
\newcommand*\mat[1]{\bm{#1}}
%\newcommand*\mat[1]{\textsf{#1}}

%The following snippets mostly originated with the \TeX Book and were adapted for \LaTeX{} from Karl~Berry's torture test for plain \TeX{} math fonts.

$x + y - z$, \quad $x + y * z$, \quad $z * y / z$, \quad
$(x+y)(x-y) = x^2 - y^2$,

$x \times y \cdot z = [x\, y\, z]$, \quad $x\circ y \bullet z$, \quad
$x\cup y \cap z$, \quad $x\sqcup y \sqcap z$, \quad

$x \vee y \wedge z$, \quad $x\pm y\mp z$, \quad
$x=y/z$, \quad $x \coloneq y$, \quad $x\le y \ne z$, \quad $x \sim y \simeq z$
$x \equiv y \nequiv z$, \quad $x\subset y \subseteq z$

$\sin2\theta=2\sin\theta\cos\theta$, \quad
$\hbox{O}(n\log n\log n)$, \quad
$\Pr(X>x)=\exp(-x/\mu)$,

$\bigl(x\in A(n)\bigm|x\in B(n)\bigr)$, \quad
$\bigcup_n X_n\bigm\|\bigcap_n Y_n$

% page 178

In text matrices $\binom{1\,1}{0\,1}$ and $\bigl(\genfrac{}{}{0pt}{}{a}{1}\genfrac{}{}{0pt}{}{b}{m}\genfrac{}{}{0pt}{}{c}{n}\bigr)$

% page 142

\[a_0+\frac1{\displaystyle a_1 +
{\strut \frac1{\displaystyle a_2 +
{\strut \frac1{\displaystyle a_3 +
{\strut \frac1{\displaystyle a_4}}}}}}}\]

% page 143

\[\binom{p}{2}x^2y^{p-2} - \frac1{1 - x}\frac{1}{1 - x^2}
=
\frac{a+1}{b}\bigg/\frac{c+1}{d}.\]

%% page 145

\[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+x}}}}}\]

%% page 147

\[\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right)
\bigl|\varphi(x+iy)\bigr|^2=0\]

%% page 149

% \[\pi(n)=\sum_{m=2}^n\left\lfloor\biggl(\sum_{k=1}^{m-1}\bigl
% \lfloor(m/k)\big/\lceil m/k\rceil\bigr\rfloor\biggr)^{-1}\right\rfloor.\]

\[\pi(n)=\sum_{m=2}^n\left\lfloor\Biggl(\sum_{k=1}^{m-1}\bigl
\lfloor(m/k)\big/\lceil m/k\rceil\bigr\rfloor\Biggr)^{-1}\right\rfloor.\]

% page 168

\[\int_0^\infty \frac{t - i b}{t^2 + b^2}e^{iat}\,dt=e^{ab}E_1(ab), \quad
a,b > 0.\]

% page 176

\[\mat{A} \coloneq \begin{pmatrix}x-\lambda&1&0\\
0&x-\lambda&1\\
0&0&x-\lambda\end{pmatrix}.\]

\[\left\lgroup\begin{matrix}a&b&c\\ d&e&f\\\end{matrix}\right\rgroup
\left\lgroup\begin{matrix}u&x\cr v&y\cr w&z\end{matrix}\right\rgroup\]

% page 177

\[\mat{A} = \begin{pmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\
a_{21}&a_{22}&\ldots&a_{2n}\\
\vdots&\vdots&\ddots&\vdots\\
a_{m1}&a_{m2}&\ldots&a_{mn}\end{pmatrix}\]

\[\mat{M}=\bordermatrix{&C&I&C'\cr
C&1&0&0\cr I&b&1-b&0\cr C'&0&a&1-a}\]

