!<arch>
et.badvec       491009911   286   2     100644  125       `
.EQ (5)
psi sub {nu k vec } ~=~ sum from ljm ~ C sub ljm sup {nu k vec } ~
phi sub ljm ~ ( r vec , E sub {nu k vec }) ~.
.EN

et.bar          491009917   286   2     100644  128       `
.EQ
{{{{{ A bar B bar } bar } bar } bar ~
{ A bar B bar }} bar ~
{{A bar B bar} bar ~
{{ A bar B bar} bar } bar } bar } bar
.EN
et.baugh        491009972   286   2     100644  538       `
.EQ
V bar sub cc ~=~  left [ 1+ 51 over { beta  bar ( beta  bar +1)  } right ]
I bar sub mu R bar sub 1 ~+~ { I bar sub mu R bar sub E } over alpha bar
~+~  lambda left [ 2~ ln ( I bar sub mu over I bar sub s  )
+ ln (51  alpha  bar  over  beta  bar  ) right ]    
~~~~(2)
.EN
.sp
.EQ
A bar sub mu I bar sub mu ~=~
alpha  bar 
left [ 1- (  alpha  bar  over   alpha  bar  sub L  )
e  sup { - V bar sub i / lambda }
right ]
sum from j=1 to N
w sub j
~+~ alpha  bar
e  sup { - V bar sub i / lambda }
sum from j=N+1 to M
w sub j
~~~~~(3)
.EN
et.bcw          491009961   286   2     100644  1182      `
.ta .5i
.EQ
define -+ X ^"\v'.2m'\z\(pl\v'-.4m'\(mi\v'.4m'\v'-.2m'"^ X
define uh "U sub h"
define phi "PHI sub i"
define phr "PHI sub r"
define psi "PSI sub i"
define psr "PSI sub r"
define p4 "pi over 4"
define v0 "V sub 0"
define vh "V sub h"
define u0 "U sub 0"
define a0 "|A sub 0 | sup 2~~"
.EN
.EQ
delim $$
.EN
.EQ
I sub 1~=~a0 left {
1~-~4~{ u0  uh }over{( u0 + uh ) sup 2 } ~sin sup 2 delta over 2
right}
^^
{[ sinh sup 2 phi + cos sup 2 ( phr - p4 )]}
over {d^ sqrt{1-Y sup 2}}
.EN
.sp4
.EQ
I sub 2~=~4 a0~{v0 sup 2 sin sup 2 delta over 2~ 
[ sinh sup 2 psi + cos sup 2 (| psr |- p4 )]} over
{( v0 + vh ) sup 2 |d' sub A -d| ( v0  vh ) sup 0.5 }
.EN
.sp4
.EQ
I sub 3~=~  a0 ~v0 over {  (d|d' sub A -d|) sup 0.5
(1-Z sup 2 ) sup 0.25 (1-Y sup 2 ) sup 0.25 ( v0 + vh )
}
.EN
.sp 1.5
.EQ
~~~~~~~~~~ "{" 2~sin sup 2 ( delta over 2 )[ mark cos ( phr - psr - p4 +- p4 )
cosh ( phi + psi )
.EN
.EQ
~~~~~~~~~~ lineup +~cos ( phr + psr - p4 -+  p4 ) cosh ( phi - psi )]
~{u0 - uh} over {u0 + uh}
.EN
.sp1
.EQ
~~~~~~~~~~ ~~+~sin~delta [ mark sin ( phr - psr - p4 +- p4 ) sinh ( phi + psi )
.EN
.sp 1
.EQ
~~~~~~~~~~ lineup +~ sin ( phr + psr - p4 -+  p4 ) sinh ( phi - psi )]
"}"
.EN
et.blinn        491009979   286   2     100644  183       `
.ps 16
.vs 20p
.EQ
gsize 16
gfont 3
.EN
6.~~FROM EQS. 1, 4, 7 and 8,
.EQ (9)
r sub m ~=~
{q sub m}
over 
{
{
[{x under} sup t x under ~-~ {q under} sup t q under ]
}
sup half }
.
.EN

et.carolyn      491009983   286   2     100644  235       `
.EQ
matrix {
  rcol { "" above rpile { s above Calculated~u above v} }
  ccol { s ~~ u ~~ v
    above left [
    matrix {
    ccol {0 above 1 above 1}
    ccol {1 above 0 above 1}
    ccol {1 above 1 above 1}
    }
  right ]
  }
}
.EN

et.coot         491080536   286   2     100644  12822     `
.if n .ls 2
.EQ
delim $$
define lower "sub"
define upper "sup"
tdefine chi % "\v'-.2m'\(*x\v'.2m'" %
define app X\(apX
tdefine || % \(or\(or %
tdefine <wig % "\z<\v'.4m'\(ap\v'-.4m'" %
ndefine <wig %{ < from "~" }%
tdefine >wig % "\z>\v'.4m'\(ap\v'-.4m'" %
ndefine >wig %{ > from "~" }%
tdefine langle % "\s-3\b'\(sl\e'\s0" %
ndefine langle %<%
tdefine rangle % "\s-3\b'\e\(sl'\s0" %
ndefine rangle %>%
tdefine == % "\z\(eq\v'.23m'\(mi\v'-.23m'" %
tdefine star  % { down 30 size +2 * ^ } %
ndefine star %*%
tdefine hbar % "\zh\v'-.6m'\(ru\v'.6m'" %
ndefine hbar % h\u-\d %
ndefine ppd % _| %
tdefine ppd % "\o'\(ru\s-2\(or\s+2'" %
tdefine <-> % "\o'\(<-\(->'" %
ndefine <-> % "<-->" %
tdefine => % "\z\(->\v'.15m'\(->\v'-.15m'" %
tdefine <-> % "\z\(->\(<-" %
ndefine <-> % "<->" %
tdefine <=> % "\s-2\z<\h'.3m'\z\(eq\h'.6m'\(eq\h'-.6m'>\s+2" %
ndefine <=> % "<=>" %
define prop X\(ptX
tdefine ang % "\fR\zA\v'-.3m'\h'.2m'\(de\v'.3m'\fP\h'.2m'" %
ndefine ang % to o %
define disc % roman "disc" %
define notmember % "\o'\(sl\(mo'" %
.EN
.ds VN 11
.ds NU 6
.ds LH NIGEL COOTE
.ds TT INCLUSIVE ANNIHILATION IN YUKAWA THEORY
.nr LS 1.5v
.nr PS 9
.nr VS 11
.TL
Inclusive annihilation in Yukawa theory*
.AU
Nigel Coote
.AI
Joseph Henry Laboratories, Princeton University, Princeton, New Jersey  08540
.ft 1
(Received 1 August 1974)
.AB -
$gsize 8$In the model of Yukawa field theory we
consider the inclusive process
$e sup + ~+~e sup -$ $ ~->~ gamma (p) $ $ ~->~ pi (k)~+~
roman anything$.
We find, by summing up the logarithms,
that certain transforms $H sup (n)$ of the
amplitude, which in the high-energy limit
$u~==~p sup 2 ~->~ inf$
may be identified with the moments
$int lower 1 upper {0} d omega omega sup
n+1 F bar ( omega ,p sup 2 )$
of the structure functions
$F bar ( omega ,p sup 2 )$,
have simple leading behavior.
At a fixed point of the renormalization
group the $H sup (n)$ would grow like
a fractional power $delta sub n$ of
$p sup 2$.
$delta sub n$ is a property of the
underlying theory (Yukawa) rather than
the structure of the pion, treated as a bound state, and
may be calculated in perturbation theory.
