\documentclass[12pt,letterpaper]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage[left=.5in,top=.5in,right=.5in,bottom=.5in,nohead]{geometry}
\begin{document}
\section*{Math 3245}
\subsection*{The Logarithmic Function}
\paragraph{Definition}
The logarithmic function of a complex number $ z $ is defined by,
\[
\log(z) = \ln|z| + i \arg(z)
\]
\paragraph{Definition}
Principle Logarithmic Function
\[
\mathrm{Log}(z) = \ln|z| + i \mathrm{Arg}(z)
\]
\[
\frac{\mathrm{d}}{\mathrm{d}z}\left[ \mathrm{Log}(z) \right] = \frac{1}{z}
\]
\subsection*{Mapping by w = Log(z)}
\[
w = u + i v = \ln|z| + i \theta
\]
\[
u = \ln |z|
\;\;\;\;\;\;\;\;\;\;\;\;\;
v = \theta
\]
\subsection*{Complex powers of z}
\paragraph{Definition}
If $\alpha$ is a complex number and $z = x + i y$, then $z^{\alpha}$ is defined by,
\[
z^{\alpha} = e^{\alpha \log z}, \; z \neq 0
\]
\paragraph{Note}
The principle value of $e^{\alpha}$ can be obtained by setting $k = 0$.
\[
\frac{\mathrm{d}}{\mathrm{d}z}\left[ z^{\alpha} \right] = \alpha z^{\alpha - 1}
\]
\subsection*{Trigonometric and Hyperbolic Functions}
\paragraph{Definition}
For any complex number $z=x+i y$,
\[
\sin{z} = \frac{e^{i z} - e^{-i z}}{2 i} \; , \; \cos{z} = \frac{e^{i z} + e^{-i z}}{2}
\]
\[
\sinh{z} = \frac{e^{z} - e^{-z}}{2} \; , \; \cosh{z} = \frac{e^{z} + e^{-z}}{2}
\]
\subsection*{Complex Integrals}
\paragraph{Theorem}
If $f$ is continuous on a smooth curve $C$ given the parameterization $z(t) = x(t) + i y(t)$ then,
\[
\int_C f(z) \, \mathrm{d} z = \int_a^b f \left( z(t) \right) z'(t) \, \mathrm{d} t
\]
\subsection*{ML-Inequality}
\paragraph{Theorem}
If $f$ is continuous on a smooth curve $C$ and if $|f(z)| \leq M$ for all $z$ on $C$ then,
\[
\left| \int_C f(z) \mathrm{d} z \right| \leq ML
\]
where $L$ is the length of $C$.
\subsection*{Cauchy Goursat Theorem}
\paragraph{Theorem}
Any domain having the property that every closed contour $C$ can be continuously shrunk in $D$ to a point without leaving $D$ is called a simply connected domain.
\subsection*{Greens Theorem}
\paragraph{Theorem}
Let $C$ be a simply closed contour with a positive orientation, and let $R$ be the domain that forms the interior of $C$. If $P$ and $Q$ are continuous and have continuous partial derivatives, $P_x, P_y, Q_x, Q_y$ at all points in $C$ and $R$ then,
\[
\oint_C P \mathrm{d}x + Q \mathrm{d} y = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)  \mathrm{d} A
\]
\subsection*{Cauchy's Theorem}
\paragraph{Theorem}
If $f$ is analytic in a simply connected domain $D$, and $f'(z)$ is continuous in $D$, then for every closed contour $C$ in $D$,
\[
\oint_C f(z) \, \mathrm{d}z = 0
\]
\subsection*{Cauchy-Goursat Theorem}
\paragraph{Theorem}
If $f$ is analytic at all points within and on a simply closed contour $C$,
\[
\oint_C f(z) \, \mathrm{d}z = 0
\]
\subsection*{Deformation of Contours}
\paragraph{Theorem}
If $f$ is analytic in a domain $D$ (double connected) that contains the simple closed curves $C$ and $C_1$ then,
\[
\oint_C f(z) \, \mathrm{d}z \oint_{C_1} f(x) \, \mathrm{d}z
\]
\subsection*{Cauchy-Goursat Theorem for Multiply Connected Domains}
\paragraph{Theorem}
Suppose $C, C_1, C_2, \cdots, C_n$ are simple connected curved with positive orientation such that $C_1, C_2, \cdots, C_n$ are interior to $C$ but the regions interior to each $C_k, \, k = 1, 2 \cdots n$ have no points in common. If $f$ is analytic on each contour and at each point interior to $C$ but exterior to all $C_k, \, k = 1, 2 \cdots n$ then,
\[
\oint_C f(z) \, \mathrm{d}z = \sum_{k = 1}^n \oint_{C_k} f(z) \, \mathrm{d}z
\]
\subsection*{Independence of the Path}
\paragraph{Definition}
Let $z_0$ and $z_1$ be points in a domain $D$. A contour integral $ \int_C f(z) \, \mathrm{d}z$ is said to be independent of the path if its value is the same for all contours $C$ in $D$ with initial point $z_0$ and terminal point $z_1$
\paragraph{Theorem}
Suppose $f(z)$ is analytic in a simply connected domain $D$, and $C$ is any contour in $D$. Then $\int_c f(z) \, \mathrm{d}z$ is independent of the path of $C$.
