MATH 655, Review for Midterm Test 1

Test topics

1. Complex Numbers and Functions
   1. Complex Numbers, Geometric Interpretation. Functions and Continuity (1-1)
   2. Sequences and Series. Uniform Convergence (1-2)
   3. Power Series; Convergent Series, Formal Algebra of Power Series (1-3)
   4. Exponential Function, Logarithm, Complex Powers, Trigonometric Functions and their Inverses (1-4)
   5. The Spherical Representation (A:1.2.4)
   6. Fractional Linear Transformations. The Cross-Ratio. Symmetry Principle (A:3.3.1-3)
   7. Geometric Properties of Elementary Functions: Powers, Exponential, Logarithm, Trigonometric (1-5)
   8. Multi-Valued Functions. Branches, Branch Points and Branch Cuts. Elementary Riemann Surfaces (1-5)

2. Analytic Functions
   1. Differentiation. Complex and Real Derivatives. Cauchy-Riemann Equations (2-1)
   2. Integration in the Complex Plane. Cauchy's Theorem (2-2)

Test Preparation

• Read the material in the textbook(s) closely, fill in the missing details as necessary
• Work through examples in the book and on the homework.
• Practice problems as well as review problems (some of which may end up on the midterm) are available (see the main page).

General rules

• All tests are closed books/notes; graphing calculators, cell phones or other electronic devices are not permitted.
• The duration of the test is 75 minutes.
• Problems will come in the same format as on the quizzes: you will need to write a concise proof to answer each question.