# Book:Murray R. Spiegel/Theory and Problems of Complex Variables/SI (Metric) Edition

## Contents

- 1 Murray R. Spiegel:
*Theory and Problems of Complex Variables (SI (Metric) Edition)* - 2 Errata
- 2.1 Example: $\paren {z_3 - \overline {z_3} }^5$
- 2.2 Condition for Points in Complex Plane to form Parallelogram
- 2.3 Polar Form of Complex Number: $3 \, \map \cis {\dfrac {-2 \pi} 3}$
- 2.4 Complex Addition: Travel $2$
- 2.5 Complex Division: $\dfrac {\paren {3 \cis \dfrac \pi 6} \paren {2 \cis \dfrac {-5 \pi} 4} \paren {6 \cis \dfrac {5 \pi} 3} } {\paren {4 \cis \dfrac {2 \pi} 3}^2}$
- 2.6 Quadruple Angle Formula for Sine
- 2.7 5th Roots of $-16 + 16 \sqrt 3 i$
- 2.8 Roots of $z^6 + 1 = \sqrt 3 i$
- 2.9 Examples of Set Intersection and Set Union
- 2.10 Example of Set Intersection with Union
- 2.11 Condition for Quartic with Real Coefficients to have Wholly Imaginary Root
- 2.12 Cosine to Power of Odd Integer

## Murray R. Spiegel: *Theory and Problems of Complex Variables (SI (Metric) Edition)*

Published $1981$, **Schaum's Outline Series**

- ISBN 0-070-84382-1.

### Subject Matter

### Contents

- Preface

**Chapter 1 COMPLEX NUMBERS**- The real number system. Graphical representation of real numbers. The complex number system. Fundamental operations with complex numbers. Absolute value. Axiomatic foundations of the complex number system. Graphical representation of complex numbers. Polar form of complex numbers. De Moivre's theorem. Roots of complex numbers. Euler's formula. Polynomial equations. The $n$th roots of unity. Vector interpretation of complex numbers. Spherical representation of complex numbers. Stereographic projection. Dot and cross product. Complex conjugate coordinates. Point sets.

**Chapter 2 FUNCTIONS, LIMITS AND CONTINUITY**- Variables and functions. Single-and multiple-valued functions. Inverse functions. Transformations. Curvilinear coordinates. The elementary functions. Branch points and branch lines. Riemann surfaces. Limits. Theorems on limits. Infinity. Continuity. Continuity in a region. Theorems on continuity. Uniform continuity. Sequences. Limit of a sequence. Theorems on limits of sequences. Infinite series.

**Chapter 3 COMPLEX DIFFERENTIATION AND THE CAUCHY-RIEMANN EQUATIONS**- Derivatives. Analytic functions. Cauchy-Riemann equations. Harmonic functions. Geometric interpretation of the derivative. Differentials. Rules for differentiation. Derivatives of elementary functions. Higher order derivatives. L'Hospital's rule. Singular points. Orthogonal families. Curves. Applications to geometry and mechanics. Complex differential operators. Gradient, divergence, curl and Laplacian. Some identities involving gradient, divergence and curl.

**Chapter 4 COMPLEX INTEGRATION AND CAUCHY'S THEOREM**- Complex line integrals. Real line integrals. Connection between real and complex line integrals. Properties of integrals. Change of variables. Simply-and multiply-connected regions. Jordan curve theorem. Convention regarding traversal of a closed path. Green's theorem in the plane. Complex form of Green's theorem. Cauchy's theorem. The Cauchy-Goursat theorem. Morera's theorem. Indefinite integrals. Integrals of special functions. Some consequences of Cauchy's theorem.

**Chapter 5 CAUCHY'S INTEGRAL FORMULAE AND RELATED THEOREMS**- Cauchy integral formulae. Some important theorems. Morera's theorem. Cauchy's inequality. Liouville's theorem. Fundamental theorem of algebra. Gauss' mean value theorem. Maximum modulus theorem. Minimum modulus theorem. The argument theorem. Rouché's theorem. Poisson's integral formulae for a circle. Poisson's integral formulae for a half plane.

**Chapter 6 INFINITE SERIES. TAYLOR'S AND LAURENT SERIES**- Sequences of functions. Series of functions. Absolute convergence. Uniform convergence of sequences and series. Power series. Some important theorems. General theorems. Theorems on absolute convergence. Special tests for convergence. Theorems on uniform convergence. Theorems on power series. Taylor's theorem. Some serial series. Laurent's theorem. Classification of singularities. Entire functions. Mesomorphic functions. Lagrange's expansion. Analytic continuation.

