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All chapters and problems refer to Schaum's Outline of Complex Variables, 2nd ed., Spiegel et al. Definitions of the words in bold may be asked on the exam.

Chapter 1

Definitions: absolute value (norm), conjugate, polar form, $n^{\text{th}}$ roots of unity, Euler's formula

Proofs: Trig identities (22); reverse triangle inequality 7(c)

Sample problems: 13, 16, 37, 131

Chapter 2

Definitions: multiple-valued functions, $\exp \left(z\right)$, $\ln \left(z\right)$, $\cos \left(z\right)$, $\sin \left(z\right)$, limits, continuity, branch lines, sequences, infinite series

Sample problems: 4, 8, 13, 16, 30; find real and imaginary parts of the functions above

Chapter 3

Definitions: derivative, analytic, Cauchy–Riemann equations, harmonic functions, harmonic conjugates, singular points, orthogonal families

Proofs:

• Cauchy–Riemann equations (3.5)
• If $f=u+iv$ is analytic, show that $u$ and $v$ are harmonic if they have continuous second partial derivatives (3.6)
• Orthogonal families (3.27)

Sample problems: 1, 2, 4, 7, 11; be able to verify C–R in polar form

Chapter 4

Definitions: complex line integral, indefinite integral, simply and multiply connected regions, Cauchy's Theorem, Morera's Theorem, Theorems 4.1–4.5 (p. 117)

Proofs: Cauchy's Theorem (4.11)

Sample problems: 2, 17, 21, 22, 23, 27, 43

Chapter 5

Definitions: Cauchy Integral formulas, Cauchy's Inequality, Liouville's theorem, Fundamental Theorem of Algebra, Gauss Mean Value Theorem, Poisson Integral formulas

Proofs: Cauchy Integral formulas; Liouville's theorem; Gauss Mean Value Theorem

Sample problems: 5.2, 5.5, 5.29, 5.52, 5.70

Chapter 6

Definitions: singularities, poles, Taylor series, Laurent series

Skills: Be able to compute Taylor and Laurent series, identify orders of poles, and give examples. Know the Taylor series of $\frac{1}{1-z}$, $e^z$, $\sin \left(z\right)$, $\cos \left(z\right)$.

Proofs: Taylor's Theorem (6.22)

Sample problems: 6.23a, 6.26, 6.27

Chapter 7

Definitions: Residue, Cauchy Principal value. Know and be able to apply the Residue Theorem, calculate residues, and use residue theory to evaluate real integrals.

Proofs: ${\oint }_{C}f\left(z\right)dz=2\pi i{a}_{-1}$  (p. 205, 7.7)

Sample problems: 7.9, 7.10, 7.12, 7.18

Chapter 8

Definitions: conformal mapping, Jacobian, fractional transformation

Proofs:

• If $f\left(z\right)=u+iv$ is analytic in $R$, then $\frac{\partial (u,v)}{\partial (x,y)}={\left|f^{\prime }\right(z\left)\right|}^2$  (p. 259, 8.5)
• If $f\left(z\right)$ is analytic and $f^{\prime }\left(z\right)\ne 0$ in a region $R$, then the mapping $w=f\left(z\right)$ is conformal at all points in $R$.

Sample problems: 8.12, 8.16

Chapter 9

Definitions: Laplace's equation, Dirichlet problem, Poisson formulas

Sample problems: 9.2, 9.7, 9.8