{VERSION 3 0 "SGI MIPS UNIX" "3.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 
0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 
0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
261 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE 
"" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 
2 2 0 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 
0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE 
"Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }
0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Error" 7 8 1 {CSTYLE "" -1 
-1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 
-1 0 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 
0 0 1 2 2 2 2 2 2 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple O
utput" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 
1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 40 "Integration for a logarit
hmic extension." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 251 "Let K be some differential field, and assume that we are
 able to compute anti-derivatives of elements of K whenever the anti-d
erivative is elementary (i.e. whenever the anti-derivative can be expr
essed in terms of: log, exp, and algebraic extensions)." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Let theta be an elem
ent of some differential field extension of K." }}{PARA 0 "" 0 "" 
{TEXT -1 61 "Say for example: theta = log(a) where a is some element o
f K." }}{PARA 0 "" 0 "" {TEXT -1 63 "Then theta' = a'/a.  This leads u
s to the following definition:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 16 "theta is called " }{TEXT 261 19 "logarith
mic over K " }{TEXT -1 29 "if the following things hold:" }}{PARA 0 "
" 0 "" {TEXT -1 68 "*)   theta is an element of a differential field e
xtension of K, and" }}{PARA 0 "" 0 "" {TEXT -1 42 "*)  theta' = a'/a f
or some element a of K," }}{PARA 0 "" 0 "" {TEXT -1 273 "*)   and ther
e is no b in K for which b' = a'/a   (note that theta' = a'/a, so this
 condition means that theta is not in K, in other words, theta is real
ly a new function we didn't already have inside K, so the differential
 field K(theta) really has more functions than K)." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 157 "Assume that we know how \+
to integrate elements of K. Let theta be logarithmic over K, so theta'
 = a'/a for some a in K. How to integrate elements of K(theta)?" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Suppose K
=" }{TEXT 257 1 "C" }{TEXT -1 36 "(x,exp(x)). Let theta1=exp(x), so K=
" }{TEXT 258 1 "C" }{TEXT -1 131 "(x,theta1). Now suppose we want to i
ntegrate elements of K(theta) where theta=log(x^2+exp(x)). So we want \+
to integrate elements of:" }}{PARA 0 "" 0 "" {TEXT -1 9 "K(theta)=" }
{TEXT 259 1 "C" }{TEXT -1 17 "(x,theta1,theta)=" }{TEXT 260 1 "C" }
{TEXT -1 26 "(x,exp(x),log(x^2+exp(x))." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 405 "Note that at this point we do not k
