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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 32 "Introduction to Computer \+
Algebra" }}{PARA 19 "" 0 "" {TEXT -1 15 "Jacques Carette" }}{PARA 256 
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{TEXT -1 17 "Number Arithmetic" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "2^238-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"coV\\:B.b,/`RZ!*=U(
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34 "evalf(exp(Pi-gamma)/sqrt(2), 100);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#$\"_qXVoI!=XcQ+5XdE#4z\"Q'>\")Q74\"QRR`7q9$)GbqVO5c'>:[$pY\\AGr=
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" 1 "" {XPPMATH 20 "6#\"^hq+++++++++++++++++++++++++++++++++++++++++++
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!QbfPB<G(*[2QD9TYIuHoh:J+]*e%eUaP>\\_7\"zoW;pmw.i&=\\)*)G!>ZI6`K,1\\mI
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{MPLTEXT 1 0 11 "ifactor(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*\\[l)
-%!G6#\"\"#\"$;(\"\"\")-F&6#\"\"$\"$c$F*)-F&6#\"\"&\"$y\"F*)-F&6#\"\"(
\"$=\"F*)-F&6#\"#6\"#qF*)-F&6#\"#8\"#fF*)-F&6#\"#<\"#WF*)-F&6#\"#>\"#Q
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\"$>(F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Automatic taking of GC
Ds:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "25/10;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"\"&\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "3!!!/700!;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"en++kcAEwUU'*HY='*
pxd!z[KU/$Ro%yR2\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "%/mul
(i,i=701..720) - 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Amusing almost-integer from Ramanu
jan:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "frac(evalf(exp(Pi*sqrt(163)
),30));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-************!#7" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Polynomials" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "expand( (x-1)^30 );" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#,jn\"\"\"F$*&\"#IF$%\"xGF$!\"\"*&\"$N%F$)F'\"\"#F$F$*&
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'\"\"$F$F(*&\"(DH&eF$)F'\"\")F$F$*&\"(+e.#F$)F'\"\"(F$F(*&\"'vPfF$)F'
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oF$)F'\"#BF$F(*&FcoF$)F'\"#CF$F$*&FgoF$)F'\"#DF$F(*&FYF$)F'\"#EF$F$*&F
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "p := (x-1)^2 - x^2 +2*x -1;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,**$),&\"\"\"!\"\"%\"xGF)\"
\"#F)F)*$)F+F,F)F**&F,F)F+F)F)F)F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "expand(p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "q := (x-1)^2/(x^2-2*x+1);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG*&,&%\"xG\"\"\"F(!\"\"\"\"#,
(*$)F'F*F(F(*&F*F(F'F(F)F(F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 10 "normal(q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}
{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "degree( (x^2-x+3)^30
, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#g" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 37 "Computed structurally (syntactically)" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "degree( p, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Computed semantically \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "coeff( p, x, 2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Calculus" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 48 "int( 1/sqrt(1-x^2) + sin(x*y) - arctan(1/
x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,-%'arcsinG6#%\"xG\"\"\"*&
%\"yG!\"\"-%$cosG6#*&F'F(F*F(F(F+*&F'F(-%'arctanG6#*&F(F(F'F+F(F+-%#ln
GF3F(*&#F(\"\"#F(-F66#,&F(F(*&F(F(*$)F'F9F(F+F(F(F+" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 26 "int( BesselJ(1,x)^2/x,x );" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,(#\"\"\"\"\"#F%*&#F%F&F%*$)-%(BesselJG6$\"\"!%
\"xGF&F%F%!\"\"*&#F%F&F%*$)-F,6$F%F/F&F%F%F0" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 59 "int( x^alpha*(1-x)^beta,x=0..1) assuming alpha>-
1, beta>-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%BetaG6$,&%%betaG\"\"
\"F(F(,&%&alphaGF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "d
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aG\"\"\"F*F*F*-F&6#,(F)F*\"\"#F*%%betaGF*!\"\"F*-%%BetaG6$F(,&F/F*F*F*
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x)^beta,x=0..1), alpha );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$
*()%\"xG%&alphaG\"\"\"-%#lnG6#F(F*),&F*F*F(!\"\"%%betaGF*/F(;\"\"!F*" 
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" {TEXT -1 22 "Differential Equations" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "de := diff(y(x),x,x) + y(x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#deG,&&%\"yG6$%\"xGF)\"\"\"-F'6#F)F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dsolve(de);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&%$_C1G\"\"\"-%$sinGF&F+F+*&%$_C2GF+-%
$cosGF&F+F+" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "DEto
ols[expsols](de, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$-%$expG6#
*&%\"xG\"\"\"^#F)F)-F%6#*&^#!\"\"F)F(F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "de2 := x^2*diff(y(x),x,x) + y(x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$de2G,&*&)%\"xG\"\"#\"\"\"&%\"yG6$F(F(F*F*-F,6#F(F*
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ans := dsolve(de2, y(x)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ansG/-%\"yG6#%\"xG,&*(%$_C1G
\"\"\"F)#F-\"\"#-%$sinG6#,$*&F.F-*&\"\"$F.-%#lnGF(F-F-F-F-F-*(%$_C2GF-
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{MPLTEXT 1 0 18 "odetest(ans, de2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 16 "Zero-Equivalence" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "f := (x-abs(x))*(x+abs(x));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG*&,&%\"xG\"\"\"-%$absG6#F'!\"\"F(,&F'F(F)F(F(" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(f);" }}{PARA 11 "" 
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}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(f);" }}{PARA 11 "
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}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(f) assuming x::r
eal;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 33 "g := piecewise( x<0, 1, -1)^2 -1;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"gG,&*$)-%*PIECEWISEG6$7$\"\"\"2%\"xG\"\"!7$!
