1 The complex number system

1.1 The field of complex numbers

Definition 1.1 Complex numbers and their arithmetic operations [Palka1991, Sec. I.1.1]

The set of complex numbers is $\mathbb {C}= \mathbb {R}\times \mathbb {R}$, i.e., the set of pairs $(x,y)$ of real numbers $x,y \in \mathbb {R}$.

The operations of addition and multiplication on $\mathbb {C}$ are defined by the formulas

\begin{align*} (x_1,y_2) + (x_2,y_2) = \; & (x_1+x_2, y_1+y_2) \\ (x_1,y_1) \cdot (x_2,y_2) = \; & (x_1 x_2 - y_1 y_2 , \, x_1 y_2 + y_1 x_2) . \end{align*}

Denote $0 = (0,0) \in \mathbb {C}$ and $1 = (1,0) \in \mathbb {C}$.

For $z = (x,y) \in \mathbb {C}$, denote $-z = (-x,-y) \in \mathbb {C}$ and if $z \ne 0$ then denote $z^{-1} = \Big(\frac{x}{x^2 + y^2} , \frac{-y}{x^2 + y^2} \Big) \in \mathbb {C}$.

We write a complex number $(x,y)$ as $x + \mathfrak {i}\, y$. The compex number $\mathfrak {i}= (0,1) \in \mathbb {C}$ is called the imaginary unit.

Typically used variable names for complex number are $z,w,\zeta \in \mathbb {C}$ etc.

Theorem 1.2 The field of complex numbers [Palka1991, Sec. I.1.1]

The set $\mathbb {C}$ of compex numbers with its operations of addition and multiplication, is a field, i.e., the following properties hold for all $z, w, z_1, z_2, z_3 \in \mathbb {C}$:

$z + w = w + z$ (commutativity of addition)
$z w = w z$ (commutativity of multiplication)
$z_1 + (z_2 + z_3) = (z_1 + z_2) + z_3$ (associativity of addition)
$z_1 (z_2 z_3) = (z_1 z_2) z_3$ (associativity of multiplication)
$0 = 0 + 0 \, \mathfrak {i}= (0,0) \in \mathbb {C}$ satisfies $z + 0 = z$ (neutral element for addition)
$1 = 1 + 0 \, \mathfrak {i}= (1,0) \in \mathbb {C}$ satisfies $z \cdot 1= z$ (neutral element for multiplication)
$z + (-z) = 0$ for any $z \in \mathbb {C}$ (opposite element / additive inverse)
$z \, z^{-1} = 1$ for any $z \in \mathbb {C}\setminus \left\{ 0 \right\} $ (multiplicative inverse)
$(z_1 + z_2) w = z_1 w + z_2 w$ (distributivity).

Proof ▶

Straightforward calculations using the definitions of the operations (Definition 1.1).

1.2 Conjugate, modulus, and argument

Definition 1.3 Complex conjugate [Palka1991, Sec. I.1.2]

The complex conjugate of a complex number $z = x + \mathfrak {i}y$ (where $x,y \in \mathbb {R}$) is the complex number $\overline{z} = x - \mathfrak {i}y$.

Lemma 1.4 Properties of complex conjugate [Palka1991, Sec. I.1.2 (1.1)]

For any $z , w \in \mathbb {C}$, we have

\begin{align*} \overline{\overline{z}} = z , \qquad \overline{z + w} = \overline{z} + \overline{w} , \qquad \overline{z w} = \overline{z} \, \overline{w} , \end{align*}

\begin{align*} \Re \mathfrak {e}(z) = \frac{z + \overline{z}}{2} , \qquad \Im \mathfrak {m}(z) = \frac{z - \overline{z}}{2 \mathfrak {i}} . \end{align*}

Proof ▶

Direct calculations.

Definition 1.5 Absolute value (modulus) [Palka1991, Sec. I.1.2]

The absolute value (or modulus) of a complex number $z = x + \mathfrak {i}y$ (where $x,y \in \mathbb {R}$) is the nonnegative real number $|z| = \sqrt{x^2 + y^2} \ge 0$.

Lemma 1.6 Properties of absolute value [Palka1991, Sec. I.1.2 (1.2)]

For any $z, w \in \mathbb {C}$, we have

\begin{align*} |z|^2 = z \, \overline{z} , \qquad |z w| = |z| \, |w| , \end{align*}

\begin{align*} \Re \mathfrak {e}(z) \le |z| , \qquad \Im \mathfrak {m}(z) \le |z| , \end{align*}

\begin{align*} |z + w| \le |z| + |w| , \qquad |z + w| \ge \big| |z| - |w| \big| . \end{align*}

Also, if $z \ne 0$, then

\begin{align*} z^{-1} = \frac{\overline{z}}{|z|^2} , \qquad \qquad \Big| \frac{w}{z} \Big| = \frac{|w|}{|z|} . \end{align*}

Proof ▶

Straightforward.