\subsection*{Fundamental Theorem of Contour Integrals}
\paragraph{Definition}
Suppose a function $f$ is continuous on a domain $D$. If there exists a function $F$ such that $F'(z) = f(z)$, then $F$ is called an anti-derivative of $f$.
\paragraph{Theorem}
Suppose that a function $f$ is continuous on a domain $D$ and $F$ is an anti-derivative of $f$ in $D$. Then, for any contour $C$ in $D$ with initial point $z_0$ and terminal point $z_1$,
\[
\int_C f(z) \, \mathrm{d}z = F(z_1) - F(z_0)
\]
\paragraph{Note}
If a continuous function $f$ has an anti-derivative $F$ in $D$, then $\int_C f(z) \, \mathrm{d}z$ is independent of the path.
\subsection*{Sufficient Condition for the existence of an anti-derivative}
\paragraph{Theorem}
If $f$ is continuous and $\int_C f(z) \, \mathrm{d}z$ is independent of the path $C$ in a domain $D$, then $f$ has an anti-derivative everywhere in $D$.
\subsection*{Cauchy's Integral Formula}
%% \[ \oint_C \frac{f(z)}{z-z_0} \, \mathrm{d}z \]
\paragraph{Theorem}
Let $f$ be analytic in a simple connected domain $D$, and let $C$ be a simple closed contour lying entirely within $D$. If $z_0$ is any point within $C$ then,
\[
f(z_0) = \frac{1}{2 \pi i} \oint_C \frac{f(z)}{z - z_0} \, \mathrm{d}z
\]
\subsection*{Cauchy's Integral Formula for Derivatives}
\paragraph{Theorem}
Let $f$ be analytic on and inside a closed contour $C$ lying in a simply connected domain $D$. If $z_0$ is any point in $D$ (inside $C$), then,
\[
\frac{n!}{2\pi i} \oint_C \frac{f(z)}{(z-z_0)^{n+1}} \, \mathrm{d} z = f^{(n)}(z_0)
\]
\[
\oint_C \frac{f(z)}{(z-z_0)^{n+1}} \, \mathrm{d} z = \frac{2\pi i}{n!} f^{(n)}(z_0)
\]









\subsection*{Cauchy's Inequality}
\paragraph{Theorem}
Suppose that $f$ is analytic in a simply connected domain $D$ and $C$ is a circle defined by $|z-z_0| = r$ that lies entirely in $D$. If $|f(z)| \leq M$ for all points in $z$ in $C$ then,
\[
\left| f^{(n)}(z_0) \right| \leq \frac{n! M}{r^n}
\]
\subsection*{Liouville's Theorem}
\paragraph{Theorem}
If $f$ is an entire function and is bounded for all values of $f$ in the complex plane, then $f$ is a constant.
\subsection*{Fundamental Theorem of Algebra}
\paragraph{Theorem}
If $p(z)$ is a non-constant polynomial, then the equation $p(z)=0$ has at least one root.
\subsection*{Morera's Theorem}
\paragraph{Theorem}
If $f$ is continuous in a  simply connected domain $D$ and if $\oint_C f(z) \, \mathrm{d} z = 0$ for every closed curve $C$ in $D$, then $f$ is analytic.
\subsection*{Maximum Modules Theorem}
\paragraph{Theorem}
Suppose that $f$ is analytic and non-constant in a closed region $R$ bounded by a simple closed curve $C$. Then the modulus $|f(z)|$ attains its maximum on $C$.
\end{document}