Complex Analysis     MATH 305

Exam 1

Thursday, March 3, in class

The first midterm exam covers the material we have discussed in Sections 1 through 23 in the course text. You should go over Homework Assignments #1 - 4, Quiz #1, Computer Lab #1, in-class handouts, and your class notes.

We will review for the exam on Tuesday, March 1 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 the practice 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. Complex Numbers: complex plane, addition, multiplication, basic algebraic properties, multiplicative and additive inverses, conjugate

2. Vectors and Moduli: addition and subtraction of complex numbers vectorially, modulus of a complex number, equation of a circle using the modulus, triangle inequality, properties of the modulus (e.g., the modulus of a product equals the product of the moduli)

3. Complex Conjugate: properties of, connection with modulus

4. Exponential Form: polar coordinates, Euler's formula, argument versus Argument, products and powers in exponential form, de Moivre's formula, arguments of products and quotients

5. Roots of Complex Numbers: how to find the nth roots of a complex number, geometric description of the location of the roots, roots of unity

6. Functions: writing functions in terms of real and imaginary parts, mappings from the z-plane to the w-plane, mapping the exponential function

7. Limits: epsilon-delta definition, approaching from different directions, CR Limit Theorem, Big Limit Theorem (BLT), limits involving infinity, the Riemann Sphere, LIPI Theorem, definition of a continuous function

8. Derivative: definition of the derivative, basic differentiation formulas, Cauchy-Riemann equations (in both rectangular and polar coordinates), SCD Theorem

Some Practice Problems:

Chapter 1
Sections 1 and 2 (p. 5): #1b, 2, 4
Section 4 (p. 12): #1b, 1c, 6
Section 5 (pp. 14 - 16): #1b, 1d, 7, 10
Sections 6, 7 and 8 (pp. 22 - 24): #5a, 6, 9
Sections 9 and 10 (pp. 29 - 31): #1, 2a, 7

Chapter 2
Section 12 (pp. 37 - 38): #1a, 1b, 4
Section 13 and 14 (pp. 44 - 45): #5
Sections 15 - 18 (pp. 55 - 56): #1b, 2b, 11
Sections 19 and 20 (pp. 62 - 63): #1a, 1d, 8
Sections 21 - 23 (pp. 71 - 73): #1b, 1c, 3a, 4c, 5