now how to integrate elements of this field K, however, Maple does kno
w, and furthermore, in a few weeks we will know as well. We will not s
how how to integrate elements of K in this worksheet. We will only sho
w how to integrate elements of K(theta) assuming that we already know \+
how to integrate elements of K. To integrate elements of K, we'll use \+
Maple's int command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "res
tart;  theta1:=exp(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'theta1G-%
$expG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "theta:=log(
x^2+theta1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaG-%#lnG6#,&*$)
%\"xG\"\"#\"\"\"F--%$expG6#F+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 307 "UseVariables:=proc(f) # this procedure replaces the functions
 theta1 and theta by variables\n\n    # indicate to Maple that these a
re globally used\n    global theta,theta1,Theta,Theta1;\n\n    # Note:
 do the substitution for the last extension (theta=Theta) first. \n   \+
 subs( theta=Theta, theta1=Theta1, f)\nend;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%-UseVariablesGR6#%\"fG6\"F(F(-%%subsG6%/%&thetaG%&The
taG/%'theta1G%'Theta1G9$F(6&F-F0F.F1F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 108 "UseFunctions:=proc(f)\n    global theta,theta1,Theta
,Theta1;\n    subs( Theta=theta, Theta1=theta1, f)\nend;   " }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%-UseFunctionsGR6#%\"fG6\"F(F(-%%subsG6%/%&
ThetaG%&thetaG/%'Theta1G%'theta1G9$F(6&F.F1F-F0F(" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 256 "We will use functions theta=exp(x), theta1=log(x^
2+exp(x)) whenever we differentiate. We use variables theta=Theta, the
ta1=Theta1 whenever we do things where Maple only accepts polynomials \+
or rational functions as input, such as factor, gcd, resultant etc." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "F:=(theta1+theta+theta^2)/
theta^3/(theta^2-x)^2+2*log(theta)+theta^2+log(x)+theta1;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"FG,,*&,(-%$expG6#%\"xG\"\"\"-%#lnG6#,&*$)F+
\"\"#F,F,F(F,F,*$)F-F3F,F,F,*&)F-\"\"$F,),&F4F,F+!\"\"F3F,F;F,*&F3F,-F
.6#F-F,F,F4F,-F.F*F,F(F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 
"f:=normal(diff(F,x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"fG*&,bq*
(-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F0-%$expG6#F.F0F0F-F0F1F0!\"#**\"\"&F0)F
(\"\"%F0F.F0F1F0F0**F/F0F1F0F.F0)F(F/F0!\"\"**\"\"(F0F.F0)F1F/F0F:F0F0
**F/F0F(F0)F.\"\"$F0F1F0F;**F/F0F(F0F.F0F>F0F;**F/F0)F(\"\"*F0F.F0F1F0
F;**\"\"'F0)F(F=F0F-F0F1F0F0**FGF0)F(F6F0F@F0F1F0F;**F/F0)F(FAF0)F.F8F
0F1F0F0**F/F0)F(\"#6F0F.F0F1F0F;**FGF0FDF0F-F0F1F0F0**FGF0FHF0F@F0F1F0
F;**F/F0FJF0FMF0F1F0F0**FAF0)F(\"\")F0F.F0F1F0F0**FAF0)F(FGF0F1F0F-F0F
;*(F7F0F1F0F@F0F0*()F(\"#5F0F@F0F1F0F;**FAF0FUF0FMF0F1F0F0**FAF0FXF0)F
.F6F0F1F0F;*(F7F0)F.FGF0F1F0F0*(FenF0F.F0F>F0F;**FAF0FUF0F-F0F>F0F0**F
AF0FXF0F@F0F>F0F;*(F7F0FMF0F>F0F0*(FAF0F>F0F-F0F;*(F/F0FLF0F@F0F;*(F8F