%% page 186

\[\sum_{n=0}^\infty a_nz^n\qquad\hbox{converges if}\qquad
|z|<\Bigl(\limsup_{n\to\infty}\root n\of{|a_n|}\,\Bigr)^{-1}.\]

\[\frac{f(x+\Delta x)-f(x)}{\Delta x}\to f'(x)
\qquad \hbox{as $\Delta x\to0$.}\]

\[\|u_i\|=1,\qquad u_i\cdot u_j=0\quad\hbox{if $i\ne j$.}\]

%% page 191

\[\it\hbox{The confluent image of}\quad
\begin{Bmatrix}\hbox{an arc}\hfill\\\hbox{a circle}\hfill\\
\hbox{a fan}\hfill\\\end{Bmatrix}
\quad\hbox{is}\quad
\begin{Bmatrix}\hbox{an arc}\hfill\\
\hbox{an arc or a circle}\hfill\\
\hbox{a fan or an arc}\hfill\end{Bmatrix}.\]

%% page 191

\begin{align*}
T(n)\le T(2^{\lceil\lg n\rceil})
&\le c(3^{\lceil\lg n\rceil}-2^{\lceil\lg n\rceil})\\
&<3c\cdot3^{\lg n}\\
&=3c\,n^{\lg3}.
\end{align*}

%\begin{align*}
%\left\{%
%\begin{gathered}\alpha&=f(z)\\ \beta&=f(z^2)\\ \gamma&=f(z^3)
%\end{gathered}
%\right\}
%\qquad
%\left\{%
%\begin{gathered}
%x&=\alpha^2-\beta\\ y&=2\gamma
%\end{gathered}
%\right\}%
%\end{align*}

%\[\left\{
%\begin{align}
%\alpha&=f(z)\cr \beta&=f(z^2)\cr \gamma&=f(z^3)\\
%%\end{align}
%\right\}
%\qquad
%\left\{
%%\begin{align}
%x&=\alpha^2-\beta\cr y&=2\gamma\\
%\end{align}
%\right\}.\]
%%% page 192

\begin{align*}
\begin{aligned}
(x+y)(x-y)&=x^2-xy+yx-y^2\\
&=x^2-y^2\\
(x+y)^2&=x^2+2xy+y^2.
\end{aligned}
\end{align*}

%% page 192

\begin{align*}
\begin{aligned}
\biggl(\int_{-\infty}^\infty e^{-x^2}\,dx\biggr)^2
&=\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(x^2+y^2)}\,dx\,dy\\
&=\int_0^{2\pi}\int_0^\infty e^{-r^2}\,dr\,d\theta\\
&=\int_0^{2\pi}\biggl(e^{-\frac{r^2}{2}}
\biggl|_{r=0}^{r=\infty}\,\biggr)\,d\theta\\
&=\pi.
\end{aligned}
\end{align*}

%% page 197

\[\prod_{k\ge0}\frac{1}{(1-q^kz)}=
\sum_{n\ge0}z^n\bigg/\!\!\prod_{1\le k\le n}(1-q^k).\]

\[\sum_{\substack{\scriptstyle 0< i\le m\\\scriptstyle0<j\le n}}p(i,j) \,\ne
%
% \[\sum_{i=1}^p \sum_{j=1}^q \sum_{k=1}^r a_{ij} b_{jk} c_{ki}\]
%
\sum_{i=1}^p \sum_{j=1}^q \sum_{k=1}^r a_{ij} b_{jk} c_{ki} \,\ne
%
\sum_{\substack{\scriptstyle 1\le i\le p \\ \scriptstyle 1\le j\le q\\
\scriptstyle 1\le k\le r}} a_{ij} b_{jk} c_{ki}\]

\[\max_{1\le n\le m}\log_2P_n \quad \hbox{and} \quad
\lim_{x\to0}\frac{\sin x}{x}=1\]

\[p_1(n)=\lim_{m\to\infty}\sum_{\nu=0}^\infty\bigl(1-\cos^{2m}(\nu!^n\pi/n)\bigr)\]