.AE
.\" if t .2C /* mg: Equations too wide for 2C */
.if n .ls2
.SH
I.  INTRODUCTION
.PP
.rm TL
.rm AU
.rm AI
.rm AB
.rm AE
.rm AX
.rm NH
.rm AY
.rm EA
$gsize 9$The object of this paper is to study
the scaling behavior of the amplitude
$H(k,p)$ (see Fig. 1),
relevant to
.BS
.sp 2.125i
.ce
FIG. 1. The amplitude $ size 8 {H = | gamma ( p ) -> pi (k) + roman anything | sup 2}$.
.BE
.EQ I
gamma (p)~->~ pi (k)~+~ roman anything
.EN
in the high-energy limit
.EQ I (1)
u~==~p sup 2 ~->~ inf ,
.EN (1)
with
.EQ L (2)
H sub {mu nu} (k,p sup 2 ) = 
sum from {n}~ langle 0|J sub mu (0)|
pi (k),n rangle ~ langle pi (k),n|J sub nu (0)|0 rangle 
times ~delta sup 4 (p-k-p sub n )
(2 pi ) sup 7 ,
.EN (2)
when the underlying theory is the
Yukawa field theory
.KS
.EQ L (3)
bold L ~ =~ psi bar (
i gamma cdot partial -M sub F ) psi
+( half )( partial sub mu
phi ) sup 2 -( half )m sub S sup 2 phi
sup 2 - g psi bar psi phi
- ~( lambda /4!) phi sup 4 ~+~
roman counterterms
.EN (3)
.KE
(for unexplained notation and the
relation of $H$ to the usual structure functions
see Appendix B).
.PP
To this end we follow the method
Mueller used for $phi sup 4$
field theory.$"" sup 1$  We
believe the extension of Mueller's
method to the richer Yukawa theory 
reveals more of the structure which would be
found in a more realistic theory.
In particular, we can treat the
pion as a bound state rather than a
fundamental field, and see how this fits
in with conventional parton model
ideas.$"" sup 2$
We can also see to what extent
``symmetry'' principles hold in this
model; however, we only touch on this
question briefly in this paper.
We believe that a realistic theory
would be asymptotically free.
Therefore, the final discussion proceeds as if
Yukawa theory had a fixed point
of the renormalization group,
$(g sub 0 , lambda sub 0 )$,
and the anomalous dimensions were small.
.PP
The amplitude $H$ is approached
in two steps.  We start from an amplitude
$T sup ij (k,p)$,
two-particle irreducible in the $t$ channel
(see Fig. 2), and proceed to the
full kernels $K sup ij$, $i,j=0,1$
via the system of integral equations of the generic form
.BS
.sp 2.00i
.ti +2n
FIG. 2. Disallowed decompositions (a) for the two-scalar irreducible amplitude
$size 8  {S sup 11}$;
(b) for the two-Fermi irreducible
amplitude
$size 8 {F sup 11} . ~~ ->$ = the Fermi line (particle or antiparticle);
\*>  = the scalar line.
T denotes an amplitude irreducible with respect to both scalars
and fermions.
In this figure and in subsequent diagrams of this type
the variables
$ size 8 {k sup 2 ,  p sup 2}$ are off the mass shell.
.BE
shown in Fig. 3.
Here
the
$K sup ij$ are defined analogously
to Eq. (2), save that the
external particles $pi , gamma$ are
replaced by the fundamental fields
$psi , phi$ off the mass shell;
e.g., $i=1,~j=0$ means that
lines carrying momentum $k,p$ are
Fermi, scalar, respectively:
.EQ L (4)
B~ == ~K sub {alpha beta} sup 10
(k,p)
== sum from {n} mark delta sup 4
(p-k-p sub n ) int~d sup 4 ye sup
{ik cdot y} langle 0| T bar psi bar sub
alpha (0) phi (y)|n rangle sub A (
2 pi ) sup 4
~ times ~ int d sup 4 ze sup
{-ik cdot z} langle n|T psi sub beta (0)
phi (z)|0 rangle sub A ,
.EN (4)
with subscript $A$ denoting the amputated matrix
element; the propagator for $psi , phi$
is factored out.
In the spacelike analog $(p sup 2 ~<~0,~k
sup 2 ~<~0)$ relevant to electron-pion scattering,
.BS
.sp 1.125i
.ti +2n
FIG. 3. Generic integral equation:
$ size 8 {k sup 2 = v,~ q sup 2 = u ',~ p sup 2 = u}$.
An example of spin notation
(we are only interested at present in cases $size 8{l sub k = 0 ~roman or~ half}$)
is $size 8{l sub k}$ = magnitude of spin on lines
carrying momentum $size 8 k$
with $size 8 {m sub k , m' sub k}$ components along
the $size 8 z$ axis.
.BE
we would have
.EQ I
B~prop~roman Disc sub
{(p-k) sup 2} B tilde ,
.EN
where
.EQ I (5)
B tilde ~==~i ~int~ d sup 4
xd sup 4 yd sup 4 ze sup
{ip cdot x+i(y-z) cdot k}
times~ langle 0|T psi bar sub alpha (x)
phi (y) phi (z) psi sub beta (0)|0 rangle  sub A ,
.EN (5)
and we could obtain the scaling
behavior of $B$, as
$p sup 2 ~->~inf$, by means of a
Wilson expansion in $B tilde$.
However, in the timelike case,
$p sup 2 ~>~0,~k sup 2 ~>~0$,
it is essential to use differing ``$i epsilon$''
prescriptions for incoming and outgoing
lines carrying momentum $p$ in
(5).$ "" sup 3$
Such a prescription will be obtained if we
appropriately analytically continue a
nonforward scattering amplitude; but
for this a Wilson expansion is not
available.
.PP
Equations of the form in Fig. 3
have been the object of much study by,
e.g., those involved with the multiperipheral model since,
for $k -> ~-k$, they describe
the forward scattering of two protons.
An explicit diagonalization for
general spin, whereby they are reduced to 
one-dimensional integral equations, has been
given by Abarbanel and Saunders$"" sup 4$
for the case $k sup 2 ~<~0,~p sup 2 ~<~0$.
However, the timelike case,
$k sup 2 ~>~0,~p sup 2 ~>~0$,
needed here does not appear to have been set out
explicitly.
Therefore, in Sec. II we give this
diagonalization; we follow Ref. 4
closely and use the same notation where
possible.
For later sections only the fact that
Eq. (7) diagonalizes to Eq. (28) is
needed.
The resulting one-dimensional equations involve
certain transforms $A sup chi$ of the original
amplitudes $A$.
We give an interpretation of $A sup chi$.
.PP
Section III gives [Eq. (35)] the ``new
improved renormalization group equations''$"" sup 5$
for Yukawa theory, a homogeneous form of
the original Callan-Symanzik$"" sup 6,7$
equations.
This means we do not need to appeal to Weinberg's
theorem$"" sup 8$ initially.
This form is essentially the same as that
recently obtained by 't Hooft.$"" sup 9$
It does, unlike Weinberg's better known
version, apply to scalars.
.PP
In Sec. IV we study the limit
$u~->~inf$ for the two-particle
irreducible amplitudes $T$.
According to conventional wisdom,$"" sup 10$
they should behave as if they were not evaluated
at the exceptional momentum $t~=~0$.
We are thus able to reach the key equation
(70):  The transforms
$T sup chi (v, u)$ are sensible
``zero mass'' quantities in this limit; they
can be calculated by setting
$k sup 2 ~==~ v ~=~0~=~M sub F ~=~m sub S$.
.PP
In Sec. V we study the limit
$u~->~inf$ for kernels
$K sup ij$, obtaining an expansion in terms
of functions of the form
.EQ I (6)
f(v)g(u),
.EN (6)
where the $g$ are defined in the
zero-mass theory, satisfy a Callan-Symanzik
equation, and so scale as a fractional
power of $u$ at a fixed point of the
renormalization group.
