\documentclass{beamer}
\usepackage{ beamerthemesplit,amssymb,latexsym,amsmath,amscd,epsfig,amsthm, amsfonts, multirow, multicol}
\usepackage{wrapfig}
\usepackage{graphicx} 
\usepackage{tikz}

\theoremstyle{theorem}
%\newtheorem{lemma}{Lemma}
%\newtheorem{theorem}{Theorem}
%\newtheorem{corollary}{Corollary}
\newtheorem*{theorem1}{Theorem 1$'$}
\newtheorem*{theorem3}{Theorem 3$'$}
\newtheorem{conjecture}{Conjecture}
\newtheorem{proposition}{Proposition}
\newtheorem*{theorem2}{Theorem 2$'$}

%\theoremstyle{remark}
%\newtheorem{example}{Example}
%\newtheorem{problem}{Problem}
%\newtheorem*{remark}{Remark}
%\newtheorem*{definition}{Definition}

\newcommand{\C}{{\mathbb C}}
\newcommand{\T}{{\mathbb T}}
\newcommand{\D}{{\mathbb D}}
\newcommand{\N}{{\mathbb N}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\Harm}{{\rm Harm}}
\newcommand{\const}{{\rm const}}
\newcommand{\conv}{{\rm conv}}
\newcommand{\dd}{{\rm d}}
\newcommand{\tr}{{\rm tr}}
\newcommand{\dist}{{\rm dist}}
\newcommand{\supp}{{\rm supp}}
 \newcommand{\HH}{{\mathcal K}}
\newcommand{\h}{{\mathcal H}^{n-1}}
\newcommand{\A}{{\mathcal A}}
\newcommand{\I}{{\mathcal I}}
\newcommand{\M}{{\mathcal M}}
\newcommand{\RR}{{\mathcal R}}
\newcommand{\cL}{{\mathcal L}}

 \author{ Yurii Lyubarskii, NTNU}
\title[TMA4120, Lecture 17] { 17.1 Geometry of analytic functions.  Conformal mapping}
\date{October 16, 2016}
 

\begin{document}

\frame{\titlepage}


\frame{
\frametitle{ Geometrical viewpoint}

Function of a complex variable: $w=f(z)$, $z\in D$.

\

$D$ - {\em range} of  $f$,  $f(D)=\{w=f(z), z\in D$ - image of $D$ under action $f$.
  
 Examples: image of curves, image of domains.
 
 Remind:  $f(z)=e^z$, images of the cartesian coordinate net, images of strips etc.
 
 Our goal for today: get intuition (and some rigorous) knowledge  about mapping by analytic functions. 
 
}
 
 
 \frame{
\frametitle{ Simplest examples}

 
 \begin{itemize}
 \item \underline{Shift:} Fix $a\in \C$.  Let $w=f(z)=z+a$. This is a shift of the complex plane  
  \item \underline{Rotation:} Fix $b\in \C$.  Let $w=f(z)=bz$. This is a rotation of the complex plane
   \item \underline{Linear mapping:}  $w=f(z)=a+bz$. This is  shift and  rotation of the complex plane
 \end{itemize}
 
 \
 \fbox{ Linear mappings preserve the angles between straight lines}
 
 \
 
 In order to have this fact for more general mappings  we need definition of angle between curves. 
}

%\end{document}
 \frame{
\frametitle{ Angles between curves } 

Let $C_1$, $C_2$ be two curves which intersect at the point $z_0$.

\ 

 { \em Angle between $C_1$ and $C_2$ at $z_0$ is angle between their  tangents at this point}
 
 \

Remind definition of curve (in complex notation):

\centerline{$C=\{z(t)=x(t)+iy(t), t\in (\alpha, \beta)\subset \R\}$.}

\ 

Examples: arcs, segments  in  $\C$, arbitrary curves. 

\ 
Natural parametrisation: by the arc length.

\

Given $t_0\in (\alpha,\beta)$ and the corresponding point $z_0=z(t_0)\in \C$,  $\dot{z}(t_0)$ is  directed along the  tangent
at $z_0$.




 
}

 
 \frame{
\frametitle{Conformal mapping}

 Mapping is \textcolor{green}{conformal} if it preserves angles between curves. 
 
