Book:Walter Ledermann/Complex Numbers

Walter Ledermann: Complex Numbers

Published $\text {1960}$, Routledge & Kegan Paul

Subject Matter

• Complex Analysis

Contents

Preface

1. Algebraic Theory of Complex Numbers

1. Number Systems

2. The Algebraic Theory

2. Geometrical Representations

3. Roots of Unity

4. Elementary Functions of a Complex Variable

1. Introduction

2. Sequences

3. Series

4. Power Series

5. The Functions $e^z, \cos z, \sin z$

6. The Logarithm

Answers to Exercises

Index

Next

Rating

Speed	2 - Trundle
Clarity	3 - Terse
Density	3 - Meaty
Level	2 - Hilly
Scope	2 - Limited
Solutions	3 - Solutions, but no working

Errata

Argument of $-1 - i$

Chapter $2$: Geometrical Representations: Example $\text{(iv)}$

$\map \arg {-1 - i} = -\dfrac \pi 4$

Equation relating Points of Parallelogram in Complex Plane

Chapter $2$: Geometrical Representations: Exercise $7$

The vertices parallelogram $ABVU$ are represented by the complex numbers $a, b, v, u$ respectively. The angle $UAB$ is equal to $\alpha$ and $\cmod {UA} = \lambda \cmod {AB}$. Prove that $u = \paren {1 - q} a + q b$ and $v = -q a + \paren {1 + q} b$, where $q = \lambda e^{i \alpha}$.

Triple Angle Formula for Cosine: $2 \cos 3 \theta + 1$

Chapter $3$: Roots of Unity: Example $6$

Show that

$z^6 + z^3 + 1 = \paren {z^2 - 2 z \cos \dfrac {2 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {4 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {8 \pi} 9 + 1} \qquad (3.14)$

and deduce that

$2 \cos 3 \theta + 1 = \paren {\cos \theta - \cos \dfrac {2 \pi} 9} \paren {\cos \theta - \cos \dfrac {4 \pi} 9} \paren {\cos \theta - \cos \dfrac {8 \pi} 9} \qquad (3.15)$

Use $(3.13)$ when $n = 9$ and observe that the factor that corresponds to $r = 3$, is $z^2 - 2 z \cos \dfrac {2 \pi} 3 + 1 = z^2 + z + 1$, the remaining three quadratic factors being as on the right hand side of $(3.14)$. This expression is therefore equal to $\paren {z^9 - 1} / \paren {z - 1} \paren {z^2 + z + 1} = \paren {z^9 - 1} / \paren {z^3 - 1} = z^6 + z^3 + 1$, which proves $(3.14)$. Next, divide $(3.14)$ throughout by $z^3$ and then put $z = e^i$. With this value of $z$, $z + z^{-1} = 2 \cos \theta$, $z^3 + z^{-3} = 2 \cos 3 \theta$, and $(3.15)$ is an immediate consequence.

Radius of Convergence of Power Series Expansion for Cosine Function

Chapter $4$: Elementary Functions of a Complex Variable: Section $4$: Power Series: Example $\text{(iii)}$

The series $C \paren z = 1 - \dfrac {z^2} {2!} + \dfrac {z^4} {4!} - z \dfrac 6 {6!} + \cdots$ and $S \paren z = z - \dfrac {z^3} {3!} + \dfrac {z^5} {5!} - \dfrac {z^7} {7!} + \cdots$ also converge for all $z$, ...