Definition 1.7 Argument [Palka1991, Sec. I.1.2]

A real number $\theta \in \mathbb {R}$ is an argument of a complex number $z \in \mathbb {C}$ if

\begin{align*} z = |z| \, \Big( \cos (\theta ) + \mathfrak {i}\, \sin (\theta ) \Big) . \end{align*}

(Note/warning: For a nonzero complex number $z$, it is convenient to denote $\theta = \arg (z)$, but this is an abuse of notation, the argumentis defined only modulo addition of integer multiples of $2 \pi $.)

The principal argument of a nonzero complex number $z \in \mathbb {C}$ is its unique argument on the interval $(-\pi , \pi ]$, and it is denoted by $\mathrm{Arg}(z)$.

Lemma 1.8 Discontinuity of the principal argument

The principal argument $\mathrm{Arg}\colon \mathbb {C}\setminus \left\{ 0 \right\} \to (-\pi ,\pi ]$ is continuous on the subset $\mathbb {C}\setminus (-\infty ,0]$, but it is discontinuous on the negative real axis $(-\infty ,0]$.

1.3 The polar form

Definition 1.9 Complex exponential function

We define the complex exponential function $\exp : \mathbb {C}\to \mathbb {C}$ by

\begin{align*} \exp (x + \mathfrak {i}y) = e^x \, \big( \cos (y) + \mathfrak {i}\, \sin (y) \big) \qquad \text{ for } x,y \in \mathbb {R}, \end{align*}

where $e^x$ is the usual real exponential. We also use the notation $e^z = \exp (z)$ for complex exponentials.

The exponential with purely imaginary argument takes the form of Euler’s formula

\begin{align*} e^{\mathfrak {i}\theta } = \cos (\theta ) + \mathfrak {i}\, \sin (\theta ) \qquad \text{ for } \theta \in \mathbb {R}. \end{align*}

Lemma 1.10 Properties of the complex exponential

For any $z,w \in \mathbb {C}$ we have

\begin{align*} e^{z + w} = e^z \, e^w. \end{align*}

For any $z \in \mathbb {C}$ we have

\begin{align*} e^{\overline{z}} = \overline{e^z}, \quad |e^z| = e^{\Re \mathfrak {e}(z)}, \quad \arg (e^z) = \Im \mathfrak {m}(z) \; \; (\mathrm{mod}~ 2\pi ). \end{align*}

For $z,w \in \mathbb {C}$ we have $e^{z} = e^{w}$ if and only if $z = w + 2 \pi \mathfrak {i}n$ for some $n \in \mathbb {Z}$.

Proof ▶

…

Lemma 1.11 Polar form

Every complex number $z \in \mathbb {C}$ can be written in the polar form

\begin{align*} z = r \, e^{\mathfrak {i}\theta } \qquad \text{ where $r \ge 0$ and $\theta \in \mathbb {R}$.} \end{align*}

The modulus of $z$ is the number $r = |z|$ above. If $z \ne 0$, then $\theta $ above is a choice of the argument of $z$, i.e., $\theta = \mathrm{Arg}(z) + 2 \pi m$ for some $m \in \mathbb {Z}$.

Proof ▶

…

Lemma 1.12 Multiplication in polar form [Palka1991, Sec. I.1.2 (1.6)]

For any $z,w \in \mathbb {C}$, written in polar form as $z = r e^{\mathfrak {i}\theta }$ and $w = r' e^{\mathfrak {i}\theta '}$, the product can be written in polar form as

\begin{align*} z w = r r’ \, e^{\mathfrak {i}(\theta + \theta ')} . \end{align*}

In other words,

\begin{align*} |z w| = |z| \, |w| \qquad \text{ and } \qquad \arg (z w) = \arg (z) + \arg (w) \; \; (\mathrm{mod}~ 2\pi ). \end{align*}

Proof ▶

…

Theorem 1.13 De Moivre’s formula [Palka1991, Sec. I.1.2 (1.7)]

For any $\theta \in \mathbb {R}$ and $n \in \mathbb {Z}$, we have

\begin{align*} \big( \cos (\theta ) + \mathfrak {i}\, \sin (\theta ) \big)^n = \cos (n \theta ) + \mathfrak {i}\, \sin (n \theta ) . \end{align*}

Proof ▶

Induction using Lemma 1.12.

Lemma 1.14 Roots of unity

For any $n \in \mathbb {N}$, the solutions $z \in \mathbb {C}$ to the equation

\begin{align*} z^n = 1 \end{align*}

are the $n$ distinct complex numbers

\begin{align*} z_j \, = \, e^{\mathfrak {i}2 \pi j / n} \, = \, \cos \Big( \frac{2 \pi j}{n} \Big) + \mathfrak {i}\, \sin \Big( \frac{2 \pi j}{n} \Big) \qquad \text{ where } j = 0, 1, \ldots , n-1 . \end{align*}

These solutions are called the (complex) $n$th roots of unity.

In particular, we have the polynomial factorization

\begin{align*} z^n - 1 \, = \, \prod _{j=0}^{n-1} \Big( z - e^{\mathfrak {i}2 \pi j / n} \Big) . \end{align*}

Proof ▶

…

1.4 Functions of a complex variable

1.4.1 Polynomials and rational functions

Definition 1.15 Polynomial

Polynomial functions are functions $p : \mathbb {C}\to \mathbb {C}$ of the form

\begin{align*} p(z) = a_n z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0 \end{align*}

where $a_0,a_1,\ldots ,a_{n-1},a_n \in \mathbb {C}$ are coefficients.