 Examples: \\
  
  --  Linear mapping \\
  
  -- Exponential mapping ({\small so far we checked this just  for horizontal and vertical lines }) \\
  
  \pause
  
  -- World map
 
 \pause
 
 Conformity is a local  property of the  mapping 
 
 \pause
  
 \underline{Fact}  \textcolor{green}{The mapping $w=f(z)$ by an analytic function is conformal at each  point $z$  where $f'(z)\neq 0$  }

 
\pause

Actually the inverse statement is also true
}

%\end{document}

 
 
 \frame{
\frametitle{Idea of the proof}
 
If $f$ is analytic near $z_0$, then locally (i.e. when $z$ is close to $z_0$)
$$
f(z)=\underbrace{f(z_0)+f'(z_0)(z-z_0)}_{\mbox{this is the main part}}+ o(|z-z_0|),
$$
 The main part is a linear mapping!
 
 \pause
 
 Locally each analytic function is a shift and  a rotation (where $f'\neq 0$).  
 
 \
 
 \underline{Definition}  $f(z)$ is analytic at $z_0$ if $f'(z_0)=0$.
 
 \textcolor{green}{Q:}  What happens in the critical points  ?


}

%\end{document}
 
 \frame{
\frametitle{Power  function}
 
\centerline{ $w=f(z)=z^\alpha$, $\alpha > 0$.}  
 
 Mention that this is an analytic function if $z\neq 0$ due to our definition of power function
 $z^\alpha=e^{\alpha \ln z}$ and $f'(z)=\alpha z^{\alpha-1}$ 
 
 It also can be written as
 
 \
 
 $z=re^{i\phi}$ $\Rightarrow$ $w=r^\alpha e^{i\alpha \phi}$ - this mapping opens angles (if $\alpha >1$) or compress angles (if $\alpha<1$).
 
 \
 
 Spacial case: $w=f(z)=z^\alpha$, $n > 0$, integer.  Then $f(z)$ is an analytic function at $z=0$ as well, $z=0$ is a critical point. Each angle with vertex at zero 
 increases $n$ times. 
 
 \
 
 Question: How many times does $f(z)=z^n$ takes each value?
 
 
 
}
 
\frame{
\frametitle{More examples }

\begin{itemize}
\item Exponential function  $f(z)=e^z$.
\item Logarithmic function $f(z)=\ln z$
\end{itemize}

\underline{Definition}  Let $f$ maps a domain $S\subset \C$ {\em one to one}  onto a domain $T\subset \C$, (i.e. $T=\{w=f(z), z\in S\}$ and 
for each $w\in T$ there is just one $z\in S$ such that $f(z)=w$) we can define the inverse mapping $f^{-1}:T\to S$:
$$
z=f^{-1}(w) \ \mbox{if} \ w=f(z)
$$

\

\underline{Principle of inverse mapping:}  If $f$ is conformal then $f^{-1}$ is conformal as well 



 }

\frame{
\frametitle{ More examples }

\underline{Inversion:}  $w=f(z)= \frac 1 z$

 Make pictures. Inversion maps the unit disc  onto exterior of the unit disc.
 
 \
 
 \underline {Rule}  In order to trace the imager of a domain we have to look at the image of its  boundary.
 
 \
 
 \underline{Zoukovskii mapping:} $w=f(z)=z+1/z$.
 
 \begin{itemize}
 \item Derivative and  the critical  points 
 \item Exterior (and interior)  of the unit disc onto exterior of the segment
 \item Bigger discs onto ellipses
 \item Shifted discs onto "Zoukovskii airfoil"
 \end{itemize}

}


\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\frame{
\frametitle{Basic sets}
{\bf{Circular domains}} We fix a complex number $z_0$. Then
\begin{itemize}
\item $\{z: |z-z_0|=R\}$ is a circle of radius $R$ centered at $z_0$,
\item $\{z:|z-z_0|<R\}$ is an open disk of radius $R$ and center $z_0$, 
\item $\{z:|z-z_0|\le R\}$ is a closed disk of radius $R$ and center $z_0$, 
\item $\{z:r<|z-z_0|<R\}$ is an open circular ring (annulus) of radii $r<R$ and center $z_0$. 
\end{itemize}
{\bf{Half-Planes}} Let $z=x+iy$
\begin{itemize}
\item The upper half-plane is the set of points with $y>0$ and the lower half-plane is the set where $y<0$.
\item The right half-plane is the set where $x>0$, the left half-plane is where $x<0$.
\end{itemize}}
\frame{
\frametitle{Point sets (Punktmengder) (vocabulary)}
Let  $S$ be a set of points on the complex plane.
\begin{itemize}
\item $S$ is called {\bf{open}} (\aa pen) if for each point $z\in S$ there is a disk centered at $z$ which is contained in $S$
\item $S$ is called {\bf{linearly connected}} (sammenhengende) if for any two points $z_1$ and $z_2$ in $S$ there is a continuous curve $\gamma$ with end-points $z_1$ and $z_2$ which is contained in $S$ ( a continuous curve is a continuous mapping $\gamma:[0,1]\to\mathbb{C}$)
\item $S$ is called a {\bf{domain}} (omegn) if $S$ is open and linearly connected.
\end{itemize}}
\frame{
\frametitle{Continuous functions}
Let $D$ be a domain in $\C$, consider a function $f:D\to\mathbb{C}$. It is called continuous at point $z_0$ if for any $\epsilon>0$ there exists $\delta>0$ such that if $|z-z_0|<\delta$ then $z\in D$ and $|f(z)-f(z_0)|<\epsilon$.
\pause

