Math 601-602 — Final Exam Review

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. There will be 6 to 8 questions, some with multiple parts. The problems will be similar to ones done for homework, and as examples in class and in the text.

Topics Covered in Complex Analysis

Algebra of complex numbers. Chapter 1 of Matthias Beck, Gerald Marchesi, and Dennis Pixton, A First Course in Complex Analysis covers complex numbers per se — addition, multiplication, real and imaginary parts, absolute value (modulus), complex conjugate, argument (polar angle), and polar form. In particular, be able to work with the principal argument, Arg(z). Be able to find any of the quantities listed for a complex number. Be able to solve algebraic equations involving complex numbers; for example, z⁴ = -16.

Analytic functions. Know what an analytic function is and be able to use the Cauchy-Riemann conditions to establish that a function f(z) = u+iv is analytic. Most of the functions that you encountered in elementary calculus are analytic, except for isolated singularities or, in the case of Log(z), everything except Im(z) ≤ 0. Here is the list of functions that we encountered:

polynomials

rational functions (ratios of polynomials)

e^z = e^xcos(y) + i e^xsin(y) (z=x+iy)

Log(z) = ln(|z|) + iArg(z)

a^z = e^{z Log(a)}

sin(z) = (e^iz − e^−iz)/(2i)

cos(z) = (e^iz + e^−iz)/2

sinh(z) = (e^z − e^−z)/2

cosh(z) = (e^z + e^−z)/2

other trigonometric functions; e.g., tan(z), cot(z), csc(z), sec(z)

algebraic combinations of analytic functions — sums, products, quotients

composition — f(g(z)); for example, sin(e^z)

Integration. This is covered in chapter 4 of the online text. There are three main formulas: the Fundamental Theorem (Theorem 4.2), Cauchy's Theorem (∮_C f(z)dz = 0, where C is any simple closed curve), Cauchy's integral formula (Theorem 4.8), and Cauchy's integral formulas for the derivatives of f(z) (cf. chapter 8, Theorem 8.6).

Taylor and Laurent series. Taylor series can be computed in the same as they were in elementary calculus. You should know the Taylor series for e^z, cos(z), sin(z), Log(1+z), (1-z)⁻¹. Laurent series are series having negative powers as well as positive ones; that is, they have the form
∑_n a_n(z−z₀)ⁿ, where n runs from −∞ to ∞.
The sum doesn't have to have all of the negative or positive powers. For example,
z⁻³cos(z) = z⁻³−z⁻¹/2 + z/24 + z⁻²/ 720 + ...
We will only be concerned with Laurent series valid near an "isolated singularity." We will only work with Laurent series that are similar to the one given in the example above.

Singularities. This material is covered in chapter 9. Isolated singularities can be classified as one of three types: removable, pole, or essential. Poles have orders, m = 1, 2, 3, etc. For orders m = 1 and m = 2, the poles are also called simple or double, respectively. The classification can be done in several ways, but the easiest way is through the Laurent expansion valid near an isolated singularity z₀.

Suppose that f has the Laurent expansion
f(z) = ∑_n a_n(z−z₀)ⁿ.
If there are no negative powers in the expansion, the singularity is removable. For example, sin(z)/z has an isolated singularity at z = 0 because it is not defined there. However, near z = 0, we have
sin(z)/z = 1 − z²/6 + ...,
so the singularity is removable. Really, this means that by defining sin(z)/z to be 1 at z = 0, we can make it analytic there; i.e., we can remove the singularity.

If there are a finite negative powers in the expansion, the singularity is a pole. the order of the pole is the highest negative power in the expansion. The function z⁻³cos(z) has the Laurent series
z⁻³cos(z) = z⁻³−z⁻¹/2 + z/24 + z⁻²/ 720 + ...,
so it has a pole of order 3.

If there are an infinite number of negative terms in the Laurent series, i.e. the negative part series doesn't "cut off" at any finite value, then the singularity is essential. The function e^1/z has an essential singularity at z = 0 because
e^1/z = 1 + z⁻¹ + z⁻²/2! + z⁻³/3! + ...

Residues. In the Laurent series for a function f(z) with an isolated singularity at z = z₀, the residue of f at z₀ is the the coefficient of (z−z₀)⁻¹. In symbols, we write
Res_z0(f) := a₋₁.
The residues for the functions listed above are the following:

• z⁻³cos(z), Res ₀ = −1/2.
• sin(z)/z, Res ₀ = 0.
• e^1/z, Res ₀ = 1.

The Residue Theorem. The residue theorem (Theorem 9.4) states that if f(z) is analytic on a positively oriented (counterclockwise) simple closed curve C, and if f has only isolated singularities in C, then
∮_C f(z)dz = 2π i ∑ residues enclosed in C
In the notes on Residues and Contour Integration Problems, there are a number of problems involving singularities, residues, and applications, some with complete solutions and others with just answers. Consult these notes for further details.

Updated 5/12/09