**Chapter 7 THE RESIDUE THEOREM. EVALUATION OF INTEGRALS AND SERIES**- Residues. Calculation of residues. The residue theorem. Evaluation of definite integrals. Serial theorems used in evaluating integrals. The Cauchy principal value of integrals. Differentiation under the integral sign. Leibnitz's rule. Summation of series. Mittag-Leffler's expansion theorem. Some special expansions.

**Chapter 8 CONFORMAL MAPPING**- Transformations or mappings. Jacobian of a transformation. Complex mapping functions. Conformal mapping. Riemann's mapping theorem. Fixed or invariant points of a transformation. Some general transformations. Translation. Rotation. Stretching. Inversion. Successive transformations. The linear transformation. The bilinear or fractional transformation. Mapping of a half plane on to a circle. The Schwarz-Christoffel transformation. Transformations of boundaries in parametric form. Some special mappings.

**Chapter 9 PHYSICAL APPLICATIONS OF CONFORMAL MAPPING**- Boundary-value problems. Harmonic and conjugate functions. Dirichlet and Neumann problems. The Dirichlet problem for the unit circle. Poisson's formula. The Dirichlet problem for the half plane. Solutions to Dirichlet and Neumann problems by conformal mapping. Applications to fluid flow. Basic assumptions. The complex potential. Equipotential lines and streamlines. Sources and sinks. Some special flows. Flow around obstacles. Bernoulli's theorem. Theorems of Blasius. Applications to electrostatics. Coulomb's law. Electric field intensity. Electrostatic potential. Gauss' theorem. The complex electrostatic potential. Line charges. Conductors. Capacitance. Applications to heat flow. Heat flux. The complex temperature.

**Chapter 10 SPECIAL TOPICS**- Analytic continuation. Schwarz's reflection principle. Infinite products. Absolute, conditional and uniform convergence of infinite products. Some important theorems on infinite products. Weierstrass' theorem for infinite products. Some special infinite products. The gamma function. Properties of the gamma function. The beta function. Differential equations. Solution of differential equations by contour integrals. Bessel functions. Legendre functions. The hypergeometric function. The zeta function. Asymptotic series. The method of steepest descents. Special asymptotic expansions. Elliptic functions.

**INDEX**

## Errata

### Example: $\paren {z_3 - \overline {z_3} }^5$

Chapter $1$: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(c)}$:

*If ... $z_3 = \sqrt 3 - 2 i$, evaluate ...:*

*$\paren {z_3 - \overline {z_3} }^5$*

*Ans. $1024 i$*

### Condition for Points in Complex Plane to form Parallelogram

Chapter $1$: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $65$:

*Let $z_1, z_2, z_3, z_4$ be the position vectors of the vertices for quadrilateral $ABCD$. Prove that $ABCD$ is a parallelogram if and only if $z_1 - z_2 - z_3 + z_4 = 0$.*

### Polar Form of Complex Number: $3 \, \map \cis {\dfrac {-2 \pi} 3}$

Chapter $1$: Supplementary Problems: Polar Form of Complex Numbers: $84 \ \text{(f)}$:

*Graph each of the following and express in rectangular form.*- ... $\text {(f)} \ \ 3 e^{-2 \pi i / 3}$

*Ans. ... $\text {(f)} \ -3 \sqrt 3 / 2 - \paren {3 / 2} i$*

### Complex Addition: Travel $2$

Chapter $1$: Supplementary Problems: Polar Form of Complex Numbers: $85$:

*An airplane travels $150 \, \mathrm {km}$ southeast, $100 \, \mathrm {km}$ due west, $225 \, \mathrm {km}$ $30 \degrees$ north of east, and then $323 \, \mathrm {km}$ northeast. Determine ... how far and in what direction it is from its starting point.*

*Ans. $\ 375 \, \mathrm {km}$, $23 \degrees$ north of east (approx.)*

### Complex Division: $\dfrac {\paren {3 \cis \dfrac \pi 6} \paren {2 \cis \dfrac {-5 \pi} 4} \paren {6 \cis \dfrac {5 \pi} 3} } {\paren {4 \cis \dfrac {2 \pi} 3}^2}$

