE ONLY: Picard's Theorem

28) Meromorphic functions on Riemann sphere (Prop. 11.8)

29) Cauchy's Residue Theorem

30)Type II integral theorem (p. 219)

31) Type III integral theorem (p. 222)

32) Series summation theorem (class notes)

33) Counting zeros theorem (12.4)

34) Rouche's theorem