\

In other words, $f$ is continuous at $z_0$ if 
\[\lim_{z\to z_{0}}f(z)=f(z_0).\] 

\pause

Let $f(z)=u(z)+iv(z)$, where $u,v:D\to\R$. Then $f$ is continuous at   $z_0=(x_0,y_0)$  if and only if $u$ and $v$ are continuous at this point.}
\frame{
\frametitle{Examples of continuous functions}
\begin{itemize}
\item $f(z)=|z|$ is continuous everywhere,
\item $f(z)=\rm{Arg}(z)$ is discontinuous at points $z=x+0i,x\le 0$,
\item $f(z)=\Re(z)=x, f(z)=\Im(z)=y, f(z)=z,f(z)=\bar{z}$ are continuous everywhere,
\item $f(z)=e^z=e^xe^{iy}$ is continuous everywhere.
\end{itemize} 

Combinations of continuous functions (sums, differences, products, compositions) are continuous. Everything is as for real-valued functions of two variables.}
\frame{
\frametitle{Derivative}
Let $f:D\to\C$ be a continuous function. We say that $f$ is differentiable at some point $z_0\in D$ if the following limit exists
\[
f'(z_0)=\lim_{w\to 0}\frac{f(z_0+w)-f(z_0)}{w}
\]
Remember that $w$ here is  a complex number!
\pause
This limit if exists is called the derivative of $f$ at the point $z_0$.
\pause

\

There are standard rules for the derivatives as in Calculus 1. 

\

\pause
 \textcolor{red} {However this is VERY different from the  derivatives in Calculus 1}.
}
\frame{
\frametitle{ Good old examples}
\begin{itemize}
\item $f(z)=C$,  then $\frac{f(z)-f(z_0)}{z-z_0}=0$, the constant function is differentiable with $f'(z_0)=0$.
\item $f(z)=z$, then $\frac{f(z)-f(z_0)}{z-z_0}=1$, the function is differentiable and $f'(z_0)=1$,
\item $f(z)=z^2$ then $\frac{f(z_0+w)-f(z_0)}{w}=2z_0+w\to 2z_0$ as $w\to 0$, $f'(z_0)=2z_0$\item $f(z)=c_kz^k+c_{k-1}z^{k-1}+...+c_1z+c_0$ is a polynomial, then $f$ is differentiable at each point and $f'(z)=kc_kz^{k-1}+(k-1)c_{k-1}z^{k-2}+...+c_1$
\end{itemize}}
\frame{
\frametitle{ Bad new examples}
\begin{itemize}
\item $f(z)=\Re (z)=x$, then $\frac{f(z+w)-f(z)}{w}=\frac{\Re(w)}{w}$ has no limit as $w\to 0$! This function is not differentiable,
\item $f(z)=\bar{z}$, then $\frac{f(z+w)-f(z)}{w}=\frac{\bar{w}}{w}$ has no limit as $w\to 0$, not differentiable
\item $f(z)=|z|^2$, then $f(z)=z\bar{z}$ and $\frac{f(z+w)-f(z)}{w}=\frac{z\bar{w}+\bar{z}w}{w}=\bar{z}+z\frac{\bar{w}}{w}$, the limit exists only when $z=0$, $f'(0)=0$ but $f$ is not differentiable at $z\neq 0$.
\item $f(z)=|z|$, at which points  is it differentiable? 
\end{itemize}
\

\pause

\textcolor{green}{Q:} What should we demand from $f=u+iv$ in order to grant existence of $f'$?

}
\end{document}