\part{Fundamentals}

\chapter{Complex numbers}

These new numbers 'complete' our understanding of algebra; they are the appropriate extension of the real numbers such that every degree $n$ polynomial has $n$ roots (up to multiplicity) as stated by the fundamental theorem of algebra.


$(\mathbb{C},+,\cdot)$ is an unorderable, algebraically closed field extension $(\mathbb{R}(i),+,\cdot)$.
$(\mathbb{C},| \cdot |)$ is a complete metric space isometric and bijective to $( \mathbb{R}^2,\|\cdot\| )$.



One interesting property we loose when considering complex numbers is the inability to form an order consistent with its algebraic rules. To see this, we have $0\leq i\implies 0 \leq 1 \implies \bot$. In the other case, $i \leq 0 \implies 1 \leq 0 \implies \bot$. This means that no partial order is compatible with the algebraic properties of the complex numbers.

\section{Origins}


Complex numbers were first though of in the context of solving cubic equations by Gerolamo Cardano. 
- cubic equation
- discriminant of degree 3

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.


\section{Complex numbers}

\[ \mathbb{C} = \{ x+iy : x,y \in \mathbb{R} \}\]
\[ \mathbb{C} = \mathbb{R}(i) \]

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}
\[ \forall z \in \mathbb{C} [ \exists! x,y \in \mathbb{R} [ z=x+iy ]] \]
\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}



\section{Field of complex numbers}

Recalling how we called algebra on real numbers the 'field of real numbers', we similarly have the 'field of complex numbers'. If you're familiar with ring theory, the field of complex numbers is the 'field extension' of the field of real numbers by the imaginary unit, symbolically represented by $\mathbb{C}=\mathbb{R}(i)$. This just means this field is the result of appending $i$ to our real number field.

For the most part, the same properties of the field of real numbers apply, along with the previously discussed consequences. That said, the algebraic properties of $i$ takes away the ability to order numbers.

\begin{proposition}
\[ \nexists \leq [ \leq \text{ is a partial order on } ( \mathbb{C},+,\times ) ]\]
\end{proposition}


This means that for sets of complex numbers, many notions from real analysis are not directly applicable, notably, we can't have supremums and infimums of sets or sequences of complex numbers and there is no least upper bound property for the complex numbers.

This is another reason why the modulus of a complex number is so important; though these ideas cannot translate directly from real analysis, the fact that the modulus returns a real number means that many analogues of these ideas survive in some sense.

For example, though a convergent sequence $(z_n)_{n\in \mathbb{N}}$ of complex numbers can't have a supremum, the sequence $(|z_n|)_{n\in \mathbb{N}}$ definitely does. Soon we'll delve into the topological consequences of the modulus, further reiterating its important place in complex analysis.


One thing that we've noted so far is how similar complex numbers are to $\mathbb{R}^2$ vectors; though they aren't the exact same since a multiplication operation on $\mathbb{R}^2$ vectors is not defined. However, in terms of their addition operations they are exactly the same mathematically!

\begin{proposition}
The group of $\mathbb{R}^2$ vectors over addition is isomorphic to the group of complex numbers over addition.
\[ (\mathbb{R}^2,+) \cong (\mathbb{C},+) \]
\end{proposition}



\section{Complex Euclidean topology}

\begin{proposition}
The $\mathbb{C}$ Euclidean topology is homeomorphic to the $\mathbb{R}^2$ Euclidean topology.
\[ (\mathbb{R}^2,\|\cdot\|) \equiv ( \mathbb{C} , |\cdot|) \]
\end{proposition}

This means that much of what vector analaysis does to generalize real analysis can prove useful in the study of complex analysis.
































\chapter{Complex Sequences and series}


\section{Complex sequences}

Sequences are first covered in Set Theory as an ordered list of (possibly infinite) values, where repetition is allowed. It is then formally defined as a function mapping integers to elements of an arbitrary set. Set theory being as fundamental as it is, it didn't have much more to say about sequences.