.ce
.sp
.SH
.ne 5
II. DIAGONALIZATION OF THE BETHE-SALPETER EQUATION IN THE TIMELIKE CASE
.sp
.ce
A.  Kinematics
.PP
Consider the generic equation of Fig. 3,
where the notation is appropriate to form
(11) below.  We must start from the
``primeval'' form, in which the lines carry
Greek spinor indices $beta sub 1$
etc.,
.if t 1C
.EQ L (7) 
A sub {beta sub 1 beta sub 2 alpha sub 1
alpha sub 2} (k,p)~=~I sub
{beta sub 1 beta sub 2 alpha sub 1 alpha sub 2}
~+~[1/(2 pi ) sup 4 ]~sum from
{nu sub 1 nu sub 2 mu sub 1 mu sub 2}
~ int ~d sup 4 qA sub
{beta sub 1 beta sub 2 nu sub 1 nu sub 2}
(k,q)S sub {nu sub 1 mu sub 1} (q)
S sub {mu sub 2 nu sub 2} (q)I sub
{mu sub 1 mu sub 2 alpha sub 1 alpha sub 2}
(q,p),
.EN (7)
where $S sub {nu mu} (q)$ is a
full propagator.
We may write, in the case of spin $half$,
.EQ I (8)
S sub {nu sub 1 mu sub 1} ~ = ~
sum from {+- , mu}~f sub +- 
(q sup 2 ) u sub {nu sub 1} sup {( mu , +- )} u bar
sub {mu sub 1} sup {( mu , +- )}
=~sum from {+- , mu}~f sub +-
(q sup 2 )[(-1) sup {mu - half}
u sub {nu sub 1} sup {(- mu , +- )} ]
[(-1) sup {mu - half} u bar sub {mu sub 1}
sup {(- mu , +- )} ],
.EN (8)
where the $f$ are scalar functions of
$q sup 2$, and our spinors
$u sup {( mu , +- )}$ are as
defined in Appendix A and satisfy
.EQ I (9a)
R sub z ( phi )u sup {( mu , epsilon )}
~=~e sup {-i mu phi} u sup {( mu , epsilon )} roman {~~(rotation
~angle~ phi ~about~the}~z~ roman axis),
.EN (9a)
.EQ I (9b)
R sub x ( beta ) u sup {( mu , epsilon )}
~=~sum from {mu '}~d sub {mu mu '} sup half
( beta )u sup {( mu ', epsilon )} roman {~~(rotation~
angle}~ beta ~roman {about~the}~x~roman axis),
.EN (9b)
.EQ I (9c)
u( lambda q)~=~ lambda sup half u(q)~roman
{(for~arbitrary~constant~} lambda ).
.EN (9c)
.\" if t .2C 
.PP
We state trivial facts as (8) and (9)
because, although written explicitly here
for spin $half$, the procedure of
Ref. 4 is valid for arbitrary spin.
Equation (9c) is special to spin
$half$ because we wish to ensure that the
$f$ have dimension $-2$.
.PP
In our applications the
$A(k,p),~I(k,p)$ in Eq. (7) will be
absorptive amplitudes defined by
equations such as (4), so
$v~==~k sup 2 ~>~0,~u~==~p sup 2 ~>~0$.
Necessarily, $u'~==~q sup 2 ~>~0$.
Equation (7) has symmetry under arbitrary Lorentz
transformations of $q$.
Thus three parameters in the integration,
specifying the direction of $q$, may be
factored out by using the theory of the
Lorentz group $roman SL (2,C)$.
Namely, we Fourier-transform with respect to
the representation functions $d sup chi$ of
$roman SL (2,C)$ on the SU(2) subgroup.
.PP
We now note the following.
.PP
(i)  The remainder of this kinematical
part expresses $A$ in standard form (12).
If $LAMBDA ~=~RBR'~ \(mo ~ roman SL (2,C)$,
$R,R'~ \(mo ~ roman SU (2)$,
and $B~==~B( theta )$
is a boost along the $z$ axis we define
.EQ I
A( LAMBDA , v,u)~=~A(k sub s ,
LAMBDA p sub s ),
.EN
.EQ I
A(v,u, theta )~=~A(k sub s ,B p sub s ),
.EN
and have
.KS
.EQ I
A sub {J sub k M sub k J sub p M sub p}
( LAMBDA ,v,u)~=~sum from M,M' ~d sub
{MM sub k} sup {J sub k} (R)d sub {M sub p M'}
sup {J sub p} (R')
times~A sub {J sub k MJ sub p M'}
(v,u, theta ),
.EN
.KE
which expresses that A in the form (12) is an
``SU(2) bicovariant distribution.''  Here
$p sub s ~=~u sup half (1,0,0,0)$ is ``at rest.''
.PP
(ii) The remainder also transforms (7) into (17),
where the dependence on SU(2)
(rotations) is explicitly factored out.
The subsequent Fourier transform will
factor out the dependence on boosts
$theta$.
We now form ``spin projected'' amplitudes,
.EQ L (10)
A sub {m sub k m' sub k m sub p m' sub p} ~=~
sum from {beta sub 1 beta sub 2 alpha sub 1 alpha sub 2}~
u bar sub {beta sub 1} sup {(m sub k )} (-1)
sup {(1/2-m' sub k )} u sub {beta sub 2} sup
{(-m' sub k )}
~ times ~A sub {beta sub 1 beta sub 2 alpha sub 1
alpha sub 2} u sub {alpha sub 1} sup {(m sub p )} (-1)
sup {( half -m' sub p )} u bar sub {alpha sub 2} sup
{(-m' sub p )} ,
.EN (10)
where factors $(-1) sup {half -m}$
have arisen because we wish to consider
the $t$-channel angular momentum;
we have suppressed $epsilon$
dependence.
Thus we rewrite Eq. (7) in the form .\^.\^.
