1 Complex Numbers

1.1 Definitions

Construct \(\mathbb{C}\) by adding an element \(i\) to real numbers \(\mathbb{R}\), with \[\begin{align*} i^2 = -1. \end{align*}\] Any complex numbers \(z \in \mathbb{C}\) has the form \(z = x + iy\) with \(x, y \in \mathbb{R}\); \(x = \operatorname{Re}(z)\), real part; \(y = \operatorname{Im}(z)\), imaginary part.

\(\mathbb{R} \subset \mathbb{C}\) consisting of elements \(x + i 0 = x\)

In the following, use the notation above and \(z_1 = x_1 + i y_1\), \(z_2 = x_2 + i y_2\) etc.

  1. Addition ( and subtraction) \[\begin{align*} z_1 \pm z_2 = (x_1 \pm x_2) + i (y_1 \pm y_2) \end{align*}\]

  2. Multiplication \[\begin{align*} z_1 z_2 = (x_1 x_2 - y_1 y_2) + i(x_1 y_2 + x_2 y_1) \end{align*}\]

If \(z \neq 0\), observe from the definition \[\begin{align*} z^{\text{-}1} = \frac{x}{x^2 + y^2} - i \frac{y}{x^2 + y^2} \end{align*}\] satisfies \(z z^{\text{-}1} = 1\)

  1. Complex conjugate \[\begin{align*} \overline{z} &= z^{*} = x - iy \\ \operatorname{Re}(z) &= \frac{1}{2} (z + \overline{z}) \\ \operatorname{Im}(z) &= \frac{1}{2i} (z - \overline{z}) \\ \overline{(\overline{z})} &= z \\ \overline{z_1 + z_2} &= \overline{z_1} + \overline{z_2} \\ \overline{z_1 z_2} &= (\overline{z_1}) (\overline{z_2}) \end{align*}\]

  2. Modulus is defined by \(r = |z|\), real and \(\geq 0\), with \[\begin{align*} r^2 = |z|^2 = z \overline{z} = x^2 + y^2 \end{align*}\]

  3. Argument \(\theta = \arg{z}\), real, defined for \(z \neq 0\) by \[\begin{align*} z = r(\cos \theta + i \sin \theta) \hspace{2cm} \textbf{polar form} \end{align*}\] for some real \(\theta\)

\[\begin{align*} \cos \theta &= \frac{x}{(x^2 + y^2)^{1/2}},\ \sin \theta = \frac{y}{(x^2 + y^2)^{1/2}}, \\ & \implies \tan \theta = \frac{y}{x} \end{align*}\]

\(\arg(z)\) is determined only \(\operatorname{mod} 2 \pi\), i.e. can change \(\theta \to \theta + 2n \pi\) where \(n \in \mathbb{Z}\). To make it unique we can restrict the range, e.g. principal value defined by \(- \pi < \theta \leq \pi\)

  1. Argand diagram and Complex Plane Plot \(\operatorname{Re}(z)\) and \(\operatorname{Im}(z)\) on orthogonal axes, then \(r = |z|\) and \(\theta = \arg(z)\) are the length and angle shown.

Example 1.1 For \(z = -1 + i \sqrt{3} = 2 (- \frac{1}{2} + i \frac{\sqrt{3}}{2})\), we have \(|z| = 2\) and \(\arg(z) = \frac{2 \pi}{3} + 2 n \pi\) where \(n \in \mathbb{Z}\).

Note \[\begin{align*} \tan \theta = - \sqrt{3} \\ \implies \theta = \frac{2 \pi}{3} + 2 n \pi \\ = \arg(z) \\ \textbf{or } \theta = - \frac{\pi}{3} + 2 n \pi \\ = \arg(-z) \end{align*}\]

1.2 Basic properties and Consequences

  1. \(\mathbb{C}\) with operations \(+, \times\) is a field, i.e. 
    1. \(\mathbb{C}\) with \(+\) is an abelian group.
    2. \(\mathbb{C} \setminus \{0\}\) with \(\times\) is an abelian group.
    3. distributive laws hold, e.g. \(z_1(z_2 + z_3) = z_1 z_2 + z_1 z_3\).
  2. Fundamental Theorem of Algebra
    A polynomial of degree \(n\) with coefficients in \(\mathbb{C}\) can be written as a product of \(n\) linear factors. \[\begin{align*} p(z) &= c_n z^n + \ldots + c_1 z + c_0 \ \ \text{ where } c_i \in \mathbb{C}, c_n \neq 0 \\ &= c_n ( z - \alpha_1) \ldots (z - \alpha_n) \text{ where } \alpha_i \in \mathbb{C}. \end{align*}\]

Hence \(p(z) = 0\) has at least one root and \(n\) roots connected with multiplicity (if we count duplicates as separate).

