\chapter{Complex numbers}


\section{Cubic equations}


Complex numbers were first though of in the context of solving cubic equations by Gerolamo Cardano. 

\subsection{Depressed cubic}

\[x^3+px+q=0\]

\[x= \sqrt[3]{\frac{-q}{2} + \sqrt{\frac{q^2}{4}+ \frac{p^3{27}}  }} + \sqrt[3]{\frac{-q}{2} - \sqrt{\frac{q^2}{4}+ \frac{p^3{27}} }} \]

Make the substitution $x=u+v$ and set $3uv+p=0$, then solve the set of simultaneous equations.

\subsection{Cubic formula}

\[f(\]

With the study of the depressed cubic in our repertoire, we can analyze a general cubic with more ease, since any cubic can be transformed into a depressed cubic.

By assuming solution $x+t$ and substituting,one sees that setting $t$ to $\frac{-b}{3a}$ will cancel out the $x^2$ term. This means that the following solution transforms a general cubic equation into a depressed cubic.
\[x=t-\frac{b}{3a}\]


\[at^3+(\frac{-b^2}{3a}+c)x+(\frac{-b^3}{27a^3}+\frac{b^3}{9a^2}+ \frac{-cb}{3a}+d)\]

We can use our theory on depressed cubics to solve this polynomial, noting the following.

\[p = \frac{-b^2}{3a}+c,q=\frac{-b^3}{27a^3}+\frac{b^3}{9a^2}+ \frac{-cb}{3a}+d\]



Although this factor of $i$ was only used as placeholder in calculation, it seemed to produce viable results. In fact, for some polynomials there was no getting around the use of this 'imaginary' factor.






\section{Imaginary unit}

\[i^2=-1\]

It's also worth mentioning that this definition has also implicitly created a second solution to the equation $z^2=-1$, $z=\pmi$.

\[(-i)^2=-1\]

Although we define $i$ to be the imaginary unit, $-i$ is important in the idea of \emph{complex conjugacy}; we'll develop this soon.

\[(x+yi)(x-yi)=x^2 + y^2\]

\section{Complex numbers}


\begin{definition}[Complex numbers]
\emph{complex numbers} are numbers of the form $x+iy$ where $x,y$ are real.  $\mathbb{C}$ is the collection (set) of all complex numbers.
\end{definition}

Complex numbers can therefore be interpreted as two real numbers; a \emph{real part}, and an \emph{imaginary part} (the part multiplied by $i$). This is called the \emph{cartesian form} of a complex number, and each complex number has a unique representation in this form.

\begin{proposition}
All complex numbers are uniquely represented as $x+iy$
%\[ \forall z \in \mathbb{C} [ \exists! x,y \in \mathbb{R} [ z=x+iy ]] \]
\end{proposition}

Since cartesian forms are unique for each complex number, we can formally define unique \emph{real and imaginary} parts of complex number.

\begin{definition}
\[\Re(x+iy)=x\]
\[\Im(x+iy)=y\]
\end{definition}


\section{Modulus}


\[|z|= \sqrt{\Re(z)^2+\Im(z)^2}\]

\begin{proposition}
\[|zw|=|z||w|\]
\end{proposition}




\section{Complex conjugate}


\[z^{*} = \Re(z)-i\Im(z) \]

\begin{proposition}
\[\Re(z) = \frac{z + z^{*}}{2} \]
\[\Im(z) = \frac{z - z^{*}}{2i} \]
\end{proposition}

\[(z+w)^{*} = z^{*} + w^{*}\]
\[(z-w)^{*} = z^{*} - w^{*}\]
\[(zw)^{*} = z^{*} w^{*}\]
\[(\frac{z}{w})^{*} = \frac{z^{*}}{w^{*}}\]
\[(z^{*})^{*}=z\]
\[|z^{*}|=z\]

\[zz^{*}= |z|^2\]

\[  z = \frac{|z|^2}{z^{*}}\]

\begin{corollary}
\[ |z|=1 \implies z = \frac{1}{z^{*}}\]
\end{corollary}

\section{Properties of complex numbers}
The introduction of the imaginary unit has many notable consequences on algebra.


\[z^2 + w^2 = (z+iw)(z-iw)\]

\[k \in \mathbb{Z} \implies i^{4k}=1\]


\section{Polar form}

Polar coordinates
\begin{proposition}
All complex numbers except for $0$ can be uniquely represented in the following form.
	\[ \forall z \in \mathbb{C} \setminus \{ 0 \}\exists! r \in [0,\infty),\theta \in (-\pi,\pi] [z =r\cos(\theta)+ i \sin(\theta)] \]
\end{proposition}
This representation is called the \emph{polar form} of a complex number.