0FDF0F-F0F;*(\"#7F0FHF0F@F0F0*(FdoF0FJF0FMF0F;*(F8F0FLF0FinF0F0*(F8F0F
OF0F-F0F;*(FdoF0FDF0F@F0F0*(FdoF0FHF0FMF0F;*(F8F0FJF0FinF0F0*&FenF0F-F
0F;*(FAF0FUF0F@F0F0*(FAF0FXF0FMF0F;*&F7F0FinF0F0*&FenF0F1F0F;*(FGF0F1F
0F@F0F;*(FdoF0FLF0F-F0F0*(F8F0F(F0F@F0F;*(FfnF0F7F0F-F0F0*(F8F0F:F0F@F
0F;**\"#8F0F1F0F-F0F:F0F0*(FLF0F@F0F1F0F;*(F(F0FMF0F1F0F0*(FLF0F.F0F>F
0F;*(F(F0F-F0F>F0F0**F8F0FLF0F.F0F1F0F0F0**F+F0F7F0),&*$F:F0F;F.F0FAF0
F.F0F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "UseVariables(f);
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,bq*&)%&ThetaG\"\"'\"\"\")%\"xG
\"\"%F)!\"$*(\"#7F))F'\"\"$F))F+\"\"#F)F)*(F,F)F'F))F+F1F)!\"\"*(\"#5F
))F'F,F)F2F)F)*(F,F))F'F3F)F5F)F6*&)F'F8F)%'Theta1GF)F6*(F1F))F>F3F)F2
F)F6*(F(F)F>F)F5F)F6*&F9F))F+\"\"&F)F)*(F3F)F0F)F5F)F6*(F,F))F'\"\"*F)
F2F)F6*(F/F))F'\"\"(F)F5F)F)*(F/F))F'FDF)F*F)F6*(F,F)F0F)FCF)F)*(F,F))
F'\"#6F)F2F)F6*(F/F)FGF)F5F)F)*(F/F)FJF)F*F)F6*(F,F)FMF)FCF)F)*&F=F)F2
F)F6*(F1F))F'\"\")F)F5F)F)**F3F)F'F)F2F)F>F)F6**FDF)F9F)F+F)F>F)F)**F3
F)F>F)F+F)F;F)F6**FKF)F+F)F@F)F;F)F)**F3F)F'F)F5F)F>F)F6**F3F)F'F)F+F)
F@F)F6**F3F)FGF)F+F)F>F)F6**F(F)FJF)F2F)F>F)F)**F(F)FMF)F5F)F>F)F6**F3
F)F0F)F*F)F>F)F)**F3F)FPF)F+F)F>F)F6**F(F)FGF)F2F)F>F)F)**F(F)FJF)F5F)
F>F)F6**F3F)FMF)F*F)F>F)F)**F1F)FWF)F+F)F>F)F)**F1F)F&F)F>F)F2F)F6*(F9
F)F>F)F5F)F)*(F=F)F5F)F>F)F6**F1F)FWF)F*F)F>F)F)**F1F)F&F)FCF)F>F)F6*(
F9F))F+F(F)F>F)F)*(F=F)F+F)F@F)F6**F1F)FWF)F2F)F@F)F)**F1F)F&F)F5F)F@F
)F6*(F9F)F*F)F@F)F)**\"#8F)F>F)F2F)F;F)F)*(F0F)F5F)F>F)F6*(F'F)F*F)F>F
)F)*(F0F)F+F)F@F)F6*(F'F)F2F)F@F)F)**F,F)F0F)F+F)F>F)F)F)**,&*$F2F)F)F
>F)F)F9F)),&*$F;F)F6F+F)F1F)F+F)F6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "sqrfree(denom(%),Theta);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$,$*&,&*$)%\"xG\"\"#\"\"\"F+%'Theta1GF+F+F)F+!\"\"7$7$
%&ThetaG\"\"%7$,&*$)F0F*F+F+F)F-\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "%[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$%&ThetaG
\"\"%7$,&*$)F%\"\"#\"\"\"F,%\"xG!\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "d:=Theta; pole_order:=4;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"dG%&ThetaG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+po
le_orderG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "A1:=add(c
[i]*Theta^i,i=0..degree(d,Theta)-1)/d^(pole_order-1);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#A1G*&&%\"cG6#\"\"!\"\"\"*$)%&ThetaG\"\"$F*!\"\"
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We're going to have to differ
entiate A1, so:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A1:=UseF
unctions(A1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G*&&%\"cG6#\"\"!
\"\"\"*$)-%#lnG6#,&*$)%\"xG\"\"#F*F*-%$expG6#F3F*\"\"$F*!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f-diff(A1,x);" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#,&*&,bq*(-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F/-%$expG6
#F-F/F/F,F/F0F/!\"#**\"\"&F/)F'\"\"%F/F-F/F0F/F/**F.F/F0F/F-F/)F'F.F/!
