\chapter{Introduction (04/14/2008)}
\label{c1}

We rapidly review complex numbers and more particularly complex
analysis. Our discussion is brief, and concentrates on building up as
quickly as possible the tools that we will use. In lieu of extended
discussions, we try and provide one example of everything.
 
\section{Complex numbers}

\subsection{Definition}

We start be reviewing complex numbers. We define two operations,
``addition'' and ``multiplication'', on pairs $(x,y)$ of reals as
follows:
\begin{eqnarray}
(x_1,y_1) + (x_2,y_2) & = & (x_1 + y_1, x_2 + y_2), \\
(x_1,y_1) \times (x_2,y_2) & = & (x_1x_2 - y_1y_2, x_1y_2 + x_2y_1).
\end{eqnarray}
One can verify that $(0,0)$ is the identity for addition, and every
element $(x,y)$ has the inverse $(-x,-y)$ under addition. Hence
subtraction is defined. The element $(1,0$) is the identity under
multiplication. Every element except for $(0,0)$ has
an inverse as can be seen from
\begin{equation}
(x,y) \times \left( \frac{x}{x^2+y^2}, -\frac{y}{x^2+y^2} \right) =
(1,0).
\end{equation}

We see that $(\Cset,+,\times)$ form a field, and drop the $\times$
sign from now on.  By identifying $\Rset \times \Rset$ with $\Cset$
under the map $(x,y) \to z = x + \ci y$, we have defined the complex
numbers. We identify elements of the form $(x,0)$ with real
numbers. While $\ci$ in this approach can initially be viewed as an
abstract symbol, considering the element $(0,1)$ shows that $\ci^2 =
-1$ shows that $\ci$ is the complex unit.

The reason $\ci$ was introduced is that the set $\Cset$ is
algebraically closed, i.e.~ every polynomial has a zero in
$\Cset$. (This is of course not true in $\Rset$, as shown by the
polynomial $x^2 + 1$.) This is a theorem, given the above definition:
the fundamental theorem of algebra.

Note that $\Cset$ is also a vector space under multiplication by
reals.

\subsection{Geometry}

The identification of a complex number with a pair of real numbers
leads immediately to the Argand diagram, where $z = x + \ci y$ is
associated with the point in the plan with coordinates $(x,y)$. Some
of the algebraic operations acquire geometric meanings then: $-z$ is
the reflection of $z$ through the origin, $z_1 + z_2$ is the vector
sum of the two vectors associated with points (arrowhead at the point,
tail at the origin), and we can now think about distances and angles.

We write the distance from the origin of the point $z$ as $|z| \equiv
\sqrt{x^2 + y^2}$; this is the modulus of $z$ (sometimes written as
$r$ when polar coordinates are being used). Defining the complex
conjugate of $z$, $\bar z$, by $\bar z = x - \ci y$ gives $|z|^2 = z
\bar z$. The angle made by the line (vector, whatever) to $z$ is the
argument of $z$; clearly $(x,y) = |z|
(\cos{\theta},\sin{\theta})$. Then $\tan{\theta} = y/x$ in general,
but one has to be careful about cases like $x = 0$ and what happens in
different quadrants of the complex plane. These may seem like trivial
issues, but can lead to real problem in numerical implementations.

\begin{warning}
An equation that is true or ``obvious'', as a relation between complex
entities, can fail as part of a numerical implementation.
\end{warning}

The argument $\theta$ is safe whenever it appears inside a sine,
cosine, or exponential function. Recalling the series for these, we
see that in polar coordinates
\begin{equation}
z = r\cos{\theta} + \ci r \sin{\theta} = r \ee^{\ci\theta}.
\end{equation}
Now we blithely use the results we know about the exponential
function, obtaining some trigonometric identities on the way. This
shows that multiplication is a ``twist and scale'' operation:
\begin{equation}
z_1 z_2 = (r_1 \cos{\theta_1}, r_1 \sin{\theta_1}) (r_2
\cos{\theta_2}, r_2 \sin{\theta_2}) = r_1 r_2 \ee^{\ci(\theta_1+\theta_2)}.
\end{equation}
The modulus of the product is the produce of the moduli. The argument
of the product is the sum of the arguments (watch out for the modulo
$2\pi$ aspect).

It is sometimes useful to think of the complex plane as a
simply-connected set, i.e.~as a sphere. Then the point at infinity is
part of this sphere, which is called the Riemann sphere. This
geometric picture can be very useful when thinking about multivalued
functions.

\begin{exercise}
Write a two-paragraph discussion of the Riemann sphere.
\end{exercise}

\section{Analyticity}

A central notion in this book will be that of analyticity. A number of
the techniques that we consider will explicitly be concerned with
constructing analytic functions. This will usually be the most
difficult part of the problem. Once a certain function has been
constructed with the appropriate analyticity properties, one is
usually home and dry.

To begin, we need the notion of a function. A function $f(z)$ is a
rule that to the complex number $z$ associates another complex number
$f$. We will return to this issue when talking about multivalued
functions (sometimes called multifunctions). The range of $f$ is the
subset of $\Cset$ of values $f(z)$; $f$ is one-to-one if $f(z_1) =
f(z_2) \Rightarrow z_1 = z_2$.

We will often write $f = u + \ci v$. We will also be alarmingly
slapdash about the arguments. When it suits us, $f$ will be a function
of $z$, of $x$ and $y$, or of $z$ and $\bar z$.

\subsection{Differentiation}

The complex function $f$ is differentiable at the point $z$ if the
limit
\begin{equation}
f'(z) \equiv \lim_{h \to 0} \frac{f(z+h)-f(z)}{h}
\label{eq:diffdef}
\end{equation}
exists \emph{for all complex $h$ as $h \to 0$}. We have not defined limits
and so on: these notions are taken to be understood, if only
informally, for real numbers, and we are discussing the extension to
the complex case.

The fact that (\ref{eq:diffdef}) has to hold for all $h$, i.e.~in all
directions about the origin, is fantastically important. It leads to
entirely unexpected consequences; to be brief we can say that complex
differentiable functions are nice.

We can use this property to derive in a very simple manner the
Cauchy--Riemann equations. First take $h$ to be real. Then $f(z+h) =
f(x+h,y)$ and $\dd f/\dd z = \partial/\partial x f(x,y)$. Similarly is
we write $h = \ci k$, $\dd f/\dd z = \partial/\partial (\ci y)
f(x,y)$. Identifying real and imaginary parts gives the Cauchy--Riemann
equations:
\begin{equation}
\pard{u}{x} = \pard{v}{y}, \qquad \pard{u}{y} = -\pard{v}{x}.
\label{eq:CR}
\end{equation}

\begin{theorem}
If $f$ is analytic, then (\ref{eq:CR}) hold. Conversely, if $u$ and
$v$ are continuous and (\ref{eq:CR}) hold, then $f$ is analytic.
\end{theorem}

It is sometimes useful to consider $z$ and $\bar z$ as independent
variables. We can change variables for an arbitrary function by our
usual abuse of notation, $f(z,\bar z) = f(x,y)$. Then the chain rule
gives, at least formally,
\begin{equation}
\pard{}{x} = \pard{}{z} + \pard{}{\bar z}, \qquad \pard{}{y} =
\ci \pard{}{z} - \ci \pard{}{\bar z}.
\end{equation}
Note that this leads to
\begin{equation}
\nabla^2 = \frac{\partial^2}{\partial x^2} +
\frac{\partial^2}{\partial y^2} = 4 \frac{\partial^2}{\partial
  z \partial \bar z}.
\label{eq:del2}
\end{equation}

The term analytic is sometimes used interchangeably with
differentiable, and sometimes not. So is the word holomorphic. We will
just use analytic from now on. So the complex function $f$ is analytic
in a subset of the complex plane $S$ is (\ref{eq:diffdef}) holds at
all points of $S$.

We can check analyticity by hand to build intuition:
\begin{example}
The function $f(z) = z^m$, with $m$ integer, is differentiable
everywhere in $\Cset$. We write
\begin{equation}
\frac{(z+h)^m-z^m}{h} = m z^{m-1} + \frac{m(m-1)}{2} h z^{m-2} +
\cdots \to m z^{m-1},
\end{equation}
which holds for all $z$.
\end{example}
We can see where it fails in harder situations:
\begin{example}
The function $f(z) = |z|^2$ is differentiable only at the
origin. We have
\begin{equation}
\frac{(z+h)(\bar z + \bar h) - z^2}{h} = \bar z + \frac{\bar h}{h} z.
\end{equation}
This does not have a unique limit as $h \to 0$ (since the second term
depends on the argument of $h$), except when $z = 0$.
\end{example}

However, in most cases  we use the following shortcut:
\begin{tip}
The obvious functions that we know and love (polynomials, exponential,
sine, logarithm, etc\ldots) are analytic except at points where
there's an obvious problem.
\end{tip}

\subsection{Taylor and Laurent series}

An equivalent definition of analyticity is by use of Taylor series:
the function $f$ is analytic at $z_0$ if it has a convergent
expansion of the form
\begin{equation}
f(z_0+h) = a_0 + a_1 h + \half a_2 h^2 + \cdots,
\end{equation}
where we identify $a_1$ with $f'(z_0)$. Given our definition
(\ref{eq:diffdef}), this is a theorem. We do not prove it.

Unlike the real-variable case, we have another, very useful, kind of
series: the Laurent series. If the function $f$ is analytic in the
annulus $0 < |z - z_0| < d$, then it can be written as
\begin{equation}
f(z_0+h) = \cdots + a_{-1} h^{-1} + a_0 + a_1 h + \half a_2 h^2 +
\cdots,
\end{equation}
where the sum can in principle extend infinitely far to the left. In
that case the point $z_0$ is called an essential
singularity. Otherwise, it is a pole.

This definition excludes points of non-analyticity where are not
isolated. These includes points of accumulation such as
$1/(\sin{z^{-1}})$ which is non-analytic at $z = (n\pi)^{-1}$ for all
integer $n$ and branch points. We will have more to say about the
latter.

An entire function has no singularities in the complex plane
(excluding the point at infinity). This means no poles, no
non-isolated singularities, no branch points.


\section{Multivalued functions}

Our definition in words of function as a rule implies that to each $z$
we associate one $f(z)$. This is the usual procedure, and certainly
necessary for computer implementation. However, there are times when a
more general definition is useful.

A multivalued function or multifunction is a rule that to each $z$
associates one or more $f(z)$. For example, we can define the
multifunction $\sqrt{z}$ to be one of the complex number that, when
squared, give $z$. The usual real-valued definition of $\sqrt{x}$
picks the positive root of $y^2 = x$. However for complex numbers, we
want square roots of all $z$.

Dealing with multifunctions leads naturally to the idea of Riemann
surfaces: $\Cset$ is mapped to a more complicated geometric structure,
with different copies of $\Cset$ stitched together.

From a practical perspective, we will not use multifunctions in their
raw form (it can be done), but instead introduce branch cuts to make
the functions single-valued. We pay a price: functions are no longer
continuous in places where they were previously. For now, we defer the
discussion to Chapter~\ref{c2}.