Chapter $1$: Supplementary Problems: De Moivre's Theorem: $89 \text {(d)}$:

*Evaluate each of the following:*- ... $\dfrac {\paren {3 e^{\pi i / 6} } \paren {2 e^{- 5 \pi i / 4} } \paren {6 e^{5 \pi i / 3} } } {\paren {4 e^{2 \pi i / 3} }^2}$

*Ans. $3 \sqrt 2 / 2 - \paren {3 \sqrt 3 / 2} i$*

### Quadruple Angle Formula for Sine

Chapter $1$: Supplementary Problems: De Moivre's Theorem: $93 \ \text {(a)}$:

*Prove that:*- $\dfrac {\sin 4 \theta} {\sin \theta} = 8 \cos^3 \theta - 4 = 2 \cos 3 \theta + 6 \cos \theta - 4$

### 5th Roots of $-16 + 16 \sqrt 3 i$

Chapter $1$: Supplementary Problems: Roots of Complex Numbers: $96 \ \text {(c)}$:

*Find all the indicated roots and locate them in the complex plane.**... $\text {(c)}$ fifth roots of $-16 + 16 \sqrt 3 i$, ...*

*Ans. ... $\text {(c)}$ $2 \cis 48 \degrees$, $2 \cis 120 \degrees$, $2 \cis 192 \degrees$, $2 \cis 264 \degrees$, $2 \cis 336 \degrees$ ...*

### Roots of $z^6 + 1 = \sqrt 3 i$

Chapter $1$: Supplementary Problems: Roots of Complex Numbers: $97 \ \text {(b)}$:

*Solve the equations ... $\text{(b)}$ $z^6 + 1 = \sqrt 3 i$*

*Ans. ... $\set {\sqrt [6] 2 \cis 40 \degrees, \sqrt [6] 2 \cis 1000 \degrees, \sqrt [6] 2 \cis 160 \degrees, \sqrt [6] 2 \cis 220 \degrees, \sqrt [6] 2 \cis 280 \degrees, \sqrt [6] 2 \cis 340 \degrees}$*

### Examples of Set Intersection and Set Union

Chapter $1$: Supplementary Problems: Miscellaneous Problems: $123$:

*If $A$, $B$ and $C$ are the point sets defined by $\cmod {z + i} < 3$, $\cmod z < 5$, $\cmod {z + 1} < 4$, represent graphically each of the following:*

- $\textit {(a)} \quad A \cap B \cap C$, $\quad \textit {(b)} \quad A \cup B \cup C$, $\quad \textit {(c)} \quad A \cap B \cup C$, $\quad \textit {(d)} \quad C \paren {A + B}$, $\quad \textit {(d)} \quad \paren {A \cup B} \cap \paren {B \cup C}$, $\quad \textit {(e)} \quad AB + BC + CA$, $\quad \textit {(f)} \quad A \tilde B + B \tilde C + C \tilde A$

### Example of Set Intersection with Union

Chapter $1$: Supplementary Problems: Miscellaneous Problems: $123 \ \text{(c)}$:

*If $A$, $B$ and $C$ are the point sets defined by $\cmod {z + i} < 3$, $\cmod z < 5$, $\cmod {z + 1} < 4$, represent graphically ... :*

- $\textit {(c)} \quad A \cap B \cup C$

### Condition for Quartic with Real Coefficients to have Wholly Imaginary Root

Chapter $1$: Supplementary Problems: Miscellaneous Problems: $129$:

- $\text{(a)} \quad$
*Show that the equation $z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4 = 0$ where $a_1, a_2, a_3, a_4$ are real constants different from zero, has a**pure imaginary root if ${a_3}^2 + {a_1}^2 a_4 = a_1 a_2 a_3$.*

### Cosine to Power of Odd Integer

Chapter $1$: Supplementary Problems: Miscellaneous Problems: $130 \ \text{(a)}$:

*Prove that $\cos^n \phi = \dfrac 1 {2^{n - 1} } \set {\cos n \phi + n \cos \paren {n - 2} \phi + \dfrac {n \paren {n - 1} } 2 \cos \paren {n - 4} \phi + \cdots + R_n}$*

*where $R_n = \begin{cases} \cos \phi & \textit {if $n$ is odd} \\ \dfrac {n!} {\sqbrk {\paren {n / 2}!}^2} & \textit {if $n$ is even.}\end{cases}$*