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0 12 "simplify(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart; PDE
tools[declare](); ON;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1Nothing~dec
laredG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 368 "sys := subs(Z=la
mbda,[diff(diff(f(r,x,phi),r),r) =\n((-2*Z*r+l^2+l)*n^2+Z^2*r^2)/r^2/n
^2*f(r,x,phi),\ndiff(diff(f(r,x,phi),r),x) =\ndiff(f(r,x,phi),r)*diff(
f(r,x,phi),x)/f(r,x,phi),\ndiff(diff(f(r,x,phi),x),x) =\n(2*x-2*x^3)/(
x^4+1-2*x^2)*diff(f(r,x,phi),x)+((l+l^2)*x^2-l-l^2+m^2)/(\nx^4+1-2*x^2
)*f(r,x,phi), diff(f(r,x,phi),phi) = I*f(r,x,phi)*m]): PDEtools[declar
e](sys);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%\"fG6%%\"rG%\"xG%$phiG
\"\"\"%9will~now~be~displayed~asGF*F%F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "sys;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&/&%\"fG6$%\"
rGF(**,&*&,(*(\"\"#\"\"\"%'lambdaGF/F(F/!\"\"*$)%\"lGF.F/F/F4F/F/)%\"n
GF.F/F/*&)F0F.F/)F(F.F/F/F/F(!\"#F6F:F&F//&F&6$F(%\"xG*(&F&6#F(F/&F&6#
F>F/F&F1/&F&6$F>F>,&*(,&*&F.F/F>F/F/*&F.F/)F>\"\"$F/F1F/,(*$)F>\"\"%F/
F/F/F/*&F.F/)F>F.F/F1F1FBF/F/*(,**&,&F4F/F2F/F/FSF/F/F4F1F2F1*$)%\"mGF
.F/F/F/FNF1F&F/F//&F&6#%$phiG*(F&F/FZF/^#F/F/" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "ans := pdsolve(sys);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$ansG<#/%\"fG*(,&*&%$_C2G\"\"\"-%*LegendrePG6%%\"lG%
\"mG%\"xGF,F,*&%$_C3GF,-%*LegendreQGF/F,F,F,,&-%+WhittakerMG6%%\"nG,&F
0F,#F,\"\"#F,,$**F>F,%'lambdaGF,F;!\"\"%\"rGF,F,F,*&%$_C1GF,-%+Whittak
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"" {MPLTEXT 1 0 18 "pdetest(ans, sys);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7&\"\"!F$F$F$" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 6 "But..." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "x/x;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Consider
 the following function " }{XPPEDIT 18 0 "h(a)" "6#-%\"hG6#%\"aG" }
{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "h := a-> int(exp(-
a*x^4)*ln(x),x=0..infinity);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Now h converges for all a such tha
t " }{XPPEDIT 18 0 "Re(a)>0" "6#2\"\"!-%#ReG6#%\"aG" }{TEXT -1 1 "." }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "hh := h(a+1)+h(1-a);" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>%#hhG,&*&#\"\"\"\"#KF(*(,***\"\"#F(%#PiGF(F-#F(
F--%#lnG6#,&%\"aGF(F(F(F(!\"\"**F-F(F.F(F-F/%&gammaGF(F5**\"\"'F(F.F(F
-F/-F16#F-F(F5*&)F.F-F(F-F/F5F(F3#F5\"\"%-%&GAMMAG6##\"\"$F?F5F(F(*&F'
F(*(,***F-F(F.F(F-F/-F16#,&F(F(F4F5F(F5**F-F(F.F(F-F/F7F(F5**F9F(F.F(F
-F/F:F(F5F<F5F(FKF>F@F5F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 108 "This mak
es no sense as there is no region of the complex plane where both inte
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "
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n imperative language and slowly drifted to become more and more funct
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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "print(solve);" }}{PARA 
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ig := ToInert(eval(g));" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#igG-%,_Inert_PROCG6*-%0_Inert_PARAM
SEQG6$-%,_Inert_NAMEG6#Q\"x6\"-F,6#Q\"yF/-%0_Inert_LOCALSEQG6#-F,6#Q\"
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F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "ig2 := eval(ig, _Iner
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1 0 21 "g2 := FromInert(ig2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g2
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O*$)-%$cosG6#*&%#PiGF4F.F4F3F4F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG }}{EXCHG }{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "?index,packages" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "9 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }