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# Examples from Henrici's Applied Complex Analysis

## p. 421

$$\int^1_\kappa \left[\bigl(1-w^2\bigr)\bigl(\kappa^2-w^2\bigr)\right]^{-1/2} dw = \frac{4}{\left(1+\sqrt{\kappa}\,\right)^2} K \left(\left(\frac{1-\sqrt{\kappa}}{1+\sqrt{\kappa}}\right)^{\!\!2}\right)$$

## p. 425

$$\mathop{\rm grd} \phi(z) = \left(a+\frac{2d}{\pi}\right) v_\infty\, \overline{f'(z)} = v_\infty \left[ \pi a + \frac{2d}{\pi a + 2dw^{-1/2}(w-1)^{1/2}} \right]^-$$

## p. 455

$$-\sum^n_{m=1} \left(\,\sum^\infty_{k=1} \frac{ h^{k-1} }{\left(w_m-z_0\right)^2} \right) = \sum^\infty_{k=1} s_k\, h^{k-1}$$

# Examples from Struik's Lectures on Classical Differential Geometry

## p. 23

This procedure is simply a generalization of the method used in Sects. 1-3 and 1-4 to obtain the equations of the osculating plane and the osculating circle. Let $f(u)$ near $P(u=u_0)$ have finite derivatives $f^{(i)}(u_0)$, $i = 1, 2, \ldots, n+1$. Then if we take $u=u_1$ at $A$ and write $h = u_1 - u_0$, then there exists a Taylor development of $f(u)$ of the form (compare Eq. (1-5)):

$$f(u_1) = f(u_0) + hf'(u_0)+{h^2\over 2!}f''(u_0) + \cdots + {h^{n+1}\over (n+1)!}f^{(n+1)}(u_0) + o(h^{n+1}).$$

Here, $f(u_0)=0$ since $P$ lies on $\Sigma_2$, and $h$ is of order $AP$ (see theorem Sec. 1-2); $f(u_1)$ is of order $AD$. Hence necessary and sufficient conditions that the surface has a contact of order $n$ at $P$ with the curve are that at $P$ the relations hold:

$$f(u) = f'(u) = f''(u) = \cdots = f^{(n)}(u) = 0;\quad f^{(n+1)}(u) \ne 0.$$

## p. 40

The converse problem is somewhat more complicated: Find the curves which admit a given curve $C$ as involute. Such curves are called evolutes of $C$ (German: Evolute; French: développées). Their tangents are normal to $C({\bf x})$ and we can therefore write the equation of the evolute ${\bf y}$ (Fig. 1-34): $${\bf y} = {\bf x} + a_1{\bf n} + a_2{\bf b}.$$ Hence $${d{\bf y}\over ds} = {\bf t}(1-a_1\kappa) + {\bf n}\left({da_1\over ds}-\tau a_2\right) + {\bf b}\left({da_2\over ds}+\tau a_1\right)$$ must have the direction of $a_1{\bf n} + a_2{\bf b}$, this tangent to the evolute: $$\kappa = 1/a, \qquad R= a_1,$$ and $${{da_1\over ds} - \tau a_2\over a_1} = {{da_2\over ds}+\tau a_1\over a_2},$$

which can be written in the form:

$${a_2{dR\over ds} - R{da_2\over ds} \over a_2^2 + R^2} = \tau.$$

This expression can be integrated: $$\tan^{-1}{R\over a_2} = \int \tau\,ds + {\rm const},$$ or $$a_2 = R\left[{\rm cot}\left(\int \tau\,ds + {\rm const}\right)\right].$$

The equation of the evolute is:

$${\bf y} = {\bf x} + R\left[{\bf n} + {\rm cot}\left(\int \tau\,ds + {\rm const}\right){\bf b}\right].$$