et.crc          491010017   286   2     100644  381       `
CRC Tables, p.335:
.EQ
4.05.~~ int dx over { ae sup mx - be sup -mx } ~=~
left { lpile 
{ 1 over { 2m sqrt ab } ~ log ~ { sqrt a e sup mx - sqrt b }
  over { sqrt a e sup mx  + sqrt b } 
above
 1 over { m sqrt ab } roman ~tanh sup -1 ( sqrt a over sqrt b e sup mx ) 
above
 - 1 over { m sqrt ab  } roman ~coth sup -1
  ( sqrt a over sqrt b e sup mx ), 
  ~~~~~~~~~(a>0,~b>0)
}
.EN

et.demo         491080617   286   2     100644  1199      `
.EQ 2
size 20{
roman det | roman{A- lambda cdot I}|= 
pile{n above size 24 PI above i=1} lambda sub i
}
.EN
.EQ 3
e sup -x sup 2 + e sup -x sub i sup 2 + e sup{-x sub i}sup 2
.EN
.EQ 4
{sum from i=0 to inf} over {2 pi}
{int int} from {- inf} to {+ inf}
   e sup{-{x sub i sup 2 + y sub i sup 2}over 2}
= e sup{- x over y}
.EN
.EQ 5
a over b = c over d = A over B times C over D
.EN
.EQ 6
P(x) = pile{x sub 2 above sum above x=x sub 1}
C sub x p sup x (1-p) sup n-x =
pile{x sub 2 above sum  above x=x sub 1}
C sub x p sup x q sup n-x
.EN
.EQ 7
sum from i=0 to {i= inf} 1 over{2 sup n + 2 sup N}= 2
.EN
.EQ 8
B sub a sub 2 +
B sub a sup 2 +{B sub a}sup 2 + (B sub a ) sup 2 + (B sup 2 ) sub a
+B sup 2 sup x
.EN
.EQ 9
c sub a sub 2 +
c sub a sup 2 +{c sub a}sup 2 + (c sub a ) sup 2 + (c sup 2 ) sub a
+c sup 2 sup x
.EN
.EQ 10
a sub 0 + b sub 1 over
{a sub 1 + b sub 2 over
 {a sub 2 + b sub 3 over
  {a sub 3 + ...}
 }
}
.EN
.EQ 11
{a over b + c over d} sup{x over y}over 89
+{{t times 32}over h-8}sub 2
.EN
.EQ
x = a sup 2 + sqrt a sup 2 + sqrt a
.EN
.EQ
x = sqrt{a sup 2 + b sup 2}+ sqrt a over b +
sqrt a sup 2 over b sup 2
.EN
.EQ
sqrt a+b over sqrt c+d
.EN
.EQ
1 over sqrt{ax sup 2 +bx+c}
.EN

et.dg           491082931   286   2     100644  8200      `
.ll 7.75i
.po 0
.EQ
z0
z1
z2
z(z)[z]z[(z+1)]z(z-0)2-z
z=1
1=z
z+z-z/z/z>z<z>=z<=z->z<-z,z;z:z>>z<<z
|z||z|z|
.EN
.EQ
z sub i,j z sub 1 z sub j z sub 0 z sup 0 z sup i z sup j z sup 1 z sub 1 sup 2
z sub i,j z sup 1 z sub j sub 0 z sub i sup j z sup 1 z sub 1 sup 2
z cdot
z times
z alpha
z beta
z delta
z theta
z lambda z chi
z DELTA z inf z int z sum z pi z partial z
z ,..., z ... z
.EN
.EQ
j0
j1
j2
j(j)[j]j[(j+1)]j(j-0)2-j
j=1
1=j
j+j-j/j/j>j<j>=j<=j->j<-j,j;j:j>>j<<j
|j||j|j|
.EN
.EQ
j sub i,j j sub 1 j sub j j sub 0 j sup 0 j sup i j sup j j sup 1 j sub 1 sup 2
j sub i,j j sup 1 j sub j sub 0 j sub i sup j j sup 1 j sub 1 sup 2
j cdot
j times
j alpha
j beta
j delta
j theta
j lambda j chi
j DELTA j inf j int j sum j pi j partial j
j ,..., j ... j
.EN
.EQ
i0
i1
i2
i(i)[i]i[(i+1)]i(i-0)2-i
i=1
1=i
i+i-i/i/i>i<i>=i<=i->i<-i,i;i:i>>i<<i
|i||i|i|
.EN
.EQ
i sub i,j i sub 1 i sub j i sub 0 i sup 0 i sup i i sup j i sup 1 i sub 1 sup 2
i sub i,j i sup 1 i sub j sub 0 i sub i sup j i sup 1 i sub 1 sup 2
i cdot
i times
i alpha
i beta
i delta
i theta
i lambda i chi
i DELTA i inf i int i sum i pi i partial i
i ,..., i ... i
.EN
.EQ
j0
j1
j2
j(j)[j]j[(j+1)]j(j-0)2-j
j=1
1=j
j+j-j/j/j>j<j>=j<=j->j<-j,j;j:j>>j<<j
|j||j|j|
.EN
.EQ
j sub i,j j sub 1 j sub j j sub 0 j sup 0 j sup i j sup j j sup 1 j sub 1 sup 2
j sub i,j j sup 1 j sub j sub 0 j sub i sup j j sup 1 j sub 1 sup 2
j cdot
j times
j alpha
j beta
j delta
j theta
j lambda j chi
j DELTA j inf j int j sum j pi j partial j
j ,..., j ... j
.EN
.EQ
f0
f1
f2
f(f)[f]f[(f+1)]f(f-0)2-f
f=1
1=f
f+f-f/f/f>f<f>=f<=f->f<-f,f;f:f>>f<<f
|f||f|f|
.EN
.EQ
f sub i,j f sub 1 f sub j f sub 0 f sup 0 f sup i f sup j f sup 1 f sub 1 sup 2
f sub i,j f sup 1 f sub j sub 0 f sub i sup j f sup 1 f sub 1 sup 2
f cdot
f times
f alpha
f beta
f delta
f theta
f lambda f chi
f DELTA f inf f int f sum f pi f partial f
f ,..., f ... f
.EN
.EQ
x0
x1
x2
x(x)[x]x[(x+1)]x(x-0)2-x
x=1
1=x
x+x-x/x/x>x<x>=x<=x->x<-x,x;x:x>>x<<x
|x||x|x|
.EN
.EQ
x sub i,j x sub 1 x sub j x sub 0 x sup 0 x sup i x sup j x sup 1 x sub 1 sup 2
x sub i,j x sup 1 x sub j sub 0 x sub i sup j x sup 1 x sub 1 sup 2
x cdot
x times
x alpha
x beta
x delta
x theta
x lambda x chi
x DELTA x inf x int x sum x pi x partial x
x ,..., x ... x
.EN
.EQ
10
11
12
1(1)[1]1[(1+1)]1(1-0)2-1
1=1
1=1
1+1-1/1/1>1<1>=1<=1->1<-1,1;1:1>>1<<1
|1||1|1|
.EN
.EQ
1 sub i,j 1 sub 1 1 sub j 1 sub 0 1 sup 0 1 sup i 1 sup j 1 sup 1 1 sub 1 sup 2
1 sub i,j 1 sup 1 1 sub j sub 0 1 sub i sup j 1 sup 1 1 sub 1 sup 2
1 cdot
1 times
1 alpha
1 beta
1 delta
1 theta
1 lambda 1 chi
1 DELTA 1 inf 1 int 1 sum 1 pi 1 partial 1
1 ,..., 1 ... 1
.EN
.EQ
00
01
02
0(0)[0]0[(0+1)]0(0-0)2-0
0=1
1=0
0+0-0/0/0>0<0>=0<=0->0<-0,0;0:0>>0<<0
|0||0|0|
.EN
.EQ
0 sub i,j 0 sub 1 0 sub j 0 sub 0 0 sup 0 0 sup i 0 sup j 0 sup 1 0 sub 1 sup 2
0 sub i,j 0 sup 1 0 sub j sub 0 0 sub i sup j 0 sup 1 0 sub 1 sup 2
0 cdot
0 times
0 alpha
0 beta
0 delta
0 theta
0 lambda 0 chi
0 DELTA 0 inf 0 int 0 sum 0 pi 0 partial 0
0 ,..., 0 ... 0
.EN
.EQ
X0
X1
X2
X(X)[X]X[(X+1)]X(X-0)2-X
X=1
1=X
X+X-X/X/X>X<X>=X<=X->X<-X,X;X:X>>X<<X
|X||X|X|
.EN
.EQ
X sub i,j X sub 1 X sub j X sub 0 X sup 0 X sup i X sup j X sup 1 X sub 1 sup 2
X sub i,j X sup 1 X sub j sub 0 X sub i sup j X sup 1 X sub 1 sup 2
X cdot
X times
X alpha
X beta
X delta
X theta
X lambda X chi
X DELTA X inf X int X sum X pi X partial X
X ,..., X ... X
.EN
.EQ
H0
H1
H2
H(H)[H]H[(H+1)]H(H-0)2-H
H=1
1=H
H+H-H/H/H>H<H>=H<=H->H<-H,H;H:H>>H<<H
|H||H|H|
.EN
.EQ
H sub i,j H sub 1 H sub j H sub 0 H sup 0 H sup i H sup j H sup 1 H sub 1 sup 2