Some authors prove uniqueness with $\theta \in [0,2\pi)$; any half-open interval with length $2\pi$ does the trick for uniqueness.
Since the expression $\cos(\theta)+i\sin(\theta)$ is so frequent, authors often write  $\cos(\theta) + i \sin (\theta)$ as $\cis(\theta)$.

\begin{definition}
$ \cis(\theta) = \cos(\theta) + i\sin(\theta) $
\end{definition}

The polar form is powerful as it connects the theory of trigonometry with that of complex numbers, giving futher insight as to the behaviour of complex numbers.


\begin{proposition}
\[ \forall \theta \in |\cis(\theta)|=1 \]
\end{proposition}


\begin{proposition}
\[ \arg (z) = \{ \theta \in \mathbb{R} : \frac{z}{|z|}=\cis(\theta) \]
\end{proposition}

The consequence of the uniqueness of the polar form implies that any two identical arguments are just translations of eachother by multiples of $2\pi$.
\begin{proposition}
\[ z \neq 0 \implies [  \arg (z) \cap \arg (w) \neq \emptyset \implies \arg (z)=\arg (w) ] \]
\[ z \neq 0 \implies [  \theta ,\varphi \in \arg (z)  \exists k \in \mathbb{Z} [ \theta = \varphi +2\pi k ] ] \]
\[ r \in (0,\infty) \land \theta \in \mathbb{R} \implies [  \arg(r\cis (\theta)) \{ \theta + 2\pik : k \in \mathbb{Z}\} ] \]
\end{proposition}

\begin{definition}
\[ \Arg (z) =  \argmin_{\theta \in \arg (z) }  |\theta| \]
\end{definition}

The primary $2$ reason why I chose to define the polar form with the argument on $(-\pi,\pi]$ is because it makes $\Arg$ slightly easier to formally define using conventional notation, and so the proposition $\Arg (z^{*}) = -\Arg (z)$ can be formally correct.

\begin{proposition}
\[ \Arg (z) + \Arg (w)  \in \arg (zw)\]
\end{proposition}

This proposition is rather striking; it connects the addition of arguments to multiplication of complex numbers. Keep this idea in mind for the future, but for now let's derive a few corollaries from this proposition.

\begin{corollary}
\[ \Arg (z)+ \pi \in \arg (-z) \\]
\[ \Arg (z) +\frac{\pi}{2} \in \arg (iz) \\]
\end{corollary}

These corollaries essentially means that a 180 degree turn is a change in sign and a 90 degree turn is a multiplication by $i$; giving us some geometric intuition as to how multiplication by the imaginary unit affects complex numbers.


\begin{proposition}
\[ \Arg (z^{*}) = -\Arg (z) \]
\end{proposition}


\section{Euler's formula}

\begin{theorem}[Euler's formula]
\[ e^{i x} = \cis(x)\]
\[ \forall x \in \mathbb{R} [ e^{i x} = \cis(x) ] \]
\end{theorem}

Euler's formula is the cornerstone for much of the reasoning in complex analysis. Though the fact that our complex numbers had 2 arguments hinted us towards the polar form, this maps the polar form to an even more versatile \emph{exponential form}.


\begin{theorem}[Euler's formula]
\[ e^{i x} = \cis(x)\]
\[ \forall x \in \mathbb{R} [ e^{i x} = \cis(x) ] \]
\end{theorem}

In addition to being incredibly useful in complex analysis, many also consider it a prime example of the beauty of mathematics; particularly the following identity.

\begin{corollary}[Euler's identity]
\[ e^{i\pi} + 1 =0 \]
\end{corollary}

Perhaps its beauty stems from the fact that it champions how mathematics can be connected in the most mysterious ways; note that this formula was the first time that mathematicians saw the fundamental but conceptually distinct constants $\pi$ and $e$ related together (well, at least conceptually distinct to those unaware of deeper mathematics).


\begin{corollary}
\[e^{i z} = \cis(z) \]
\[ \forall z \in \mathbb{C} [ e^{i z} = \cis(z) ] \]
\end{corollary}

Recall the process of square rooting a complex number; the following formula facilitates this greatly and extends to finding the $n$th root of a complex number!
\begin{theorem}[De Moivre's formula]
\[ \forall x \in \mathbb{R} [ \forall n \in \mathbb{Z} [ \cis(x)^n =\cis (nx) ] ] \]
\end{theorem}

Though this can be derived by various methods, it is a particularly elegant consequence of Euler's formula.

Moreover, the following proposition allows us to extract all the $q$-roots of a complex number $z$.

\begin{proposition}
	$r^{q}e^{i(\theta)}$
\end{proposition}