.DS
.sp
.EQ
gsize 9
.EN
.ps 9
.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
+B sub a sub i +
B sub a sup i +{B sub a}sup i + (B sub a ) sup i + (B sup i ) sub a
+B sup i 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
+c sub a sub i +
c sub a sup i +{c sub a}sup i + (c sub a ) sup i + (c sup i ) sub a
+c sup i sup x
.EN
.EQ 8
 PI sub a sub 2 +
 PI sub a sup 2 +{ PI sub a}sup 2 + ( PI sub a ) sup 2 + ( PI sup 2 ) sub a
+ PI sup 2 sup x
+ PI sub a sub i +
 PI sub a sup i +{ PI sub a}sup i + ( PI sub a ) sup i + ( PI sup i ) sub a
+ PI sup i sup x
.EN
.EQ 9
 pi sub a sub 2 +
 pi sub a sup 2 +{ pi sub a}sup 2 + ( pi sub a ) sup 2 + ( pi sup 2 ) sub a
+ pi sup 2 sup x
+ pi sub a sub i +
 pi sub a sup i +{ pi sub a}sup i + ( pi sub a ) sup i + ( pi sup i ) sub a
+ pi sup i 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
.DE
.bp
.EQ
define emx "{e sup mx}"
define mab "{m sqrt ab}"
define sa "{sqrt a}"
define sb "{sqrt b}"
int dx over {a emx - be sup -mx} ~=~
left { ~~ lpile {
     1 over {2 mab} ~log~ {sa emx - sb} over {sa emx + sb}
   above
     1 over mab ~ tanh sup -1 ( sa over sb emx ) 
   above
     -1 over mab ~ coth sup -1 ( sa over sb emx )
}
.EN
.EQ
roman erf (z) ~=~ 2 over sqrt pi int sub 0 sup z e sup -t sup 2 dt
.EN
.EQ
zeta (s) ~=~ sum from k=1 to inf k sup -s ~~~ ( Re~ s > 1)
.EN
.EQ
lim from {x -> inf} left ( 1 + 1 over x right ) sup x ~=~ e
.EN
.EQ
x dotdot ~=~ y bar
.EN
.EQ
max from 1<=n<=m log sub 2 P sub n
.EN
.EQ
union sub i ~~ union sup i ~~ union sub i sup 2 ~~ union sup {pi over 2}
~~~ union sup {i sub 2}
.EN
.EQ
int sub i ~~ int sup i ~~ int sub i sup 2 ~~ int sup {pi over 2}
~~~ int sup {i sub 2}
.EN
.bp
.ps 10
.EQ
gsize 10
.EN
.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
+B sub a sub i +
B sub a sup i +{B sub a}sup i + (B sub a ) sup i + (B sup i ) sub a
+B sup i 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
+c sub a sub i +
c sub a sup i +{c sub a}sup i + (c sub a ) sup i + (c sup i ) sub a
+c sup i sup x
.EN
.EQ 8
 PI sub a sub 2 +
 PI sub a sup 2 +{ PI sub a}sup 2 + ( PI sub a ) sup 2 + ( PI sup 2 ) sub a
+ PI sup 2 sup x
+ PI sub a sub i +
 PI sub a sup i +{ PI sub a}sup i + ( PI sub a ) sup i + ( PI sup i ) sub a
+ PI sup i sup x
.EN
.EQ 9
 pi sub a sub 2 +
 pi sub a sup 2 +{ pi sub a}sup 2 + ( pi sub a ) sup 2 + ( pi sup 2 ) sub a
+ pi sup 2 sup x
+ pi sub a sub i +
 pi sub a sup i +{ pi sub a}sup i + ( pi sub a ) sup i + ( pi sup i ) sub a
+ pi sup i 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
.bp
.EQ
define emx "{e sup mx}"
define mab "{m sqrt ab}"
define sa "{sqrt a}"
define sb "{sqrt b}"
int dx over {a emx - be sup -mx} ~=~
left { ~~ lpile {
     1 over {2 mab} ~log~ {sa emx - sb} over {sa emx + sb}
   above
     1 over mab ~ tanh sup -1 ( sa over sb emx ) 
   above
     -1 over mab ~ coth sup -1 ( sa over sb emx )
}
.EN
.EQ
roman erf (z) ~=~ 2 over sqrt pi int sub 0 sup z e sup -t sup 2 dt
.EN
.EQ
zeta (s) ~=~ sum from k=1 to inf k sup -s ~~~ ( Re~ s > 1)
.EN
.EQ
lim from {x -> inf} left ( 1 + 1 over x right ) sup x ~=~ e
.EN
.EQ
x dotdot ~=~ y bar
.EN
.EQ
max from 1<=n<=m log sub 2 P sub n
.EN
.EQ
union sub i ~~ union sup i ~~ union sub i sup 2 ~~ union sup {pi over 2}
~~~ union sup {i sub 2}
.EN
.EQ
int sub i ~~ int sup i ~~ int sub i sup 2 ~~ int sup {pi over 2}
~~~ int sup {i sub 2}
.EN
.nf
.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