\section{Analytic continuation}

In $|z| < 1$, we write
\begin{equation}
\frac{1}{1-z} = 1 + z + z^2 + z^3 + \cdots,
\end{equation}
and the sum converges. We can think of this another way: reading from
left to right, we see that the sum is the same as the rational
function in the unit disk. The two sides are just different ways of
writing the same function. Hence we use the equality to extend the sum
to the entire complex plane, nothing that the point $1$ has
issues. This is essentially analytic continuation in a nutshell.

\begin{quasitheorem}
If $f_1$ and $f_2$ are defined over subregions $D_1$ and $D_2$
of $\Cset$ and $f_1(z) = f_2(z)$ in $D_1 \cap D_2$, and this region is
 \emph{enough of the complex plane}, namely a
non-trivial curve or set, then $f_1$ provides an analytic
continuation of $f_2$ to $D_1$.
\end{quasitheorem}

Again, we have been vague. This is only interesting if $D_1$ is bigger
than $D_2$ (it may contain $D_2$.) In the previous example, we had
$D_1 = \Cset \setminus \{1\}$ and $D_2$ the unit disk.

\begin{example}
Consider the integral
\begin{equation}
F(z) = \int_0^\infty \ee^{-tz} \,\dd t = \frac{1}{z}
\end{equation}
when $\Re\;z > 0$ (otherwise the integral does not converge). The
function $F$ is analytic in this domain of definition, and the
equality holds there. Hence $z^{-1}$ is its analytic continuation to
the $\Cset \setminus \{0\}$. This kind of argument is very common with
Mellin and Laplace transforms.
\end{example}

\section{Liouville's theorem}

\begin{theorem}
A bounded entire function is constant.
\end{theorem}

This is amazing. More generally, a function satisfying $|f(z)| < c
|z|^m$ for some integer $m$ and $z$ sufficiently large is a
polynomial.

\begin{exercise}
Use Liouville's theorem to prove the fundamental theorem of algebra:
every polynomial has a root in $\Cset$.
\end{exercise}

\section{Integration}

\subsection{Cauchy's theorem}

Integrals in the complex plane can be defined using
parameterizations. We shall not worry about distinctions between
curves, contours, or smoothness. All the obvious properties of
integration based on linearity work.
\begin{example}
  This is a useful integral: $I_m = \int_C z^m \,\dd z.$ when $C$ is
  the unit circle. Parameterize the unit circle by $z =
  \ee^{\ci\theta}$, so that $\dd z = \ci \ee^{\ci\theta}
  \,\dd\theta$. Then
\begin{equation}
I_m = \int_0^{2\pi} \ci \ee^{\ci(m+1)\theta} \,\dd\theta = 0 \qquad (m
\neq -1)
\end{equation}
by the properties of trigonometric functions. Clearly $I_{-1} = 2\pi\ci$.
\end{example}

The fundamental theorem of calculus also works, and if one thinks
about applying it to a closed curve, one would like to argue that the
result always vanishes. This is almost true: the correct version is
Cauchy's theorem.

\begin{quasitheorem}
If the function $f$ is analytic inside a closed curve $C$, then
\begin{equation}
\int_C f(z) \,\dd z = 0.
\end{equation}
\end{quasitheorem}

This is a quasitheorem because we have not worried about the
smoothness of the curve $C$. The conditions on $f$ are fine (note that
Cauchy's original proof required $f$ to be analytic on the curve;
Darboux showed this was unnecessary).

We can hence deform contours in regions where the integrand is
analytic: adding connecting lines and checking appropriate
cancellations (running along the same arc in two different directions)
shows that deforming a contour makes no difference. Hence the result
above, $I_m = 2\pi\ci \delta_{m,-1}$, is valid for any contour
enclosing the origin.

\begin{exercise}
Prove Green's theorem in the plane from Cauchy's theorem.
\end{exercise}

\subsection{Integration by residues}

The fact that we can deform contours except when the integrand has a
pole inside it and the result for $I_{-1}$ above leads to the
following result:
\begin{quasitheorem}
\begin{equation}
\int_C f(z) \,\dd z = 2\pi\ci \sum_n \mbox{Res}(f;z_n),
\end{equation}
where $\mbox{Res}(f;z_0)$ denotes the residue, that is the $-1$ term
in the Laurent series, at the singularity $z_n$.
\end{quasitheorem}
If the integral exists, one doesn't have to worry too much about the
behavior of $f$ along $C$. I have sidestepped issues with non-isolated
singularities, which will not crop up here.