H sub i,j H sup 1 H sub j sub 0 H sub i sup j H sup 1 H sub 1 sup 2
H cdot
H times
H alpha
H beta
H delta
H theta
H lambda H chi
H DELTA H inf H int H sum H pi H partial H
H ,..., H ... H
.EN
.EQ
Y0
Y1
Y2
Y(Y)[Y]Y[(Y+1)]Y(Y-0)2-Y
Y=1
1=Y
Y+Y-Y/Y/Y>Y<Y>=Y<=Y->Y<-Y,Y;Y:Y>>Y<<Y
|Y||Y|Y|
.EN
.EQ
Y sub i,j Y sub 1 Y sub j Y sub 0 Y sup 0 Y sup i Y sup j Y sup 1 Y sub 1 sup 2
Y sub i,j Y sup 1 Y sub j sub 0 Y sub i sup j Y sup 1 Y sub 1 sup 2
Y cdot
Y times
Y alpha
Y beta
Y delta
Y theta
Y lambda Y chi
Y DELTA Y inf Y int Y sum Y pi Y partial Y
Y ,..., Y ... Y
.EN
.EQ
partial 0
partial 1
partial 2
partial ( partial )[ partial ] partial [( partial +1)] partial ( partial -0)2- partial 
partial =1
1= partial 
partial + partial - partial / partial / partial > partial < partial >= partial <= partial -> partial <- partial , partial ; partial : partial >> partial << partial 
| partial || partial | partial |
.EN
.EQ
partial sub i,j partial sub 1 partial sub j partial sub 0 partial sup 0 partial sup i partial sup j partial sup 1 partial sub 1 sup 2
partial sub i,j partial sup 1 partial sub j sub 0 partial sub i sup j partial sup 1 partial sub 1 sup 2
partial cdot
partial times
partial alpha
partial beta
partial delta
partial theta
partial lambda partial chi
partial DELTA partial inf partial int partial sum partial pi partial partial partial 
partial ,..., partial ... partial 
.EN
.EQ
pi 0
pi 1
pi 2
pi ( pi )[ pi ] pi [( pi +1)] pi ( pi -0)2- pi 
pi =1
1= pi 
pi + pi - pi / pi / pi > pi < pi >= pi <= pi -> pi <- pi , pi ; pi : pi >> pi << pi 
| pi || pi | pi |
.EN
.EQ
pi sub i,j pi sub 1 pi sub j pi sub 0 pi sup 0 pi sup i pi sup j pi sup 1 pi sub 1 sup 2
pi sub i,j pi sup 1 pi sub j sub 0 pi sub i sup j pi sup 1 pi sub 1 sup 2
pi cdot
pi times
pi alpha
pi beta
pi delta
pi theta
pi lambda pi chi
pi DELTA pi inf pi int pi sum pi pi pi partial pi 
pi ,..., pi ... pi 
.EN
.EQ
lambda 0
lambda 1
lambda 2
lambda ( lambda )[ lambda ] lambda [( lambda +1)] lambda ( lambda -0)2- lambda 
lambda =1
1= lambda 
lambda + lambda - lambda / lambda / lambda > lambda < lambda >= lambda <= lambda -> lambda <- lambda , lambda ; lambda : lambda >> lambda << lambda 
| lambda || lambda | lambda |
.EN
.EQ
lambda sub i,j lambda sub 1 lambda sub j lambda sub 0 lambda sup 0 lambda sup i lambda sup j lambda sup 1 lambda sub 1 sup 2
lambda sub i,j lambda sup 1 lambda sub j sub 0 lambda sub i sup j lambda sup 1 lambda sub 1 sup 2
lambda cdot
lambda times
lambda alpha
lambda beta
lambda delta
lambda theta
lambda lambda lambda chi
lambda DELTA lambda inf lambda int lambda sum lambda pi lambda partial lambda 
lambda ,..., lambda ... lambda 
.EN
.EQ
z sub i sup j
i sub i sup j
j sub i sup j
f sub i sup j
x sub i sup j
1 sub i sup j
0 sub i sup j
X sub i sup j
H sub i sup j
Y sub i sup j
partial sub i sup j
pi sub i sup j
lambda sub i sup j
z sub m sup k
i sub m sup k
j sub m sup k
f sub m sup k
x sub m sup k
1 sub m sup k
0 sub m sup k
X sub m sup k
H sub m sup k
Y sub m sup k
partial sub m sup k
pi sub m sup k
lambda sub m sup k
z sub pi sup 2
i sub pi sup 2
j sub pi sup 2
f sub pi sup 2
x sub pi sup 2
1 sub pi sup 2
0 sub pi sup 2
X sub pi sup 2
H sub pi sup 2
Y sub pi sup 2
partial sub pi sup 2
pi sub pi sup 2
lambda sub pi sup 2
.EN
.EQ
gfont R
z sub i sup j
i sub i sup j
j sub i sup j
f sub i sup j
x sub i sup j
1 sub i sup j
0 sub i sup j
X sub i sup j
H sub i sup j
Y sub i sup j
partial sub i sup j
pi sub i sup j
lambda sub i sup j
z sub m sup k
i sub m sup k
j sub m sup k
f sub m sup k
x sub m sup k
1 sub m sup k
0 sub m sup k
X sub m sup k
H sub m sup k
Y sub m sup k
partial sub m sup k
pi sub m sup k
lambda sub m sup k
z sub pi sup 2
i sub pi sup 2
j sub pi sup 2
f sub pi sup 2
x sub pi sup 2
1 sub pi sup 2
0 sub pi sup 2
X sub pi sup 2
H sub pi sup 2
Y sub pi sup 2
partial sub pi sup 2
pi sub pi sup 2
lambda sub pi sup 2
.EN
.EQ
gfont B
z sub i sup j
i sub i sup j
j sub i sup j
f sub i sup j
x sub i sup j
1 sub i sup j
0 sub i sup j
X sub i sup j
H sub i sup j
Y sub i sup j
partial sub i sup j
pi sub i sup j
lambda sub i sup j
z sub m sup k
i sub m sup k
j sub m sup k
f sub m sup k
x sub m sup k
1 sub m sup k
0 sub m sup k
X sub m sup k
H sub m sup k
Y sub m sup k
partial sub m sup k
pi sub m sup k
lambda sub m sup k
z sub pi sup 2
i sub pi sup 2
j sub pi sup 2
f sub pi sup 2
x sub pi sup 2
1 sub pi sup 2
0 sub pi sup 2
X sub pi sup 2
H sub pi sup 2
Y sub pi sup 2
partial sub pi sup 2
pi sub pi sup 2
lambda sub pi sup 2
.EN
.EQ
1.2 ~~ 3.4 ~~~ x.1 ~~~ 1.x ~~~ a.b ~~~ x.y ~~~ X.Z
gfont R
1.2 ~~ 3.4 ~~~ x.1 ~~~ 1.x ~~~ a.b ~~~ x.y ~~~ X.Z
gfont B
1.2 ~~ 3.4 ~~~ x.1 ~~~ 1.x ~~~ a.b ~~~ x.y ~~~ X.Z
.EN
et.diac         491071880   286   2     100644  1925      `
.EQ
( x dyad ) ~~
( x vec ) ~~
( x bar ) ~~
( x hat ) ~~
( x dot ) ~~
( x dotdot ) ~~
( x tilde ) ~~
( x under ) ~~
.EN
.EQ
( X dyad ) ~~
( X vec ) ~~
( X bar ) ~~
( X hat ) ~~
( X dot ) ~~
( X dotdot ) ~~
( X tilde ) ~~
( X under ) ~~
.EN
.EQ
( alpha dyad ) ~~
( alpha vec ) ~~
( alpha bar ) ~~
( alpha hat ) ~~
( alpha dot ) ~~
( alpha dotdot ) ~~
( alpha tilde ) ~~
( alpha under ) ~~
.EN
.EQ
( phi dyad ) ~~
( phi vec ) ~~
( phi bar ) ~~
( phi hat ) ~~
( phi dot ) ~~
( phi dotdot ) ~~
( phi tilde ) ~~
( phi under ) ~~
.EN
.EQ
( GAMMA dyad ) ~~
( GAMMA vec ) ~~
( GAMMA bar ) ~~
( GAMMA hat ) ~~