  1. Addition and subtraction can be viewed as parallelogram constructions

Complex conjugation is reflection in real axis

  1. Proposition 1.1 Modulus/ length obeys composition property \(|z_1 z_2| = |z_1| |z_2|\). Triangle inequality \(|z_1 + z_2| \leq |z_1| + |z_2|\)

Proof (Triangle inequality). Compare \[\begin{align*} \text{LHS}^2 &= (z_1 + z_2) \overline{z_1 + z_2} \\ \text{RHS}^2 &= |z_1|^2 + 2|z_1||z_2| + |z_2|^2 \end{align*}\]

Compare “cross terms”: \[\begin{align*} & z_1\overline{z_2} + z_2 \overline{z_1} \leq 2 |z_1| |z_2| \\ & \iff \frac{1}{2} (z_1 \overline{z_2} + \overline{(z_1 \overline{z_2})}) \leq |z_1| |\overline{z_2}| \\ & \iff \operatorname{Re}(z_1 \overline{z_2}) \leq |z_1 \overline{z_2}| \end{align*}\]

An alternative form of the triangle inequality: replace \(z_2\) by \(z_2 - z_1\) and rearrange to get \[\begin{align*} | z_2 - z_1 | &\geq |z_2| - |z_1| \\ \text{or} &\geq |z_1| - |z_2| \\ \text{So } |z_2 - z_1| &\geq \left| |z_2| - |z_1| \right| \end{align*}\]

  1. Proposition 1.2 \[\begin{align*} &z_1 = r_1 (\cos \theta_1 + i \sin \theta_1),\ z_2 = r_2 (\cos \theta_2 + i \sin \theta_2) \\ &\implies z_1 z_2 = r_1 r_2 (\cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2)) \end{align*}\]

    i.e. moduli multiply and args add

Proof. Direct check: calculate \(z_1 z_2\) and use standard trig formulas.

Theorem 1.1 (De Moivre's Theorem) \[\begin{align*} (\cos \theta + i \sin \theta)^n = \cos n \theta + i \sin n \theta,\ \text{ for } n \in \mathbb{Z} \end{align*}\] Note for \(z \neq 0\), \(z^0 = 1\) and \(z^{-n} = (z^{-1})^n\) for \(n > 0\).

Proof. Use 1.2 and induction

1.3 Exponential and Trigonometric Functions

Define \(\exp, \cos, \sin\) as functions on \(\mathbb{C}\) by \[\begin{align*} \exp(z) &= e^z = \sum_{n=0}^{\infty} \frac{1}{n!} z^n \\ \cos(z) &= \frac{1}{2} (e^{iz} + e^{-iz}) \\ &= 1 - \frac{1}{2!}z^2 + \frac{1}{4!}z^4 \ldots \\ \sin(z) &= \frac{1}{2i} (e^{iz} - e^{-iz}) \\ &= z - \frac{1}{3!}z^{3} + \frac{1}{5!}z^5 + \ldots \end{align*}\]

These series converge \(\forall \; z \in \mathbb{C}\) and such series can be multiplied, rearranged and differentiated. Furthermore \(e^{z}e^{w} = e^{z + w}\) and from this we can see \(e^{0} = 1\) and \((e^{z})^{n} = e^{nz}\) for \(n \in \mathbb{Z}\). For the positive integers this is trivial and for the negative integers we can know that \(e^z e^{-z} = 1\) so \(e^{-z} = (e^{z})^{-1}\).

Lemma 1.1 For \(z = x + iy\):

  1. \(e^z = e^x ( \cos y + i \sin y)\)
  2. \(\exp\) on \(\mathbb{C}\) takes all complex values except \(0\).
  3. \(e^z = 1 \iff z = 2n \pi i,\ n \in \mathbb{Z}\)

Proof.  i. \(e^{x + iy} = e^x e^{iy}\) but \(e^{iy} = \cos y + i \sin y\)

  1. \(|e^z| = e^x\), so \(|e^z|\) take all real values \(> 0\). \(\arg{e^z} = y\) taking all possible values.

  2. \[\begin{align*} e^z = 1 &\iff e^x = 1, \cos y = 1, \sin y = 0 \\ &\iff x = 0, y = 2 \pi n \end{align*}\]

Returning to polar form 1.2, this can be written \(z = r ( \cos \theta + i \sin \theta) = re^{i \theta}\) for \(r = |z|\) and \(\theta = \arg z\). De Moivre’s Theorem 1.1 now follows from \((e^{i \theta})^n = e^{i n \theta}\).

1.3.1 Roots of units

\(z\) is an Nth root of unity if \(z^N = 1\). To find all solutions: \[\begin{align*} && z &= r e^{i \theta} \text{ satisfying } z^N = 1 \\ &\iff & r^N e^{i N \theta} &= 1 \\ &\iff & r^N &= 1 \text{ and } N \theta = 2n \pi,\ n \in \mathbb{Z} \end{align*}\]

This gives N distinct solutions. \[\begin{align*} z &= e^{2 \pi i n/N} \\ &= \cos \frac{2\pi n}{N} + i \sin \frac{2 \pi n}{N} \\ \text{The only distinct cases are when } n &= 0, 1, 2 \ldots N - 1 \text{ due to periodicity} \\ &= \omega^n, \text{ where } \omega = e^{2 \pi i / N} \end{align*}\] These solutions lie at vertices of a regular N-gon.