For a function $f(z)/g(z)$ that has a simple pole at $z_0$, the
residue is given by $f(z_0)/g'(z_0)$ (assuming that common terms have
been cancelled out).

We use this to compute real integrals. The name of the game is to find
a contour such that the parameterization of the integral over part of
the contour gives what the desired real integral, while the rest of
the integral vanishes or can be dealt with somehow.

The classic example is the following. Note that versions of this which
require dealing with a simple pole at the origin, or that use Cauchy
principal values, are missing the point.
\begin{example}
Compute
\begin{equation}
I = \int_0^\infty \frac{\sin{x}}{x} \,\dd x.
\end{equation}
First note that we can extend the range to the whole real line by
adding a factor of $\half$. The next step is to come up with a contour
integral. The natural candidate is
\begin{equation}
I_1 = \half \int_C \frac{\sin{z}}{z} \,\dd z.
\end{equation}
Unfortunately, the function $\sin{z}$ blows for large positive and
negative imaginary values, so any semicircular contour will have an
unbounded integrand. Exponentials are better behaved, so let's use the
relation $\sin{z} = (\ee^{\ci z} - \ee^{-\ci z})/(2\ci)$.The real
problem is the contour then, since $\ee^{\ci z}/z$ blows up at the
origin. However, the choice of contour for the original function is
immaterial, so we can deform the contour off the real axis, either up
or down. We take it up to $C_+$. We now work with
\begin{equation}
I_2 = \int_{C_+} \frac{\ee^{\ci z}- \ee^{-\ci z}}{4\ci z} \,\dd z =
\int_{C_+} \frac{\ee^{\ci z}}{4\ci z} \,\dd z - \int_{C_+}
  \frac{\ee^{-\ci z}}{4\ci z} \,\dd z.
\end{equation}
The first integral on the right-hand side can be closed in the upper
half-plane and has no poles inside the contour, hence it vanishes. The
second can be closed in the lower half-plane. There is a simple pole
at the origin with residue $1$. Hence $I_2 = \pi/2$. Note the minus
sign from the fact that the contour in the lower half-plane is
described in the negative sense. Now $I = I_2 + \mbox{stuff that goes
  away}$, so we have $I = \pi/2$.
\end{example}

As a lesson from that example, we note the following:
\begin{warning}
Analyticity in $\Cset$ is very nice. However, functions sometimes
have different behavior in the complex plane compared to the real
line. Sine and cosine are no longer bounded (this is a classic
mistake).
\end{warning}
It's worth practising contour integration. Be efficient:
\begin{tip}
Don't worry about detailed proofs that contributions along parts a
contour vanish. Use orders of magnitude. Large circles or semicircles
with radius $R$ behave OK provided the integrand is $o(R^{-1})$. The
only case where one has to be careful is $O(R^{-1})$ along a
semicircle. There one actually has to use Jordan's lemma, which I do
not include here on principle.
\end{tip}

This tip uses order symbols. These are all valid in some limit as $z$
tends to something, so they hold for $z$ close enough to that
point. Briefly, $f = O(g)$ if $f/g$ is bounded and $f = o(g)$ if $f/g
\to 0$.

One can do many more integrals this way. Note the use we made of
symmetry in the integral there. Clearly the integral of
$x\sin{x}/(1+x^4)$ would work similarly. 

\begin{exercise}
What about the integral of $\sin{x}/(1+x^3)$? This is no longer
elementary and you will need special functions.
\end{exercise}

