Complex Analysis     MATH 305

Midterm Exam

Wednesday, Oct. 26, 7:00 - 8:30 pm, Swords 302

The Midterm Exam covers all the material we have discussed in Chapters 1, 2 and 3 of the course text. This is everything except for Sections 11, 27, 28, 35 and 36. You should go over homework assignments #1, 2, 4 and 5, computer lab #1, the textbook and your class notes.

We will review for the exam in class on Monday, Oct. 24. Please come prepared with specific questions. Some sample practice problems from the text are given 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 inequalities, 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. Derivatives and Analytic Functions: definition of the derivative, basic differentiation formulas, Cauchy-Riemann equations (in both rectangular and polar coordinates), SCD Theorem, definition of an analytic function, entire functions, singular points, finding where a function is analytic (if at all)

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

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

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

Some Practice Problems:

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

Chapter 2
Section 12 (p. 37 - 38): #1a, 1b, 4
Section 13 and 14 (p. 44 - 45): #5
Sections 15 - 18 (p. 55 - 56): #1b, 2c, 11
Sections 19 and 20 (p. 62 - 63): #1b, 1c, 8
Sections 21 - 23 (p. 71 - 73): #1b, 1c, 3a, 4c, 5
Sections 24 and 25 (p. 77 - 78): #1c, 1d, 2a, 4a
Section 26 (p. 81 - 83): #1a, 1d, 5, 6

Chapter 3
Section 29 (pp. 92 - 93): #4, 8b, 13
Sections 30 - 31 (pp. 97 - 98): #3, 5, 9a
Section 32 (p. 100): #2, 4
Section 33 (p. 104): #3, 8c, 9
Section 34 (pp. 108 - 109): #3, 7, 14