\"\"**\"\"(F/F-F/)F0F.F/F9F/F/**F.F/F'F/)F-\"\"$F/F0F/F:**F.F/F'F/F-F/
F=F/F:**F.F/)F'\"\"*F/F-F/F0F/F:**\"\"'F/)F'F<F/F,F/F0F/F/**FFF/)F'F5F
/F?F/F0F/F:**F.F/)F'F@F/)F-F7F/F0F/F/**F.F/)F'\"#6F/F-F/F0F/F:**FFF/FC
F/F,F/F0F/F/**FFF/FGF/F?F/F0F/F:**F.F/FIF/FLF/F0F/F/**F@F/)F'\"\")F/F-
F/F0F/F/**F@F/)F'FFF/F0F/F,F/F:*(F6F/F0F/F?F/F/*()F'\"#5F/F?F/F0F/F:**
F@F/FTF/FLF/F0F/F/**F@F/FWF/)F-F5F/F0F/F:*(F6F/)F-FFF/F0F/F/*(FZF/F-F/
F=F/F:**F@F/FTF/F,F/F=F/F/**F@F/FWF/F?F/F=F/F:*(F6F/FLF/F=F/F/*(F@F/F=
F/F,F/F:*(F.F/FKF/F?F/F:*(F7F/FCF/F,F/F:*(\"#7F/FGF/F?F/F/*(FcoF/FIF/F
LF/F:*(F7F/FKF/FhnF/F/*(F7F/FNF/F,F/F:*(FcoF/FCF/F?F/F/*(FcoF/FGF/FLF/
F:*(F7F/FIF/FhnF/F/*&FZF/F,F/F:*(F@F/FTF/F?F/F/*(F@F/FWF/FLF/F:*&F6F/F
hnF/F/*&FZF/F0F/F:*(FFF/F0F/F?F/F:*(FcoF/FKF/F,F/F/*(F7F/F'F/F?F/F:*(F
enF/F6F/F,F/F/*(F7F/F9F/F?F/F:**\"#8F/F0F/F,F/F9F/F/*(FKF/F?F/F0F/F:*(
F'F/FLF/F0F/F/*(FKF/F-F/F=F/F:*(F'F/F,F/F=F/F/**F7F/FKF/F-F/F0F/F/F/**
F*F/F6F/),&*$F9F/F:F-F/F@F/F-F/F:F/*&*(F@F/&%\"cG6#\"\"!F/,&F-F.F0F/F/
F/*&F6F/F*F/F:F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "UseVari
ables(%);  # because we're going to do rem" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#,&*&,bq*&)%&ThetaG\"\"'\"\"\")%\"xG\"\"%F*!\"$*(\"#7F*)
F(\"\"$F*)F,\"\"#F*F**(F-F*F(F*)F,F2F*!\"\"*(\"#5F*)F(F-F*F3F*F**(F-F*
)F(F4F*F6F*F7*&)F(F9F*%'Theta1GF*F7*(F2F*)F?F4F*F3F*F7*(F)F*F?F*F6F*F7
*&F:F*)F,\"\"&F*F**(F4F*F1F*F6F*F7*(F-F*)F(\"\"*F*F3F*F7*(F0F*)F(\"\"(
F*F6F*F**(F0F*)F(FEF*F+F*F7*(F-F*F1F*FDF*F**(F-F*)F(\"#6F*F3F*F7*(F0F*
FHF*F6F*F**(F0F*FKF*F+F*F7*(F-F*FNF*FDF*F**&F>F*F3F*F7*(F2F*)F(\"\")F*
F6F*F***F4F*F(F*F3F*F?F*F7**FEF*F:F*F,F*F?F*F***F4F*F?F*F,F*F<F*F7**FL
F*F,F*FAF*F<F*F***F4F*F(F*F6F*F?F*F7**F4F*F(F*F,F*FAF*F7**F4F*FHF*F,F*
F?F*F7**F)F*FKF*F3F*F?F*F***F)F*FNF*F6F*F?F*F7**F4F*F1F*F+F*F?F*F***F4
F*FQF*F,F*F?F*F7**F)F*FHF*F3F*F?F*F***F)F*FKF*F6F*F?F*F7**F4F*FNF*F+F*
F?F*F***F2F*FXF*F,F*F?F*F***F2F*F'F*F?F*F3F*F7*(F:F*F?F*F6F*F**(F>F*F6
F*F?F*F7**F2F*FXF*F+F*F?F*F***F2F*F'F*FDF*F?F*F7*(F:F*)F,F)F*F?F*F**(F
>F*F,F*FAF*F7**F2F*FXF*F3F*FAF*F***F2F*F'F*F6F*FAF*F7*(F:F*F+F*FAF*F**
*\"#8F*F?F*F3F*F<F*F**(F1F*F6F*F?F*F7*(F(F*F+F*F?F*F**(F1F*F,F*FAF*F7*
(F(F*F3F*FAF*F***F-F*F1F*F,F*F?F*F*F***,&*$F3F*F*F?F*F*F:F*),&*$F<F*F7