\begin{example}
Finally, we look at
\begin{equation}
L(z) = \int_1^z \frac{\dd u}{u},
\end{equation}
when $z$ is imaginary. The path of integration doesn't matter,
provided it doesn't go through the origin. Take it to be from $1$ to
$|z|$ and then along an arc from $|z|$ to $z = |z|
\ee^{\ci\theta}$. Using the obvious parameterizations, we find
\begin{equation}
L(z) = \int_1^{|z|} \frac{\dd x}{x} + \int_0^\theta \ci \,\dd\theta =
\log{|z|} + \ci\theta.
\end{equation}
The logarithm above is the usual real logarithm. The presence of
$\theta$ should immediately alert us to the possibility of
multivaluedness: we have not specified how many times around the
origin our path goes and there are many possibilities differing by
multiples of $2\pi\ci$. Clearly $L(z)$ is the complex logarithm, and
we have shown that it is a multifunction. The principal branch has
$-\pi < \theta \leq \theta$. We shall return to this later. Note that
the complex logarithm is the inverse of the complex exponential
function, which is invariant under the addition of $2\pi\ci$. The
integral $I_{-1}$ is an integral of $L(z)$ around a closed curve
encircling the origin. Then $\theta$ varies smoothly and changes by
$2\pi$, giving the required answer.
\end{example}

\subsection{Cauchy's integral formula}

\begin{theorem}
If $f(z)$ is analytic inside and on the contour $C$, then for any
interior point of $z$,
\begin{equation}
f(z) = \frac{1}{2\pi\ci} \int_C \frac{f(\zeta)}{\zeta-z} \,dd\zeta.
\end{equation}
\end{theorem}

This formula can be differentiated as often as we wish. Hence an
analytic formula is differentiable infinitely many times, which is
definitely not the case for a real differentiable function
(e.g.~$|x|x$).

We will be very interested in integrals of the form
\begin{equation}
F(z) \int_C \frac{\varphi(\zeta)}{\zeta-z} \,\dd\zeta,
\label{eq:Ctint}
\end{equation}
known as Cauchy-type integrals, in what follows.

\subsection{Cauchy principal value integrals}

We have glossed over the definition of improper integrals. Formally we
have for a real integral
\begin{equation}
\int_{-\infty}^\infty f(x) \,\dd x = \lim_{M_- \to \infty, M_+ \to
  \infty} \int_{-M_-}^{M_+}f(x) \,\dd x
\end{equation}
when the limit exists. The Cauchy principal value corresponds to
taking $M_- = M_+$ in the limit. It exists when the integral exists,
and sometimes when it does not. Similarly, when $f(x)$ has a simple
pole along the contour, say at the origin, we can do the same thing
and write
\begin{equation}
\dashint_a^b f(x) \,\dd x = \left( \int_a^\epsilon + \int_\epsilon^b \right)
f(x) \,\dd x.
\end{equation}
Often we will detour around simple poles and the limit becomes a
Cauchy principal value integral. These integrals are not limited to
integration along the real axis.

\begin{example}
If $f(z)$ is analytic at the origin and decays at infinity, then
\begin{equation}
\dashint_{-\infty}^\infty \frac{f(z)}{z} \,\dd z =
\dashint_{-\infty}^\infty \frac{f(z)-f(0)}{z} \,\dd z + f(0)
\dashint_{-\infty}^\infty \frac{\dd z}{z} = \dashint_{-\infty}^\infty
\frac{f(z)-f(0)}{z} \,\dd z.
\end{equation}
The first integral on the right-hand side is a Cauchy principal value
integral because of the behavior at infinity. The
second integral is zero:
\begin{equation}
\dashint _{\infty}^\infty \frac{\dd z}{z} = \lim_{\epsilon \to 0, M \to
    \infty} \left( \int_{-M}^{-\epsilon} + \int_\epsilon^M \right)
  \frac{\dd z}{z} = \log{\frac{|-\epsilon|}{|-M|}} +
    \log{\frac{M}{\epsilon}} = 0.
\end{equation}
\end{example}


\section{Plemelj formulae}

Consider what happens in a Cauchy-type integral $F(z)$ as in
(\ref{eq:Ctint})  as $z$ tends to the contour of integration
$C$. Write $F^+$ for the result as $z$ approaches $C$ from the left (with
respect to the direction of integration) and $F^-$ for the result as
$z$ approaches $C$ from the right.
\begin{equation}
F^\pm(z) = \pm \half \varphi(z) + \frac{1}{2\pi\ci} \dashint_C
\frac{\varphi(\zeta)}{\zeta-z} \,\dd z.
\end{equation}
These formulae underlie a lot of results on integral equations and
Riemann--Hilbert problems. 

