Book:Walter Ledermann/Complex Numbers

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Walter Ledermann: Complex Numbers

Published $1960$, Routledge & Kegan Paul.


Subject Matter


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


Errata

Argument of $-1 - i$

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

$\arg \left({-1 - i}\right) = -\dfrac \pi 4$


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 $z$, ...