## p. 154

If $P(u,v)$ and $Q(u,v)$ are two functions of $u$ and $v$ on a surface, then according to Green's theorem and the expression in Chapter 2, Eq. (3-4) for the element area:

$$\int_C P\,du + Q\, dv = \int\!\!\!\int_A \left({\partial Q\over \partial u} - {\partial P\over \partial v}\right) {1\over \sqrt{EG-F^2}}\,dA,$$

where $dA$ is the element of area of the region $R$ enclosed by the curve $C$. With the aid of this theorem we shall evaluate

$$\int_C \kappa_g\,ds,$$

where $\kappa_g$ is the geodesic curvature of the curve $C$. If $C$ at a point $P$ makes the angle $\theta$ with the coordinate curve $v = {\rm constant}$ and if the coordinate curves are orthogonal, then, according to Liouville's formula (1-13):

$$\kappa_g\,ds = d\theta + \kappa_1(\cos\theta)\,ds + \kappa_2(\sin\theta)\,ds.$$

Here, $\kappa_1$ and $\kappa_2$ are the geodesic curvatures of the curves $v = {\rm constant}$ and $u = {\rm constant}$ respectively. Since

$$\cos\theta\,ds = \sqrt{E}\,du, \qquad \sin\theta\,ds = \sqrt{G}\,dv,$$

we find by application of Green's theorem:

$$\int_C\kappa_g\,ds = \int_C d\theta + \int\!\!\!\int_A\left({\partial\over\partial u} \left(\kappa_2\sqrt{G}\,\right) - {\partial\over \partial v}\left(\kappa_1\sqrt{E}\,\right)\right)\,du\,dv.$$

The Gaussian curvature can be written, according to Chapter 3, Eq. (3-7),

$$K = -{1\over 2\sqrt{EG}} \left[{\partial\over\partial u}{G_u\over \sqrt{EG}} + {\partial\over\partial v}{E_v\over\sqrt{EG}}\right] ={1\over\sqrt{EG}}\left[ -{\partial\over\partial u} \left(\kappa_2\sqrt{G}\,\right) + {\partial\over\partial v} \left(\kappa_1\sqrt{E}\,\right)\right],$$

so we obtain the formula

$$\int_C\kappa_g\,ds = \int_C d\theta - \int\!\!\!\int_A K\,dA.$$

The integral $\int\!\!\int_A K\,dA$ is known as the total or integral curvature, or curvature integra, of the region $R$, the name by which Gauss introduced it.

# More Examples from jsMath

$$\left(\, \sum_{k=1}^n a_k b_k \right)^{\!\!2} \le \left(\, \sum_{k=1}^n a_k^2 \right) \left(\, \sum_{k=1}^n b_k^2 \right)$$ $$f(z)\cdot\mathop{\rm Ind}\nolimits_\gamma(z) = \frac{1}{2\pi i}\oint_\gamma\frac{f({\scriptstyle\xi})}{{\scriptstyle\xi}-z}\,d{\scriptstyle\xi}$$ $$a_1 + {1\over\displaystyle a_2 + {\strut 1\over \displaystyle a_3 + {\strut 1\over a_4}}}$$ $$x^{x^{x^{x^2}}}\quad x_{x_{x_{x_i}}}\quad x^2_i\quad x^{y^a_b}_{z^c_d}\quad y'_1+y''_2$$

# Examples of Extensions to TeX

## Hypertext references:

$$\lim_{x\to0} {\sin x\over x} \href{http://visualmatheditor.equatheque.net}{=} 1$$

## Colored text:

$$\color{red}{R} + \color{green}{G} + \color{blue}{B} \qquad \color{#FFAA33}{\sqrt{x^2+1}}$$

## Bounding boxes:

$$\bbox[yellow,2pt]{\bbox[#AA80FF,2pt]{x+1}\over 1-x^2} + \int_0^1 \bbox[border:2px green dotted,2pt]{x^2+2x\,\strut}\,dx$$

## Applying CSS styles:

$$1 \over \style{background-color: #FFEEAA; padding: 0 2 0 4}{x^2} + 1$$

## Applying CSS classes:

Only works in HTML/CSS MathJax mode, not in SVG mode. In VisualMathEditor menu [Options > Editor parameters...] choose [View menu MathJax] option. Then, right click on your equation and in MathJax menu [Math settings > Math Renderer] choose [HTML-CSS] option. $$\sqrt{1-y\over \lower2pt{\class{framed}{\,x^2 + 1\,}}}$$

$$n\in\unicode{x2124}\qquad a \mathrel{\unicode{x22B1}} b \qquad {\unicode{x0919} + 1\over 1 - \unicode{x092D}^2}$$

# Examples from the TeXbook (Chapter 16)

## p. 130

$$\textstyle \sqrt{x+2}\quad \underline 4\quad \overline{x+y}\quad x^{\underline n}\quad x^{\overline{m+n}} \quad \sqrt{x^2+\sqrt{\alpha}}$$ $$\textstyle \root 3 \of 2 \quad \root n \of {x^n+y^n}\quad \root n+1 \of k$$

## p. 131, Exercise 16.7

$$\textstyle 10^{10}\quad 2^{n+1}\quad (n+1)^2\quad \sqrt{1-x^2}\quad \overline{w+\bar z}\quad p^{e_1}_1\quad a_{b_{c_{d_e}}}\quad \root 3 \of {h''_n(\alpha x)}$$

## p. 133

$$x\times y\cdot z \quad x\circ y\bullet z \quad x\cup y\cap z \quad x\sqcup y \sqcap z \quad x\vee y\wedge z \quad x\pm y\mp z$$ $$K_n^+, K_n^- \quad z^*_{ij} \quad g^\circ \mapsto g^\bullet \quad f^*(x)\cap f_*(y)$$ $$x = y > z \quad x:= y \quad x \le y \ne z \quad x\sim y\simeq z \quad x\equiv y \not\equiv z \quad x\subset y\subseteq z$$

## p. 135

$$\hat a \quad \check a\quad \tilde a \quad \acute a \quad \grave a \quad \dot a \quad \ddot a \quad \breve a \quad \bar a \quad \vec a$$

## p. 136

$$\widehat x, \widetilde x \quad \widehat{xy}, \widetilde{xy} \quad \widehat{xyz}, \widetilde{xyz}$$

# Examples from the TeXbook (Chapter 17)

## p. 139

$${1\over 2} \quad {n+1\over 3} \quad {n+1 \choose 3} \quad \sum_{n=1}^3 Z_n^2$$ $${x+y^2\over k+1} \quad {x+y^2\over k} + 1 \quad x + {y^2\over k} + 1 \quad x+ {y^2 \over k+1} \quad x+y^{2\over k+1}$$ $${{a \over b} \over 2} \quad {a \over {b \over 2}} \quad {a/b\over 2} \quad {a \over {b/2}}$$

## p. 143

$${n\choose k} \quad {{n\choose k} \over 2}\quad {n \choose {k\over 2}}\quad {n\choose k/2} \quad {n\choose {1\over 2} k} \quad {1\over 2}{n\choose k} \quad \vcenter{\displaystyle {n\choose k} \over 2}$$ $${p\choose 2}x^2y^{p-2} - {1\over 1-x} {1\over 1-x^2}$$ $${\displaystyle {a\over b}\above1pt\displaystyle{c\over d}}$$

## p. 144

$${\textstyle \sum x_n}\quad {\sum x_n} \quad \sum_{n=1}^m$$ $${\textstyle\int_{-\infty}^{+\infty}} \quad \int_{-\infty}^{+\infty} \quad \int\limits_{0}^{\pi\over 2} \quad$$

## p. 145

$$\sum_{\scriptstyle 0\le i\le m\atop \scriptstyle 0 < j < n} P(i,j)\quad \sum_{i=1}^p\sum_{j=1}^q \sum_{k=1}^r a_{ij}b_{jk}c_{ki} \quad \sum_{{\scriptstyle 1\le i\le p \atop \scriptstyle 1\le j\le q} \atop \scriptstyle 1\le k\le r} a_{ij} b_{jk} c_{ki}$$ $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+x}}}}}}}$$

## p. 146

$$(\; )\; [\; ]\; \{\; \}\; \lfloor\;\rfloor\; \lceil\;\rceil\; \langle\;\rangle\; /\; \backslash\; |\;\vert\;\Vert\; \uparrow\; \Uparrow\; \downarrow\;\Downarrow\; \updownarrow\;\Updownarrow$$ $$\Bigg(\; \Bigg)\; \Bigg[\; \Bigg]\; \Bigg\{\; \Bigg\}\; \Bigg\lfloor\;\Bigg\rfloor\; \Bigg\lceil\;\Bigg\rceil\; \Bigg\langle\;\Bigg\rangle\; \Bigg/\; \Bigg\backslash\; \Bigg\vert\;\Bigg\Vert\; \Bigg\uparrow\; \Bigg\Uparrow\; \Bigg\downarrow\;\Bigg\Downarrow\; \Bigg\updownarrow\;\Bigg\Updownarrow\; \Bigg\lgroup\;\Bigg\rgroup\; \Bigg\lmoustache\;\Bigg\rmoustache$$ $$\bigl(x-s(x)\bigr)\bigl(y-s(y)\bigr)\quad \bigl[x-s[x]\bigr]\bigl[y-s[y]\bigr]\quad \bigl| |x| + |y| \bigr|\quad \bigl\lfloor\sqrt A\bigr\rfloor$$

## p. 147

$$\bigg({\partial\over \partial x^2}+{\partial\over \partial y^2}\bigg) \big|\phi(x+iy)\big|^2 = 0$$ $$\bigl(x\in A(n)\bigm| x\in B(n)\bigr) \quad {\textstyle \bigcup_n X_n\bigm\| \bigcap_n Y_n} \quad {a+1\over b}\!\bigg/{c+1\over d}$$ $$\bigl(x+f(x)\bigr)\big/\bigl(x-f(x)\bigr)$$

## p. 148

$$1+\left(1\over 1-x^2\right)^3 \quad \pi(n) = \sum_{k=2}^n \left\lfloor \phi(k)\over k-1 \right\rfloor$$

## p. 149

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

# Examples from the TeXbook (Chapter 18)

## p. 162

$$\sin 2\theta = 2\sin\theta\cos\theta\quad O(n\log n\log\log n)\quad \Pr(X>x)=\exp(-x/\mu)\quad$$ $$\max_{1\le n\le m} \log_2 P_n \quad \lim_{x\to 0} {\sin x\over x} = 1$$

## p. 163, Exercise 18.2

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

## p. 163

$$\sqrt{{\rm Var}(X)} \qquad {\textstyle x_{\rm max} - x_{\rm max}} \qquad {\rm LL}(k) \Rightarrow {\rm LR}(k)$$ $$\exp(x+{\rm constant}) \qquad x^3 + \hbox{lower order terms}$$ $$\lim_{n\to\infty} x_n {\rm\ exists} \iff \limsup_{n\to\infty} x_n = \liminf_{n\to\infty} x_n$$

## p. 164

$$\gcd(m,n) = \gcd(n,m\bmod n)\qquad x\equiv y+1\pmod{m^2}$$ $${n\choose k} \equiv {\lfloor n/p\rfloor \choose \lfloor k/p\rfloor} {{n\bmod p} \choose {k\bmod p}} \pmod p$$ $${\bf a+b = \Phi_m}\quad {\cal A, B, \ldots, M,N,\ldots, X,Y,Z}$$

## p. 166

$$F_n = F_{n-1} + F_{n-2} \qquad n \ge 2$$

## p. 168

$${\textstyle \int_0^\infty f(x)\,dx} \quad y\,dx-x\,dy \quad dx\,dy = r\,dr\,d\theta \quad x\,dy/dx$$ $$\int_1^x{dt\over t} \qquad$$ $$\int_0^\infty {t-ib\over t^2+b^2} e^{iat}\,dt = e^{ab} E_1(ab), \qquad a,b > 0$$

## p. 169

$${55\rm\,mi/hr} \qquad {g=9.8\rm\,m/sec^2} \qquad {\rm 1\,ml = 1.000028\,cc}$$ $$(2n)!/\bigl(n!\,(n+1)!\bigr) \qquad {52!\over 13!\,13!\,26!}$$ $$\sqrt2\,x\quad \sqrt{\,\log x} \quad O\bigl(1/\sqrt n\,\bigr) \quad [\,0,1) \quad \log n\,(\log\log n)^2 \quad$$ $$\textstyle x^2\!/2 \quad n/\kern-1mu\log n \quad \Gamma_{\!2} + \Delta^{\!2} \quad R_i{}^j{}_{\!kl} \quad \int_0^x\!\int_0^y dF(u,v) \quad \displaystyle \int\!\!\!\int_D dx\,dy$$

