Complex Comprehensive Review

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