F,F*F2F*F,F*F7F**&*(F2F*&%\"cG6#\"\"!F*,&F,F4F?F*F*F**&F:F*FfpF*F7F*" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "rem(numer(normal(%*d^pole
_order)),d,Theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&&%\"cG6#\"\"
!\"\"\")%\"xG\"\"&F)\"\"'*(\"\"$F))%'Theta1G\"\"#F))F+F2F)!\"\"*(F-F)F
1F))F+F/F)F4**F/F)F%F))F+\"\"%F)F1F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "coeffs(%,Theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
**&&%\"cG6#\"\"!\"\"\")%\"xG\"\"&F)\"\"'*(\"\"$F))%'Theta1G\"\"#F))F+F
2F)!\"\"*(F-F)F1F))F+F/F)F4**F/F)F%F))F+\"\"%F)F1F)F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "solve(\{%\},\{seq(c[i],i=0..degree(
d,Theta)-1)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/&%\"cG6#\"\"!*&%
'Theta1G\"\"\"*$)%\"xG\"\"#F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "A1:=subs(%,A1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#A1G*&%'Theta1G\"\"\"*&)%\"xG\"\"#F')-%#lnG6#,&*$F)F'F'-%$expG6#F*F'\"
\"$F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "A1:=UseFuncti
ons(A1); # because we're going to use diff" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A1G*&-%$expG6#%\"xG\"\"\"*&)F)\"\"#F*)-%#lnG6#,&*$F,
F*F*F&F*\"\"$F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "ne
w_f:=normal(f-diff(A1,x)); # now use sqrfree denominator, so use varia
bles to avoid error messages" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&new
_fG*&,`r*$)%\"xG\"\"&\"\"\"\"\"%**\"\"'F+F(F+)-%#lnG6#,&*$)F)\"\"#F+F+
-%$expG6#F)F+F,F+F7F+F+**F,F+)F)\"\"$F+F7F+)F0F6F+!\"\"**F6F+F;F+)F7F6
F+F=F+F>**F6F+F;F+F0F+F@F+F+**F6F+F;F+)F0\"\")F+F7F+F+**F.F+)F)F,F+)F0
F.F+F7F+F>*(F(F+)F0F<F+F7F+F>**F.F+FFF+F7F+F=F+F+**F<F+F@F+)F0F*F+F)F+
F+**\"\"*F+F@F+FIF+F5F+F>*(F)F+F@F+FGF+F>**F<F+F5F+F@F+F/F+F+*(FGF+F;F
+F7F+F>**F<F+F/F+FFF+F7F+F+**F.F+FLF+F7F+F5F+F+*(F(F+)F0FNF+F7F+F+**F<
F+FFF+)F0\"\"(F+F@F+F>*()F)FDF+FIF+F7F+F>**F6F+)F)F.F+F=F+F7F+F>**F<F+
FfnF+FWF+F7F+F>**F<F+)F)FXF+FLF+F7F+F+*(F;F+FUF+F@F+F+**F<F+F(F+FLF+F@
F+F+**F6F+F(F+F=F+F7F+F>*(FfnF+FIF+F@F+F>**F.F+F@F+F/F+F)F+F>**F.F+F@F
+F=F+F5F+F+*(F,F+F(F+F0F+F+*(\"#7F+FFF+F=F+F>*(F6F+FFF+F7F+F+*(F,F+FFF
+FCF+F+*(FboF+F(F+FGF+F>*(FboF+FfnF+F/F+F+*(F,F+FinF+F=F+F>*(F6F+F@F+F
GF+F+*(F6F+F(F+F=F+F+*(F,F+FinF+F/F+F>*&FFF+FUF+F+*(F<F+F(F+FWF+F>*(F<
