\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