( GAMMA dot ) ~~
( GAMMA dotdot ) ~~
( GAMMA tilde ) ~~
( GAMMA under ) ~~
.EN
.EQ
gfont R
( x dyad ) ~~
( x vec ) ~~
( x bar ) ~~
( x hat ) ~~
( x dot ) ~~
( x dotdot ) ~~
( x tilde ) ~~
( x under ) ~~
( X dyad ) ~~
( X vec ) ~~
( X bar ) ~~
( X hat ) ~~
( X dot ) ~~
( X dotdot ) ~~
( X tilde ) ~~
( X under ) ~~
.EN
.EQ
gfont B
( x dyad ) ~~
( x vec ) ~~
( x bar ) ~~
( x hat ) ~~
( x dot ) ~~
( x dotdot ) ~~
( x tilde ) ~~
( x under ) ~~
( X dyad ) ~~
( X vec ) ~~
( X bar ) ~~
( X hat ) ~~
( X dot ) ~~
( X dotdot ) ~~
( X tilde ) ~~
( X under ) ~~
.EN
.EQ
123 over x ~123 over x hat ~ 123 over x bar ~ 123 over x dot
.EN
.EQ
123 over X ~123 over X hat ~ 123 over X bar ~ 123 over X dot
.EN

.EQ
gfont I
W + W bar + {W sub a} bar + W vec + W dot +
.EN
.EQ
w + w bar + {w sub a} bar + w vec + w dot +
.EN

.EQ
gfont R
.EN
.EQ
W + W bar + {W sub a} bar + W vec + W dot +
.EN
.EQ
w + w bar + {w sub a} bar + w vec + w dot +
.EN

.EQ
gfont B
.EN
.EQ
W + W bar + {W sub a} bar + W vec + W dot +
.EN
.EQ
w + w bar + {w sub a} bar + w vec + w dot +
.EN
.EQ
gfont R
x ~ x bar ~ x bar bar ~ x dot ~ x dot bar ~ x dot bar bar
.EN
.EQ
X ~ X bar ~ X bar bar ~ X dot ~ X dot bar ~ X dot bar bar
.EN
.EQ
x sup 2 ~ x sup 2 bar ~ {x sup 2} bar ~ {x bar sup 2} bar
.EN
.EQ
X sup 2 ~ X sup 2 bar ~ {X sup 2} bar ~ {X bar sup 2} bar
.EN

et.doug         491010094   286   2     100644  1589      `
.EQ
define BR "bold r"
define r0 "bold r sub 0"
define cm "{ roman cosh ~ mu }"
define m0 "mu sub 0"
define c0 "{ roman cosh ~ m0 }"
.EN
.EQ
G( BR | r0 | omega ) ~=~
G sub k ( BR | r0 ) ~=~
i pi H sub 0 sup 1 (kR)
.EN
.br
.EQ
~~~~~~~~~~~~~=~ 4 pi i ~ left {
sum from m=0 to inf 
left [ { Se sub m (h,~~ cos ~ theta sub 0 ) }
 over { M sub m sup e (h) } right ] ~
Se sub m (h,~ cos ~ theta ) .
.EN
.in 1.3i
.EQ
. ~ left { pile
 { Je sub m (h,~ c0 ) He sub m (h,~ cm ) ) ; 
 above
  Je sub m (h,~ cm ) He sub m (h,~ c0 ); }
~~ pile { mu ~>~ m0 
 above  m0 ~>~ mu }
.EN
.in 1i
.EQ
+~ sum from m=1 to inf left [
 { So sub m (h,~ cos ~ theta sub 0 ) }
 over { M sub m sup 0 (h) } right ] ~ .
.EN
.in 1.3i
.EQ
. ~ left "" ~  { left { pile
 { Jo sub m ( h,~ c0 ) Ho sub m (h,~ cm ); 
 above
  Jo sub m (h,~ cm ) Ho sub m (h,~ c0 ); } 
~~ pile { mu ~>~ m0 
 above  m0 ~>~ mu }
} right }
.EN
.sp .5i
.in 0
.EQ
r(k)~=~-~ { PHI sub + (k) }
 over { PHI sub + (-k) } ~=~
{ (k-k sub 0 ) } over { (k+ k sub 0 ) }
size 28 prod from n=1 to inf
{ left [ sqrt { 1- left ( ka over { pi n } right ) sup 2 }
 - i left ( ka over { pi n } right ) right ]
over left [ sqrt { 1 - left ( ka over { pi n } right ) sup 2 }
 + i left ( ka over { pi n } right ) right ] }
~e sup { 2iak/ pi n } .
.EN
.in +1i
.sp 1i
.EQ
. ~ size 28 prod from m=1 to inf {
 left [ sqrt { 1 - left ( ka over { pi mu sub m } right ) }
 + i left ( ka over { pi mu sub m }  right ) right ]
 over
 left [ sqrt { 1 - left ( ka over { pi mu sub m } right ) sup 2 }
 - i left ( ka over { pi mu sub m } right ) right ] }
~e sup { -2iak/ pi m }
.EN

et.fat          491010097   286   2     100644  142       `
.EQ
x sub i ~~ fat {x sub i} ~~ grad ~~ fat grad
~~ x sub fat i ~~ fat lambda  ~~ fat partial
.EN
.EQ
gfont B
x ~~ fat x ~~ fat {x sub i}
.EN
et.gfont        491010114   286   2     100644  143       `
.EQ
gfont B
x sub 123 ~~~
123 roman 456 789 ~~~
(abc) ~~~
()[] ~~~
.EN
.EQ
gfont R
x sub 123 ~~~
123 italic 456 789 ~~~
(abc) ~~~
()[] ~~~
.EN

et.greek        491010106   286   2     100644  313       `
.EQ
alpha ~~~ beta ~~~ gamma ~~~
GAMMA ~~~ delta ~~~ DELTA ~~~
epsilon ~~~ zeta ~~~ eta ~~~
THETA ~~~ theta ~~~ lambda ~~~
LAMBDA ~~~ mu ~~~ nu ~~~
.EN
.EQ
xi ~~~ pi ~~~ PI ~~~
rho ~~~ sigma ~~~ SIGMA ~~~
tau ~~~ phi ~~~ PHI ~~~
psi ~~~ PSI ~~~ omega ~~~
OMEGA ~~~ del ~~~ "\(no" ~~~
partial ~~~ integral ~~~
.EN

et.int          491010141   286   2     100644  1057      `
.if n .ls 2
.EQ
delim $$
.EN
.ta 1i
Gamma	$GAMMA (z) ~~=~~ int sub 0 sup inf  t sup {z-1} e sup -t dt$
Error	$ roman erf (z) = 2 over sqrt pi int sub 0 sup z e sup {-t sup 2} dt$
.EQ
int
~~~~ int sub 0 sup pi ~~ int sub {- pi} sup pi ~~ int sub {-2 pi} sup {2 pi} f(x)dx
.EN
XXXXXX
.EQ
int sub 0 sup 1
~~~~ int sub 0 sup pi ~~ int sub {- pi} sup pi ~~ int sub {-2 pi} sup {2 pi} f(x)dx
.EN
XXXXXXXXX
.EQ
int from 0 to 1
~~~~ int sub 0 sup pi ~~ int sub {- pi} sup pi ~~ int sub {-2 pi} sup {2 pi} f(x)dx
.EN
XXXXXXXX
.EQ
= int int =
~~~~ int sub 0 sup pi ~~ int sub {- pi} sup pi ~~ int sub {-2 pi} sup {2 pi} f(x)dx
.EN
.EQ
= int int =
.EN
xxxxx
.EQ
int
~~~~ int sub 0 sup pi ~~ int sub {- pi} sup pi ~~ int sub {-2 pi} sup {2 pi} f(x)dx
.EN
xxxx
.EQ
int sub x sup x ~~ int sub 1 sup 1 ~~
int sub -1 sup 1 ~~ int sub 1 sup 2 ~~ int sub x sup 1
.EN
.ps 9
.EQ
gsize 9
.EN
.EQ L
lineup + ~ sum ~ GAMMA sub r over 2 ~ int sub
{x sub n-k+1} sup
{roman min left ( x sub n-k , x sub n + {2E sub n} over {GAMMA sub r} ~
{a sub i} over {1-a sub i} right )} ~dx
.EN

et.k92          491010147   286   2     100644  719       `
Knuth, vol  1, p92:
.sp 1i
.EQ
G(z)~mark =~ e sup { ln ~ G(z) }
~=~ exp left ( 
sum from k>=1 {S sub k z sup k} over k right )
~=~  prod from k>=1 e sup {S sub k z sup k /k}
.EN
.sp
.EQ
lineup =~ left ( 1 + S sub 1 z + 
{ S sub 2 sup 1 z sup 2 } over 2! + ... right )
left ( 1+ { S sub 2 z sup 2 } over 2
+ { S sub 2 sup 2 z sup 4 } over { 2 sup 2 . 2! }
+ ... right ) ...