Definition 1.16 Rational function

Rational functions are functions $f : D \to \mathbb {C}$ which can be written as ratios $f(z) = \frac{p(z)}{q(z)}$ of two polynomials $p, q \colon \mathbb {C}\to \mathbb {C}$ on a domain $D \subset \mathbb {C}$ where the denominator polynomial $q$ has no zeroes.

1.4.2 Exponentials and branches of logarithms

Definition 1.17 Principal complex logarithm

The principal logarithm is the function

\begin{align*} \mathrm{Log}\colon \; & \mathbb {C}\setminus \left\{ 0 \right\} \to \mathbb {C}\\ \mathrm{Log}(z) \, = \; & \log |z| + \mathfrak {i}\, \mathrm{Arg}(z) , \end{align*}

where $\log |z|$ is the usual natural logarithm of the positive real number $|z|{\gt}0$ and $\mathrm{Arg}(z) \in (-\pi ,\pi ]$ is the principal argument of the nonzero complex number $z \ne 0$.

(Directly from this definition one sees that for $z \in \mathbb {C}\setminus \left\{ 0 \right\} $ we have $e^{\mathrm{Log}(z)} = z$. All complex solutions $w$ to $e^w = z$ are of the form $w = \mathrm{Log}(z) + 2 \pi \mathfrak {i}n$ where $n \in \mathbb {Z}$.)

Definition 1.18 Branches of complex logarithm

A branch of the logarithm is a continuous function $\ell \colon U \to \mathbb {C}$ on an open set $U \subset \mathbb {C}$ such that

\begin{align*} e^{\ell (z)} = z \qquad \text{ for all } z \in U . \end{align*}

For example, the principal logarithm $\mathrm{Log}$ restricted to the open set $\mathbb {C}\setminus (-\infty ,0]$ is called the principal branch of the logarithm. Note that this principal branch cannot be extended continuously to the negative real axis.

(Note that since $e^w \ne 0$ for all $w \in \mathbb {C}$, any branch of the logarithm must exclude the origin from its domain of definition, $0 \notin U$.)

1.4.3 Complex power functions

Definition 1.19 Principal complex power function

Let $\alpha \in \mathbb {C}$. The principal (complex) $\alpha $th power function is the function $\mathbb {C}\setminus \left\{ 0 \right\} \to \mathbb {C}$ given by

\begin{align*} z \, \mapsto \, z^\alpha \, := \, e^{\alpha \, \mathrm{Log}(z)} . \end{align*}

(Note: Integer powers have more direct natural definitions. For $n \in \mathbb {N}$ we simply define $z^n$ by recursive multiplication and the function $z \mapsto z^n$ is continuous and defined in all of $\mathbb {C}$ and coincides with the principal power function with $\alpha = n$ on $\mathbb {C}\setminus \left\{ 0 \right\} $. We also define $z^{-n}$ by recursive multiplication of the inverse $z^{-1}$ of $z$, and the function $z \mapsto z^{-n}$ is continuous on $\mathbb {C}\setminus \left\{ 0 \right\} $ and coincides with the principal power function with $\alpha = -n$. For $n = 0$ we define $z^0 = 1$ for any $z \in \mathbb {C}$, and this coincides with the principal power function with $\alpha = 0$ on $\mathbb {C}\setminus \left\{ 0 \right\} $.)

Given a branch $\ell \colon U \to \mathbb {C}$ of logarithm on an open set $U \subset \mathbb {C}$, we obtain a branch of the $\alpha $th power function on $U$ by the formula $z \mapsto e^{\alpha \ell (z)}$. Using the same branch of the logarithm for the power functions, we have $z^\alpha z^\beta = z^{\alpha + \beta }$.

1.4.4 Branches of $n$th roots

Definition 1.20 Principal $n$th root function

Let $n \in \mathbb {N}$. The principal (complex) $n$th root of $z \in \mathbb {C}\setminus \left\{ 0 \right\} $ is

\begin{align*} \sqrt[n]{z} \; := \, z^{1/n} \, = \, e^{\frac{1}{n} \mathrm{Log}(z)} . \end{align*}

(It follows directly from the definition and the properties of complex exponential that $(\sqrt[n]{z})^n = z$. All complex solutions $w$ to $w^n = z$ are of the form $w = \zeta \sqrt[n]{z}$ where $\zeta = e^{2 \pi \mathfrak {i}j / n}$ with $j = 0, 1, \ldots , n-1$, i.e., $\zeta $ is one of the $n$ complex $n$th roots of unity.)

MS-C1300 Complex Analysis

1 The complex number system

1.1 The field of complex numbers

1.2 Conjugate, modulus, and argument

1.3 The polar form

1.4 Functions of a complex variable

1.4.1 Polynomials and rational functions

1.4.2 Exponentials and branches of logarithms

1.4.3 Complex power functions

1.4.4 Branches of \(n\)th roots