Complex Analysis     MATH 305

Exam 2

Thursday, April 21, in class

The second midterm exam covers the material we have discussed in Sections 24 through 50 in the course text (excluding Sections 27, 28, 35, 36, 42, and 47). You should go over Homework Assignments #5 - 7, Computer Lab #2, the worksheet for Chapter 3, and your class notes.

We will review for the exam on Tuesday, April 19 from 7:00 - 8:30 pm in Smith Labs 155. Please come prepared with specific questions. Some sample practice problems are available here. Solutions to these problems are available here.
Other sample problems from the textbook are provided below.

Note: No calculators are allowed on the exam so be prepared to answer questions without your personal calculator.

The following topics, definitions and theorems are important material for the exam. You may be asked to define some terms precisely as well as state and/or prove important theorems. Keep in mind that many of the problems in the text provide answers, either as part of the question or listed after the problem. You will not have such an advantage on the exam.

1. Harmonic Functions: Laplace's equation, definition of a harmonic function, the harmonic conjugate, finding harmonic conjugates

2. Important Functions: exponential e^z, logarithm log z, complex exponents z^c, cos z and sin z, know their definitions, properties and derivatives

3. Logarithms : log z versus Log z, branches and branch cuts of log z, the principal branch of log z, properties of log z

4. Contour Integrals: Set-up and evaluation of contour integrals, parametrizations of lines and circles, properties of contour integrals, upper bounds for the moduli of contour integrals (the ``ML Theorem'')

5. Important Integration Theorems and Formulas: AD theorem (antiderivative = path independence, etc.), Cauchy-Goursat theorem, simply versus multiply connected domains, Principle of deformation of paths (PDP), Cauchy integral formula

6. Other Important Material: polar coordinates, Euler's formula, properties of the modulus (e.g., the modulus of a product equals the product of the moduli), triangle inequalities, Cauchy-Riemann equations (in both rectangular and polar coordinates), definition of an analytic function, entire functions, finding where a function is analytic (if at all)

Some Practice Problems:

Chapter 2
Sections 24 and 25 (p. 77 - 78): #1d, 2a, 4a
Section 26 (p. 81 - 83): #1a, 1d, 6

Chapter 3
Section 29 (pp. 92 - 93): #4, 8b, 13
Sections 30 - 31 (pp. 97 - 98): #3, 5, 9a
Section 32 (p. 100): #1, 4
Section 33 (p. 104): #3, 8a, 8c
Section 34 (pp. 108 - 109): #2, 3, 14

Chapter 4
Sections 37 - 38 (p. 121): #3, 5
Sections 40 - 42 (pp. 135 - 136): #2, 4, 10
Section 43 (pp. 140 - 142): #4, 5
Sections 44 - 45 (p. 149): #1
Sections 46 - 49 (p. 160 - 163): #2
Section 50 (pp. 170 - 172): #3