Notes

documentclass{article}

usepackage{amsmath}
usepackage{amssymb}


begin{document}

author{A. U. Thor}
title{An example document}
date{}
maketitle

section*{Introduction}
This is an example document. Recreating this document as closely as possible
to the original is the homework. For that, you will need slightly more stuff than
we talked about during the class, most notably the texttt{booktabs} package (you can
easily find its documentation on the internet).


section{Mathematics}
Inline formulae:
begin{itemize}
item $a^2+b^2=c^2$
item $pi approx 3,14159$
item $forall alpha colon sin^2alpha + cos^2alpha = 1$
item $fcolon R to R$
item $frac{1}{2pi}int_{-pi}^{pi}f(x)e^{inx} textup dx$
item $ conv A$
item[] Displayed formulae:
end{itemize}
These styles textit{can be textbf{nested}}.


$g circ f colon X to Z$

begin{equation*}
int_{C}frac{z^2}{z^2+2z+2}textup{d}z = -4pi i,
end{equation*}

begin{equation}
g(z)=frac{z^2}{z^2+2z+2}.
end{equation}

[
1+frac{1}{2}+frac{1}{4}+frac{1}{8} + frac{1}{16}+ldots = sum_{n=0}^{infty}frac{1}{2^n}.
]

[
X=prod_{lambda in Lambda} X_{lambda}
]

[
int_{a}^{b} f(x) textup dx = F(b)-F(a).
]

[
iint_{Omega} (2x^2-y^3+xy)textup dx textup dy = iint_{Omega} 2x^2textup dx textup dy - iint_{Omega} y^3textup dx textup dy + iint_{Omega} xytextup dx textup dy.
]

[
A=idotsint_{Omega} f(x_1,x_2,ldots,x_n)
]

[
|langle x,y rangle| leq |x| |y|.
]

section{Table}
begin{equation}
int_{0}^{1} f(x)g(x) textup dx leq biggl( int_{0}^{1} bigl| f(x) bigr|^2 textup dxbiggr )^{frac{1}{2}} biggl( int_{0}^{1} bigl| g(x) bigr|^2 textup dx biggr)^{frac{1}{2}}.
end{equation}

begin{equation*}
sqrt{2}=1.414212563ldots
end{equation*}

[
left[ sum_{i} a_{i} left| sum_{j} x_{ij} right|^p right]^{1/p} qquad text{versus} qquad biggl[ sum_{i} a_{i} Bigl| sum_{j} x_{ij} Bigr|^p biggr]^{1/p}
]

[
mathbb{ABCDEFG}...
mathfrak{ABCDEFGHabcdefgh1234567}...
mathcal{ABCDEFGH}....
alpha beta gamma delta epsilon zeta eta
]


end{document}