Complex Analysis
Winter 2004
Hunsicker
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. The exam will have two parts, which you must take one right after the other. The first part is closed-book. You will have two hours for this part. It will consist of a number of definitions, statements of some theorems, and proofs or sketches of proofs for some theorems. The possible definitions and theorems are listed below. The second part of the exam will be an open-book problem section. You will have four hours to complete this part. The problems may involve filling in details from the book, or may involve calculations or short proofs from the first half of the end of chapter problems from any of the chapters we covered. No notes or calculators will be permitted on the exam. I will tell you when I hand out the exam to whom you should turn it in when you have completed it.
Definitions you should be able to state:
1) Modulus and argument of a complex number
2) Stereographic projection and the extended complex plane
3) Mobius tranformation
4) Open set, closed set
5) Limit point of a set, isolated point of a set
6) Closure of a set
7) Convex set, polygonally connected set
8) Region
9) Compact set
10) Limit of a complex valued function, continuity of a function
11) Curve, simple curve, closed curve, contour
12) Boundary of a set
13) f(x) is differentiable at a
14) f(x) is holomorphic at a
15) Power series and radius of convergence
16) Multifunction
17) [[ arg z ]]
18) [[ log z ]], holomorphic branches of the logarithm, branch cut
19) The integral of a function along a path
20) Zeros and their orders
21) Singularities (17.8)
22) Principal part
23) Removable singularity, pole of order m, isolated essential singularity
24) Meromorphic function
25) Residue
Theorems you should be able to state:
1) Bolzano-Weierstrass Theorem
2) Jordan curve theorem (for a contour)
3) Triangularization of a polygon
4) Radius of convergence lemma
5) The Fundamental Theorem of Calculus
6) Deformation Theorem I
7) Liouville’s theorem
8) Taylor’s theorem
9) Maximum Modulus Theorem
10) Meromorphic functions in the extended complex plane (17.20)
11) Cauchy’s Residue Theorem
Theorems you should be able to prove completely:
1.9: Inequality theorems
2.15: Circlines under Mobius Transformations
3.7: Closed sets and Closure
3.24: Boundedness Theorem for continuous functions
5.3: The Cauchy-Riemann equations
5.6: Partial converse to Cauchy-Riemann equations
6.8: Differentiation theorem for power series
10.4: The Fundamental integral theorem
10.10: The Estimation Theorem
11.3: Indefinite Integral Theorem I
13.1: Cauchy’s Integral Formula
13.4: Fundamental Theorem of Algebra
Theorems you should be able to sketch proofs for:
11.2: Cauchy’s Theorem I (for a triangle)
11.6: Cauchy’s Theorem I (for a contour)
13.9: Cauchy’s formula for derivatives
15.7-15.9: Identity and Uniqueness Theorems
17.3: Laurent’s theorem