Math 407-500 — Fall 2023

Test 1 Review

General Information

Calculators. You may use calculators to do arithmetic, although you will not need them. You may not use any calculator that has the capability of doing linear algebra or storing programs or other material.

Other devices. You may not use cell phones, computers, or any other device capable of storing, sending, or receiving information.

Structure and coverage. There will be 4 to 6 questions, some with multiple parts. In what follows, ICA stands for Silverman's book and SO for Schaum's Outline. The test will cover ICA Chapter 1, less 1.5, 1.7, 1.8; Chapter 2, less 2.12; Chapter 3, 3.14. For 3.13, see the lecture notes for 9/8. In SO, Chapter 1, less 1.6, 1.1.14-1.17; Chapter 2, less 2.4, 2.5 2.8; Chapter 3, less 3.4-3.6, 3.12-3.16. The two books overlap a lot. Many items are very similar to what you already know from calculus, with a complex variable twist -- e.g., the derivatives in SO 3.8, L'Hospital's rule and so on.

The problems will be similar to ones done for homework, and as examples in class and in the texts. There will also be questions about definitions and proofs of theorems. These are indicated below.

Topics Covered

Algebra of Complex Numbers

• Polar form, modulus $|z|$, real and imaginary parts, $arg(z)$, $Arg(z)$. (These angles have different meanings in ICA and SO. I will be clear about which I want you to use.)
• De Moivre's theorem, Euler's formula, powers and roots, roots of unity. Be able to prove both De Moivre's theorem and Euler's formula. Be able to use these to evaluate expressions like $\sqrt{\frac{3-2i}{5+i}}$.
• Point sets. See the list in SO, p. 7, items 1-11.
• Simple sequences, series and their limits.

Functions, Limits and Continuity

• Single valued, inverse functions, multiple-valued and branch of a multiple-valued function. Be able to locate branch points and branch cuts.
• Elementary functions -- trig, exponential, log, hyperbolic, powers, inverse trig, inverse hyperbolic functions. Know these and be able to work with them in problems. See SO for examples.
• Limits--be able to define $\lim_{z\to z_0} f(z)$. Be able to use standard limit theorems to evaluate limits, and be able to use L'Hospital's rule to evaluate limits. In addition, be able to prove L'Hospital's rule. (This really belongs in the section on differentiation but fits nicely here.)
• Be able to define the term continuity and known standard properties of continuous functions and be able use them to show that specific functions are continuous. For example, polynomials, exponentials and so on.

Differentiation

• Definition of the derivative of a function, and the definition of an analytic function.
• Be able to state the Cauchy Riemann conditions. Be able to prove that if a function is analytic, the CR conditions hold. (You won't be asked to prove the converse.) Be able to verify that they hold for a given function.
• Be able to use the standard rules of differentiation to compute derivatives of inverse functions.
• Be able to locate singularities and to classify them -- isolated: removable, simple pole, pole of order $m$, essential singularity; non isolated: branch points. You will only be asked about isolated singularities that are not essential. For the others, be able to determine whether it's removable or a pole. If it's a pole, be able to find its order.

Updated 9/21/23.