.EN
.sp
.EQ
lineup =~  sum from m>=0 left (
sum from
pile { k sub 1 ,k sub 2 , ... ,k sub m  >=0
  above
k sub 1 +2k sub 2 + ... +mk sub m =m}
{ S sub 1 sup {k sub 1} } over {1 sup k sub 1 k sub 1 ! } ~
{ S sub 2 sup {k sub 2} } over {2 sup k sub 2 k sub 2 ! } ~
...
{ S sub m sup {k sub m} } over {m sup k sub m k sub m ! } 
right ) z sup m
.EN

et.lax          491072009   286   2     100644  1265      `
.EQ (2.01)
bold x sup { n alpha } (t) ~->~ bold x sup alpha ( bold X ,t).
.EN
.sp
.EQ (2.02)
sum from n F( bold x sup { n alpha } (t))
~->~ 1 over OMEGA int F( bold x sup alpha ( bold X ,t))d bold \|X
.EN
.EQ (2.03)
bold x ( bold X ,t) ~==~
sum from  { alpha =1} to  N
rho sup alpha  over rho sup 0 bold x sup alpha ( bold X ,t)
.EN
.EQ (2.08)
sum from  {alpha =1} to  N
U sup { mu alpha } V sup { mu alpha } ~=~ delta sup { mu nu }
.EN
.EQ (2.06)
bold y sup { T mu } ( bold X ,t) 
~==~ sum from  {alpha =1} to  N
U sup { mu alpha }
bold x sup alpha
( bold X ,t)
.EN
.EQ  (7.02)
~ partial over {partial d} 
 ( epsilon sub 0 bold E sup T times  bold B ) sub i  
- m sub ij,\|j ~=~
-q sup D E sub i sup T
-( bold ~j sup D times bold B ) sub i
.EN
.EQ (7.04)
~ partial over {partial d} 
[ rho x dot sub i + epsilon sub 0 ( bold E sup T times bold B ) sub i ]
~+~ partial over \(pdz sub j
[ rho x dot sub i x dot sub j -t sub ij sup L
- E hat sub i sup T P sub j sup T -m sub ij ]~=~0
.EN
.EQ
M sub { k prime k } sup { n prime n } ( r vec sub o ) ~=~
1 over A int
e sup { -i k vec prime cdot rho vec }
F sub { n prime } (z) sup *
{ e sup { - lambda ( r vec - r vec sub o ) } } over
{ | r vec - r vec sub o | }
e sup { i k vec cdot l vec }
F sub n (z) d l vec dz
.EN A2

et.mark         491010166   286   2     100644  106       `
.EQ
	x mark = 1
.EN
.EQ
lineup = 2
.EN
.EQ
1 lineup = 2
.EN
.EQ
ij lineup =3
.EN
.EQ
bold k lineup =4
.EN
et.matrix       491010170   286   2     100644  1122      `
Knuth, vol 2, page 317:
.sp
.EQ 
Q sub n (x sub 1 ,x sub 2 , ... ,x sub n ) ~=~ left {
matrix {
lcol { 1, above x sub 1 , above 
x sub 1 Q sub n-1 (x sub 2 , ... ,x sub n ) +
Q sub n-2 (x sub 3 , ... ,x sub n ) }
lcol {  if above  if above  if }
lcol { n=0; above n=1; above n>1. }
}
.EN
.sp
Knuth, vol 2, p426:
.sp
.ti 3
When zero occurs, the determinant is even easier to compute;
for example, if
.EQ
x sub 11 =0 
.EN
but
.EQ
x sub 21  != 0,
.EN
we have
.sp
.nf
.EQ
det ~
left [ 
matrix {
col { 0 above x sub 21 above x sub 31 above ...  above x sub n1 }
col { x sub 12 above x sub 22 above x sub 32 above ~ above x sub n2 }
col { ...  above ...  above ...  above ~ above ...  }
col { x sub 1n above x sub 2n above x sub 3n above ~ above x sub nn }
}
right ]
.EN
.sp
.ti 5
.EQ
=~-x sub 21 det
left [
matrix {
col { 
x sub 12 above x sub 32 -(x sub 31 /x sub 21 )x sub 22
above ...  above x sub n2 -(x sub n1 /x sub 21 )x sub 22 }
col { ...  above ...  above ~ above ...  }
col {
x sub 1n above x sub 3n -(x sub 31 /x sub 21 )x sub 2n
above ...  above x sub nn -(x sub n1 /x sub 21 )x sub 2n
   }
}
right ]
~~.~~~(8)
.EN
et.mini         491010172   286   2     100644  386       `
.EQ
x
.EN
.sp
.EQ
alpha ~ beta ~ sum ~ int ~ pi ~ partial
.EN
.sp
.EQ
x sub i
.EN
.sp
.EQ
x sub i sup k
.EN
.sp
.EQ
a over bc
.EN
.sp
.EQ
a over b+c
.EN
.sp
.EQ
x sup 2 over a sup 2 ~=~ y sup 2 over b sup 2
.EN
.sp
.EQ
sum from i=0 to n x
.EN
.sp
.EQ
pile {a above b above c}
.EN
.sp
.EQ
left [
cpile {abcdef above b above cd}
right ]
.EN
.sp
.EQ
sqrt x ~~~ sqrt {ax sup 2  + bx+c}
.EN
et.motion       491010179   286   2     100644  822       `
.EQ
delim $$
.EN
.vs 20p
int sub 0 sup inf	$ int sub 0 sup inf$
int sub back 30 down 30 0 sup up 40 inf	$ int sub back 30 down 30 0 sup up 40 inf$
int sub back 30 down 30 size 7 0 sup up 40 size 7 inf	$ int sub back 30 down 30 size 7 0 sup up 40 size 7 inf$
.sp 2
.sp 2
int from 0 to 1	$int from 0 to 1$
int from back 100 0 to fwd 100 1	$int from back 100 0 to fwd 100 1$
int from back 150 0 to fwd 150 1	$int from back 150 0 to fwd 150 1$
$x up 9 y$
.nf
.EQ
gsize 9
define solid % "\b'\(bv\(bv'" %
define dotted % "\s7\b'\(or\(or\(or'\s0" %
.EN
\l'3i'
.EQ (20)
solid from { down 60 {u sub n} } ~~ solid from { down 60 {u sub n-1}} ~~ ... ~~
solid from { down 60 {u sub {n-k sub i +1}}} ~~ fat dotted from { down 60 {u- epsilon sub i}}
~~ solid from { down 60 {u sub n-k sub i}} ~~ (k sub i ~ roman defined ) ,
.EN
\l'3i'
et.over         491010183   286   2     100644  247       `
.EQ
+ a over b +
.EN
.EQ
+ ab over c +