N = 6

Figure 1.1: N = 6

1.4 Transformations; lines and circles

Consider the following transformations on \(\mathbb{C}\) (maps \(\mathbb{C} \to \mathbb{C}\)) \[\begin{align*} z &\mapsto z + a & &(\text{translation}) \\ z &\mapsto \lambda z & &(\text{scaling by } \lambda \in \mathbb{R}) \\ z &\mapsto e^{i \alpha} z & &(\text{rotation by } \alpha \in \mathbb{R}) \\ z &\mapsto \overline{z} & &(\text{reflection in the real axis}) \\ z &\mapsto \frac{1}{z} & &(\text{inversion}) \end{align*}\]

Consider a general point on a line in \(\mathbb{C}\) through \(z_0\) and parallel to \(w \neq 0\) (fixed \(z_0, w \in \mathbb{C}\)):

\[\begin{align*} z = z_0 + \lambda w,\ \lambda \in \mathbb{R} \end{align*}\]

To eliminate \(\lambda\), take the conjugate to get \(\overline{z} = \overline{z_0} + \lambda \overline{w}\) and equate \(\lambda\) to get \(\overline{w}z - w\overline{z} = \overline{w} z_0 - w \overline{z_0}\).

Consider a general point on a circle in \(\mathbb{C}\) with centre \(c \in \mathbb{C}\) and radius \(\rho \in \mathbb{R}^{++}\). \[\begin{align*} z = c + \rho &e^{i \alpha}, \text{ for any } \alpha \in \mathbb{R} \\ \text{Equivalently} \\ |z - c| &= \rho \\ \text{or } \\ |z - c|^2 &= \rho^2 \\ |z|^2 - \overline{c}z - c \overline{z} &= \rho^2 - |c|^2 \end{align*}\]

Möbius transformations are generated by translations, scalings, rotations and inversion. They can be viewed as acting on \(\mathbb{C}_\infty = \mathbb{C} \cup \{ \infty \}\) - geometrically a sphere (see IA Groups). We add \(\infty\) to deal with \(0^{-1}\)?

1.5 Logarithms and Complex Powers

Define \(w = \log z,\ z \in \mathbb{C},\ z \neq 0\) by \(e^w = \exp w = z\) i.e. \(\log\) is the inverse of \(\exp\), but \(\exp\) is many-to-one (\(e^z = e^{z + 2n \pi i}\)) and so \(\log\) is multi-valued. \[\begin{align*} z &= r e^{i \theta} \\ &= e^{\log r} e^{i \theta} \\ &= e^{\log r + i \theta} \\ \implies \log z &= \log r + i \theta \\ &= \log |z| + i \arg z \end{align*}\] Multiple values of \(\arg\) and \(\log\) are related: \[\begin{align*} \theta &\to \theta + 2n \pi,\ n \in \mathbb{Z} \\ \log z &\to \log z + 2n \pi i,\ n \in \mathbb{Z} \end{align*}\] To make them single valued we can restrict e.g. \(0 \leq \theta < 2 \pi\) or \(-\pi < \theta \leq \pi\) (principal value).

Example 1.2 \[\begin{align*} z &= -3i = 3 (-i) \\ &= e^{\log 3}e^{-i \pi / 2 + 2n \pi i} \\ &= e^{\log 3 -i \pi / 2 + 2n \pi i} \\ \log z &= \log 3 -i \pi / 2 + 2n \pi i \\ \arg z &= 3 \pi / 2 \\ &\text{or } - \pi / 2 \text{ with the restrictions above} \end{align*}\]

Define complex powers by \[\begin{align*} z^\alpha = e^{\alpha \log z},\ z \in \mathbb{C},\ z \neq 0,\ \alpha \in \mathbb{C} \end{align*}\] This is multi-valued in general under the change \(\arg z \to \arg z + 2n\pi\) \[\begin{align*} z^\alpha \to z^\alpha e^{2 \pi i n \alpha} \end{align*}\]

  1. If \(\alpha = p \in \mathbb{Z}\) then \(z^\alpha = z^p\) is unique
  2. If \(\alpha = \frac{p}{q} \in \mathbb{Q}\) then \(z^\alpha = z^{\frac{p}{q}}\) takes finitely many values.

But in general we have infinitely many values

Example 1.3 \[\begin{align*} (1 + i)^{\frac{1}{2}}: 1 + i &= \sqrt{2} e^{i \pi /4} \\ &= e^{\frac{1}{2} \log 2 + i \pi /4} \\ \log(1 + i) &= \frac{1}{2} \log 2 + i \pi / 4 + 2 n \pi i \\ \implies (1 + i)^{\frac{1}{2}} &= e^{\frac{1}{2} \log (1 + i)} \\ &= e^{\frac{1}{4} \log 2 + i \pi / 8 + n \pi i} \\ &= 2^{\frac{1}{4}} e^{i \pi / 8} (-1)^n \end{align*}\]

Example 1.4 \[\begin{align*} (-3i)^i &= e^{i \log (-3i)} \\ &= e^{i (\log 3 - i \pi /2 + 2 n \pi i)} \\ &= e^{i \log 3} e^{\pi /2 - 2 n \pi},\ n \in \mathbb{Z} \end{align*}\]