## p. 171

$$|-x| = |+x| \qquad \left|-x\right| = \left|+x\right| \qquad \lfloor -x\rfloor = -\lceil+x\rceil$$

## p. 172

$$x_1 + \cdots + x_n\qquad {\rm and} \qquad (x_1,\ldots,x_n)$$ $$x_1 = \cdots = x_n = 0 \qquad A_1 \times \cdots \times A_n \qquad f(x_1,\ldots,x_n)$$ $$x_1x_2 \ldots x_n \qquad (1-x)(1-x^2)\ldots(1-x^n) \qquad n(n-1)\ldots (1)$$

## p. 174

$$\{a,b,c\} \qquad \{1,2,\ldots,n\} \qquad {\{\rm red,white,blue\}} \qquad \{\,x\mid x > 5\,\} \qquad \{\,x:x > 5\,\}$$ $$\bigl\{\,\bigl(x,f(x)\bigr)\bigm| x\in D\,\bigr\} \qquad \bigl\{\,x^3\bigm| h(x)\in\{-1,0,+1\}\,\bigr\}$$

## p. 176

$$\overbrace{x+\cdots+x\,}^{k\rm\;times} \qquad \underbrace{x+y+z\,}_{>\,0}$$ $$A = \pmatrix{x-\lambda & 1 & 0\cr 0 & x-\lambda & 1\cr 0 & 0 & x-\lambda\cr} \qquad\qquad \left\lgroup\matrix{a&b&c\cr d&e&f}\right\rgroup \left\lgroup\matrix{u&x\cr v&y\cr w&z}\right\rgroup$$

## p. 177

$$A = \pmatrix{a_{11} & a_{12} & \ldots & a_{1n}\cr a_{21} & a_{22} & \ldots & a_{2n}\cr \vdots & \vdots & \ddots & \vdots\cr a_{m1} & a_{m2} & \ldots & a_{mn}} \qquad\qquad \pmatrix{y_1\cr\vdots\cr y_k}$$

## p. 178

A small matrix like $1\,1\choose 0\,1$ or a similar one like $\bigl({a\atop l}{b\atop m}{c\atop n}\bigr)$

## p. 179

$$2^{\raise1pt n}\hbox{ rather than }2^n \qquad {\rm Fe_2^{+2} Cr_2^{\vphantom{+2}}O_4^{\vphantom{+2}}} \hbox{ rather than } {\rm Fe_2^{+2} Cr_2 O_4}$$

# Examples from the TeXbook (Chapter 18, final exercises)

## Exercise 18.28

$${\bf S^{\rm-1}TS=dg}(\omega_1,\ldots,\omega_n)=\bf\Lambda$$

## Exercise 18.29

$$\Pr(\,m=n\mid m+n=3\,)$$

## Exercise 18.30

$$\sin18^\circ = {1\over 4}(\sqrt5-1)$$

## Exercise 18.31

$$k=1.38\times 10^{-16}\rm\,erg/^\circ K$$

## Exercise 18.32

$$\bar\Phi\subset NL_1^*/N=\bar L_1^* \subseteq\cdots\subseteq NL_n^*/N=\bar L_n^*$$

## Exercise 18.33

$$\textstyle I(\lambda) = \int\!\!\int_Dg(x,y)e^{i\lambda h(x,y)}\,dx\,dy$$

## Exercise 18.34

$$\textstyle \int_0^1\cdots\int_0^1 f(x_1,\ldots,c_n)\,dx_1\ldots\,dx_n$$

## Exercise 18.35

$$x_{2m}\equiv\cases{Q(X_m^2-P_2W_m^2)-2S^2&\text{(m odd)}\cr \Rule 0pt 1.2em 0pt P_2^2(X_m^2-P_2W_m^2)-2S^2&\text{(m even)}} \pmod N$$

## Exercise 18.36

$$(1+x_1z+x_1^2z^2+\cdots\,) \ldots (1+x_nz+x_n^2z^2+\cdots\,) = {1\over(1-x_1z)\ldots(1-x_nz)}$$

## Exercise 18.37

$$\prod_{j\ge0}\biggl(\sum_{k\ge0}a_{jk}z^k\biggr) = \sum_{n\ge0}z^n\,\Biggl(\sum_{\scriptstyle k_0,k_1,\ldots\ge0\atop \scriptstyle k_0+k_1+\cdots=n} a_{0k_0}a_{1k_1}\ldots\,\Biggr)$$

## Exercise 18.38

$${(n_1+n_2+\cdots+n_m)!\over n_1!\,n_2!\ldots n_m!} ={n_1+n_2\choose n_2}{n_1+n_2+n_3\choose n_3} \ldots{n_1+n_2+\cdots+n_m\choose n_m}$$

## Exercise 18.39

$$\Pi_R{a_1,a_2,\ldots,a_M\atopwithdelims[] b_1,b_2,\ldots,b_N} = \prod_{n=0}^R {(1-q^{a_1+n})(1-q^{a_2+n})\ldots(1-q^{a_M+n}) \over (1-q^{b_1+n})(1-q^{b_2+n})\ldots(1-q^{b_N+n})}$$

## Exercise 18.40

$$\sum_{p\rm\;prime} f(p) = \int_{t > 1} f(t)\,d\pi(t)$$

## Exercise 18.41

$$\{\underbrace{\overbrace{\mathstrut a,\ldots,a}^{k\;a'\rm s}, \overbrace{\mathstrut b,\ldots,b}^{l\;b'\rm s}\>}_{k+1\rm\;elements}\}$$

## Exercise 18.42

$$\pmatrix{\pmatrix{a&b\cr c&d} & \pmatrix{e&f\cr g&h}\cr \Rule 0pt 2.25em 0pt 0&\pmatrix{i&j\cr k&l}}$$

## Exercise 18.43

$$\det\left|\,\matrix{ c_0 & c_1 & c_2 & \ldots & c_{n\phantom{+1}}\cr c_1 & c_2 & c_3 & \ldots & c_{n+1}\cr c_2 & c_3 & c_4 & \ldots & c_{n+2}\cr \vdots & \vdots & \vdots & \ddots & \vdots \cr c_n & c_{n+1} & c_{n+2} & \ldots & c_{2n}} \right| > 0$$

## Exercise 18.44

$$\mathop{{\sum}'}_{x\in A}f(x)\mathrel{\mathop=\limits^{\rm def}} \sum_{\scriptstyle x\in A\atop\scriptstyle x\ne0}f(x)$$

## Exercise 18.45

$$\textstyle 2\uparrow\uparrow k \mathrel{\mathop=\limits^{\rm def}} 2^{{2^{2^{\cdot^{\cdot^{\cdot^2}}}}}{\Big\}\scriptstyle k}}$$

# Examples from the TeXbook (Chapter 18, commutative diagram)

## Exercise 18.46

$$\def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}} \def\mapdown#1{\Big\downarrow\rlap{\raise 2mu{\scriptstyle#1}}} \def\hidewidth{\kern -2em} \matrix{ &&&&&&0\cr &&&&&&\mapdown{}\cr \Rule 0pt 1.1em .6em 0&\mapright{}&{\cal O}_C&\mapright\iota&\cal E& \mapright\rho&\cal L&\mapright{}&0\cr &&\Big\Vert&&\mapdown\phi&&\mapdown\psi\cr \Rule 0pt 1.1em .6em 0&\mapright{}&{\cal O}_C&\mapright{}&\pi_*{\cal O}_D& \mapright\delta&R^1f_*{\cal O}_V(-D)&\mapright{}&0\cr &&&&&&\mapdown{\theta_i\otimes\gamma^{-1}}\cr \Rule 0pt 1.1em .6em &&&&&&\hidewidth R^1f_*\bigl({\cal O}_V(-iM)\bigr)\otimes\gamma^{-1}\hidewidth\cr &&&&&&\mapdown{}\cr &&&&&&0 }$$