.EN
.EQ
+ a over bc +
.EN
.EQ
+ ab over cd +
.EN
.EQ
+ abc over d
.EN
.EQ
+ a over bcd +
.EN
.EQ
+ ab over cdef +
.EN
.EQ
+ ab over cdefg +
.EN
.nf
XXXXXX
.EQ
1 over 2
.EN
XXXXXX
.EQ
123 over 345
.EN
XXXXXXX

et.prime        491072064   286   2     100644  1392      `
.EQ
tdefine sup' X sup "\s+3\v'.35m'\(fm\v'-.35m'\s-3" X
tdefine sup'' X sup "\s+3\v'.35m'\(fm\(fm\v'-.35m'\s-3" X
ndefine sup' X sup ' X
ndefine sup'' X sup '' X
.EN
.EQ
x sub i sup' ~~~
x sub X sup'' ~~~
x sup' ~~~
X sup'' ~~~
X sub x sup' ~~~
.EN
.if n .ls 2
.EQ
R sub L sup down 0 ' ~~~
R sub L sup down 10 ' ~~~
R sub L sup down 20 ' ~~~
R sub L sup down 30 ' ~~~
R sub L sup down 40 ' ~~~
R sub L sup down 50 ' ~~~
.EN
.EQ
R sub L sup down 0 size +3 ' ~~~
R sub L sup down 10 size +3 ' ~~~
R sub L sup down 20 size +3 ' ~~~
R sub L sup down 30 size +3 ' ~~~
R sub L sup down 40 size +3 ' ~~~
R sub L sup down 50 size +3 ' ~~~
.EN
.sp
.EQ
f(x)~=~size -1 3 over size -1 4 xxx ~~~
f(x)~=~size -2 3 over size -2 4 xxx ~~~
f(x)~=~size -3 3 over size -3 4 xxx ~~~
f(x)~=~size -4 3 over size -4 4 xxx ~~~
size -2 {3 over 4} =x~~~
size -3 {3 over 4} =x~~~
size -1 {3 over 4} =x~~~
.EN
.EQ
f(x)~=~size -1 3 over size -1 45 xxx ~~~
f(x)~=~size -2 3 over size -2 45 xxx ~~~
f(x)~=~size -3 3 over size -3 45 xxx ~~~
f(x)~=~size -4 3 over size -4 45 xxx ~~~
.EN
.EQ
gsize 9
tdefine s50 % sup size +7 down 50 ' %
tdefine s60 % sup size +7 down 60 ' %
tdefine s70 % sup size +7 down 70 ' %
tdefine s80 % sup size +7 down 80 ' %
.EN
.EQ
===
~~~ R sub i s50
~~~ R sub i s60
~~~ R sub i s70
~~~ R sub i s80
===
.EN
.EQ
gsize 10
===
~~~ R sub i s50
~~~ R sub i s60
~~~ R sub i s70
~~~ R sub i s80
===
.EN
et.scitypo      491010192   286   2     100644  609       `
.LP
.EQ
(N sub roman Sh ) sub a sup * mark = 1 over 2 left ( d sub t over L right )
  N sub roman Re N sub roman Sc
.EN
.EQ
lineup times
left [
{ 1 - sum from j=1 to {j= inf} {-4 beta sub j} over {beta sub j sup 2}
  left ( {d phi sub j} over dr sub + right ) sub {r sub + =1}
  exp left ( {- beta sub j sup 2 (x/r sub t ) } over { N sub Re N sub roman Sc} right ) 
}
over
{ 1 + sum from j=1 to {j= inf} {-4 beta sub j} over {beta sub j sup 2}
  left ( {d phi sub j} over dr sub + right ) sub {r sub + =1}
  exp left ( {- beta sub j sup 2 (x/r sub t ) } over { N sub Re N sub roman Sc} right ) 
}
right ]
.EN

et.sqrt         491010201   286   2     100644  468       `
.EQ
x = sqrt{a sup 2 + b sup 2}+ sqrt a over b +
sqrt a sup 2 over b sup 2
.EN
.EQ
sqrt a+b over sqrt c+d
.EN
.EQ
1 over sqrt{ax sup 2 +bx+c}
.EN
.EQ
sqrt{a over b}
.EN
.EQ
sqrt 2
.EN
.EQ
sqrt x+1
.EN
.EQ
sqrt { x+y sup half}
.EN
.sp .5i
.EQ
X = sqrt{A sup 2 + B sup 2}+ sqrt A over B +
sqrt A sup 2 over B sup 2
.EN
.EQ
sqrt A+B over sqrt C+D
.EN
.EQ
1 over sqrt{AX sup 2 +BX+C}
.EN
.EQ
sqrt{A over B}
.EN
.EQ
sqrt 2
.EN
.EQ
sqrt X+1
.EN
.EQ
sqrt { X+Y sup half}
.EN
et.sub          491010206   286   2     100644  834       `
.EQ
x sub i )x
x sub j )x
x sub K )x
x sub H )x
x sub 1 )x
x sub 2 )x
.EN
.EQ
x sup i )x
x sup j )x
x sup K )x
x sup H )x
x sup 1 )x
x sup 2 )x
.EN
.EQ
(x)x
(1)x
(y)x
(K)x
(H)x
(2)x
.EN
.EQ
(x )x
(1 )x
(y )x
(K )x
(H )x
(2 )x
.EN
.sp
.EQ
x sup i sup j )x
x sup i sup 1 )x
x sup i sup 2 )x
x sup i sup K )x
x sup i sup H )x
x sup i sup n )x
x sup i sup m )x
.EN
.sp
.EQ
x sup 2 sup j )x
x sup 2 sup 1 )x
x sup 2 sup 2 )x
x sup 2 sup K )x
x sup 2 sup H )x
x sup 2 sup n )x
x sup 2 sup m )x
.EN
.EQ
x sub i sub j )x
x sub i sub 1 )x
x sub i sub 2 )x
x sub i sub K )x
x sub i sub H )x
x sub i sub n )x
x sub i sub m )x
.EN
.sp
.EQ
x sub 2 sub j )x
x sub 2 sub 1 )x
x sub 2 sub 2 )x
x sub 2 sub K )x
x sub 2 sub H )x
x sub 2 sub n )x
x sub 2 sub m )x
.EN
.EQ
gfont R
(x)xxxx
(y)xxxx
(1)xxxx
(2)xxxx
(X)xxxx
(Y)xxxx
(x+y)xxxx
(X-Y)xxxx
.EN
et.under        491010208   286   2     100644  80        `
.EQ
x under ~~~ x sub i under ~~~ {x sub i} under ~~~ {x sub i sup i} under
.EN
header          491071522   286   2     100644  90        `
.de HD
'sp .5i
..
.de FT
'bp
..
.wh 0 HD
.wh -.5i FT
.nf
.de EQ
.sp
\h'5.5i'\\$1\h'|0'
..
script          491072186   286   2     100644  338       `
eqn=../a.out

for a in et.bar et.baugh et.bcw et.blinn et.carolyn et.crc et.demo et.dg et.diac et.doug et.fat et.gfont et.greek et.int et.k92 et.lax et.mark et.matrix et.mini et.motion et.over et.prime et.scitypo et.sqrt et.sub et.under
do	$eqn -Tcan $a | troff -Tcan header - > i.$a
done
$eqn -Tcan et.coot | troff -Tcan -ms > i.et.coot