Elementary Number Theory goes further and delegates a chapter to 'integer sequences'; sequences of integers and rational numbers (so much for the 'integer part)'. This book discusses properties of any real sequence and particularly focuses on their behaviour as the amount of terms approaches infinity; elementary number theory does not concern itself with such concepts. This chapter introduces sequences in the space of real numbers and aims to build a correct rigorous understand of precisely what it means for a sequence to 'converge' to a value.

This chapter additionally aims to act as a stepping stone for dealing with real functions. Real sequences are still functions, and despite being simpler, we will see that they form the fundamentals for rigorously defining ideas of analysis such as limits. Even if our sequences terms are all happen  to be rational, it is still interesting to note that the sequence might be approaching an irrational number. These are the kinds of fun things that happen when you get 'real' about mathematics (okay, that was pretty lame).


- real sequence

\begin{definition}[Complex sequence]
A \emph{complex sequence} is a sequence $(z_n)^{\infty}_{n=1}$ of complex numbers, so each $ z_n \in \mathbb{C}$
\end{definition}


The order on the field of reals cannot be extended to the field of complex numbers, so bounded above and below do not make sense in $\mathbb{C}$. Since the complex norm is always real, so a bounded complex sequence still has meaning.
\begin{definition}[Complex bounded sequence]
A \emph{complex bounded sequence} is a complex sequence whose norm is bounded for all terms, equivalently, there is some $M$ such that for all $n$ we have $|x_n| \leq M$.
\end{definition}

The notion of convergent sequences translates perfectly fine.

\begin{definition}[Complex convergent sequence]
For a complex sequence $(a_n)_{n \in \mathbb{N}}$, its \emph{limit} is a number $L$ such that for any positive $\varepsilon$, we can find an integer $N$ so that whenever $n >N$ we have  $| a_n -L| < \varepsilon$. Basically, $L$ is arbitrarily close all remaining terms of a sequence. A \emph{convergent complex sequence} is a complex sequence with a limit.
	\[ \lim_{n \to \infty} a_n = L \iff \forall \varepsilon \in (0,\infty) ( \exists N \in \mathbb{N} [ n > N \implies  |a_n - L| < \varepsilon ]  ]  )  \]
\end{definition}

\begin{definition}[Complex Cauchy sequence]
A \emph{complex Cauchy sequence} is a complex sequence $(a_n)_{n \in \mathbb{N}}$, where for any positive $\varepsilon$, we can find an integer $N$ so that whenever $n,m > N$ we have $|a_n - a_m| < \varepsilon$. Basically, the absolute difference between all remaining terms can be bouned arbitrarily close to $0$ given enough terms of the sequence have passed.
	\[ (x_n)_{n\in \mathbb{N}} \text{ is Cauchy} \iff \forall \varepsilon \in (0,\infty) ( \exists N \in \mathbb{N} [ n,m > N \implies  |a_n - a_m| < \varepsilon ]  ]  )  \]
\end{definition}

\begin{proposition}
Complex convergent sequences are complex Cauchy sequences.
\end{proposition}

We can reduce the calculation of limits of complex sequences into 2 limits of real sequences. Mapping complex analysis problems in terms of real analysis is a common technique that will be employed throughout this book.

\begin{proposition}
\[ \lim_{n \to \infty} c_n = L \iff [ \lim_{n \to \infty} \Re(c_n) = \Re(L) ] \land [ \lim_{n \to \infty} \Im(c_n) = \Im(L) ]  \]
\end{proposition}


\section{Series}

- complex series
- convergent series



- absolutely convergent series
- absolute convergence implies convergence

\subsection{Convergent series tests}

Unfortunately many convergence tests of real analysis fail to hold in $\mathbb{C}$, however we can  change our problem into a problem of 2 real series and prove convergence for those.

\begin{proposition}
	\[  \sum^{\infty}_{n=0} c_n = L \iff [ \sum^{\infty}_{n=0} \Re(c_n) = \Re(L) ] \land [ \sum^{\infty}_{n=0} \Im(c_n) = \Im(L) ]  \]
\end{proposition}

Fortunately the comparison test holds for complex series.

- complex comparison test
