MATH3401

Complex Analysis

Lecture 2

Complex Numbers

  • 1545: Cardano, roots of \(x^3+ax+b\)
    \(5+\sqrt{-15} \ra \) "mental torture"
  • 1575: \(i\), rules for operations in \(\C\)
  • 1629: \(a+\sqrt{-b}\) Girard - "Solutions impossibles"
  • 1637: Descartes - imaginary numbers
  • 1831: Complex numbers

Introduction

  • \(\N\): \(\{1, 2, 3, \ldots \}\) natural numbers
  • \(\mathbb N_0\): \(\{0, 1, 2, 3, \ldots \}\)
  • \(\Z\): \(\{0, \pm 1, \pm 2, \ldots \}\) integers
  • \(\Q\): \(\{p/q \; | \; p, q \in \mathbb Z, q\neq 0 \}\) rational numbers
  • \(\R\): real numbers
  • \(\C\): complex numbers

Equivalent representations of \(\C\)

\(z = (x,y)\) belongs to the complex plane with \(x,y\in \R\).

So \(z=x\left(1,0\right)+y\left(0,1\right)\), i. e. \[z = x+i\, y\]

where \(x\) and \(y\) are the real and imaginary parts, respectively.

Equivalent representations of \(\C\)

\(i\) is the complex number represented by \(\left(0,1\right)\).

We say \(\R \subset \C\) by identifying the complex number \(x+i\cdot 0\) with the real number \(x\).

Addition in \(\mathbb C\)

Set \[\left(x_1,y_1\right)+\left(x_2,y_2\right) = \left(x_1+x_2,y_1+y_2\right)\]

That is

\( \left(x_1 + i\, y_1\right)+\left(x_2 + i \,y_2\right) = \left(x_1+x_2\right)+i\left(y_1+y_2\right) \)

Multiplication in \(\C\)

We can use \(\times\) or \(\cdot\) or juxtaposition.

\[{\small \left(x_1, y_1\right) \cdot \left(x_2, y_2\right) = \left(x_1 x_2 - y_1 y_2, y_1 x_2 + x_1 y_2\right)}\]

That is

\( \left(x_1 + i\, y_1\right) \left(x_2 + i \,y_2\right) = \left(x_1 x_2 - y_1 y_2\right)+i\left(y_1 x_2 + x_1 y_2\right) \)

Note 2.1

The definition of \(\times\) formally applies if we use the usual rules for algebra in \(\R\) and set \(i=\sqrt{-1}\).

With the operations \(+\) and \(\times\), \(\C\) is a field.

Exercise: Check that \(\C\) is closed under \(+\) and \(\times\).

Note 2.1

(F2)(i) Existence of additive identity: \(0 = 0+i\,0\)

(F2)(ii) Existence of additive inverse: \[-z = -(x+i\,y) = (-x)+i\,(-y)\]

Note 2.1

(F5)(i) Existence of multiplicative identity: \(1 = 1+i\,0\)

(F5)(ii) Existence of multiplicative inverse: \(z = x+i\,y \neq 0\), \[ \begin{align*} z^{-1} &= \dfrac{1}{x+i\,y}\cdot \dfrac{x-i\,y}{x-i\,y} \\ &= \dfrac{x}{x^2+y^2} - i \dfrac{y}{x^2 + y^2} \end{align*} \]

Note 2.1

Since \(\C\) is a field, there holds: \[z_1 z_2 = 0 \implies \text{ either } z_1=0 \text{ or } z_2=0\]

(called null-factor or cancelation law).

Note 2.2

\[ \begin{align*} \left(z_1 z_2\right)^{-1} &= z_1^{-1}z_2^{-1} \\ &= \dfrac{1}{z_1}\cdot \dfrac{1}{z_2}=\dfrac{1}{z_1 z_2}, \text{ with } z_1,z_2\neq 0 \end{align*} \]

Note 2.3

\(i^2=-1\), \(\left(-i\right)^2=-1\).

These are the only two solutions of \(z^2=-1\) in \(\C\).

Other functions

  • Modulus: \(|z|=\sqrt{x^2+y^2}\)
    \[\text{e. g. } |7+i\,3| = \sqrt{49+9} = \sqrt{58}\] The modulus is a one-to-one function from \(\C\) to \(\R\). Indeed \(|z|:\mathbb C \ra \left[0,\infty\right) \).

Other functions

  • \(\Re (z) =\) real part of \(z\).
  • \(\Im(z) =\) imaginary part of \(z\).
  • \( \Re : \C \ra \R \) and \( \Im : \C \ra \R \)
  • Examples: \[ \Re(7+3i) = 7, \quad \Im(7+3i)= 3. \]

Credits