Examples from jsMath site displayed with MathJax javascript engine and VisualMathEditor configuration file .
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.
$$
\sqrt{1-y\over \lower2pt{\class{framed}{\,x^2 + 1\,}}}
$$
Access to any unicode character:
$$
n\in\unicode{x2124}\qquad
a \mathrel{\unicode{x22B1}} b \qquad
{\unicode{x0919} + 1\over 1 - \unicode{x092D}^2}
$$
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
$$
\def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap{\raise 2mu{\scriptstyle#1}}}
\def\hidewidth{\kern -2em}
\matrix{
&&&&&&0\cr
&&&&&&\mapdown{}\cr
\Rule 0pt 1.1em .6em
0&\mapright{}&{\cal O}_C&\mapright\iota&\cal E&
\mapright\rho&\cal L&\mapright{}&0\cr
&&\Big\Vert&&\mapdown\phi&&\mapdown\psi\cr
\Rule 0pt 1.1em .6em
0&\mapright{}&{\cal O}_C&\mapright{}&\pi_*{\cal O}_D&
\mapright\delta&R^1f_*{\cal O}_V(-D)&\mapright{}&0\cr
&&&&&&\mapdown{\theta_i\otimes\gamma^{-1}}\cr
\Rule 0pt 1.1em .6em
&&&&&&\hidewidth R^1f_*\bigl({\cal O}_V(-iM)\bigr)\otimes\gamma^{-1}\hidewidth\cr
&&&&&&\mapdown{}\cr
&&&&&&0
}
$$