\section{The principle of the argument}

Consider the integral
\begin{equation}
J = \frac{1}{2\pi\ci} \int_C \frac{f'(z)}{f(z)} \,\dd z =
[\arg{(f(z))}]_C.
\end{equation}
If $f$ is well-behaved along the contour, this will be a multiple of
$2\pi\ci$. The result will depend on the number of singularities of
the function inside the contour. When $f$ has a simple zero at $z$, we have
$f(z+h) = hf'(z) + \cdots$, so $f'(z)/f(z) \sim h^{-1} +
\cdots$. If $f$ has a zero of order $n$,  $f'(z)/f(z) \sim n h^{-1} +
\cdots$ The same argument with poles gives minus signs, so equating
the two sides, we have
\begin{equation}
\frac{1}{2\pi\ci} \int_C \frac{f'(z)}{f(z)} \,\dd z = Z - P,
\end{equation}
where $Z$ and $P$ are the numbers of zeros and poles, counted with
multiplicity.

\begin{exercise}
Discuss the integral
\begin{equation}
\int_C z \frac{f'(z)}{f(z)} \,\dd z.
\end{equation}
\end{exercise}
See Chapter~\ref{c3}


\section{Physical applications}

Complex numbers are unjustly tarred with the appellation imaginary,
and one could ascribe their creation to the mathematical aim of
solving $x^2 + 1 = 0$. However, their origin lies in the very
practical goal of solving cubic equations and obtaining three real
solutions, two of which involve complex arithmetic when using Cardan's
formula. Complex analysis is of enormous help in solving a number of
physical application, which we describe briefly.

\subsection{Waves}

The definition of the complex exponential in terms of trigonometric
functions shows that complex analysis is uniquely suited to describing
periodic systems and waves, which after all are periodic disturbances
in a medium.

The use of complex numbers ranges from efficient ways of solving $LRC$
circuits, where differentiation maps to multiplication by an imaginary
number, through Fourier transforms and FFTs, which underlie modern
signal processing, to quantum mechanics and nonlinear scattering
theory in integrable systems.

\subsection{Integral transforms}

Fourier's theorem states that any periodic function can be decomposed
into a sum of harmonics. This generalizes to Fourier transforms,
which apply to (almost) any real functions. We write
\begin{equation}
F(k) = \int_{-\infty}^\infty f(x) \ee^{-\ci k x} \,\dd x, \qquad f(x) =
\frac{1}{2\pi} \int_{-\infty}^\infty F(k) \ee^{\ci k x} \,\dd k.
\end{equation}
I will try and stick to this notation ($x$ and $k$) for transforms in
space, and use $t$ and $\omega$ for transforms in time. It can be
useful to use the opposite convection in the latter case. Sometimes I
may use $\alpha$ as the transform variable.

Many other integral transforms exist. All rely on some version or
extension of Fourier's inversion theorem to obtain an inverse
transform. Examples are Laplace transforms (one-sided in our case),
Mellin transforms, Hankel transforms, Kantorovich--Lebedev and others.

\subsection{Laplace's equation}

The other main application of complex analysis comes from
(\ref{eq:del2}): harmonic functions, i.e.~functions that satisfy
Laplace's equation, can be obtained from analytic functions. Boundary
conditions on the harmonic functions become conditions on the
analytic functions at the boundary.

Laplace's equation occurs in fluid mechanics (potential flow), heat
transfer (steady-state solutions), elasticity, potential theory and
many others. The biharmonic equation, $\nabla^4 \phi = 0$, can be
attacked using complex methods. It occurs in Stokes flow and linear
elasticity.

These approaches are limited to two dimensions usually.