F+FfnF+FLF+F+*&FinF+FIF+F>*(F,F+FFF+)F0\"#5F+F+*(FboF+F(F+FCF+F>*(FboF
+FfnF+FGF+F+*(FapF+FFF+FIF+F>**F6F+F0F+F;F+F7F+F+*(FUF+F5F+F7F+F+**F<F
+FWF+F;F+F7F+F>**F<F+FLF+FFF+F7F+F+**F6F+FGF+F7F+F5F+F+**F.F+F/F+F7F+F
;F+F>**F6F+F`pF+F;F+F7F+F+**F.F+FCF+FFF+F7F+F>**F.F+FGF+F(F+F7F+F+**F6
F+F/F+FfnF+F7F+F>**\"#BF+FIF+F;F+F7F+F>**F*F+F0F+FFF+F7F+F+F+**FIF+F3F
+),&*$F=F+F+F)F>F<F+F;F+F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "sqrfree(denom(UseVariables(%)),Theta);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$,$*&,&*$)%\"xG\"\"#\"\"\"F+%'Theta1GF+F+)F)\"\"$F+!\"
\"7$7$%&ThetaGF.7$,&*$)F2F*F+F+F)F/F." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "%[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$%&ThetaG
\"\"$7$,&*$)F%\"\"#\"\"\"F,%\"xG!\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "d:=Theta*(Theta^2-x);  pole_order:=3;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"dG*&%&ThetaG\"\"\",&*$)F&\"\"#F'F'%\"xG!\"\"F'
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+pole_orderG\"\"$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "A2:=add(c[i]*Theta^i,i=0..degree(d,
Theta)-1)/d^(pole_order-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G*
&,(&%\"cG6#\"\"!\"\"\"*&&F(6#F+F+%&ThetaGF+F+*&&F(6#\"\"#F+)F/F3F+F+F+
*&F4F+),&*$F4F+F+%\"xG!\"\"F3F+F:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "A2:=UseFunctions(A2); # prepare for diff" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#A2G*&,(&%\"cG6#\"\"!\"\"\"*&&F(6#F+F+-%#lnG6
#,&*$)%\"xG\"\"#F+F+-%$expG6#F5F+F+F+*&&F(6#F6F+)F/F6F+F+F+*&F=F+),&*$
F=F+F+F5!\"\"F6F+FB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "new_
f-diff(A2,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,**&,`r*$)%\"xG\"\"&
\"\"\"\"\"%**\"\"'F*F'F*)-%#lnG6#,&*$)F(\"\"#F*F*-%$expG6#F(F*F+F*F6F*
F***F+F*)F(\"\"$F*F6F*)F/F5F*!\"\"**F5F*F:F*)F6F5F*F<F*F=**F5F*F:F*F/F
*F?F*F***F5F*F:F*)F/\"\")F*F6F*F***F-F*)F(F+F*)F/F-F*F6F*F=*(F'F*)F/F;
F*F6F*F=**F-F*FEF*F6F*F<F*F***F;F*F?F*)F/F)F*F(F*F***\"\"*F*F?F*FHF*F4
F*F=*(F(F*F?F*FFF*F=**F;F*F4F*F?F*F.F*F**(FFF*F:F*F6F*F=**F;F*F.F*FEF*
F6F*F***F-F*FKF*F6F*F4F*F**(F'F*)F/FMF*F6F*F***F;F*FEF*)F/\"\"(F*F?F*F
=*()F(FCF*FHF*F6F*F=**F5F*)F(F-F*F<F*F6F*F=**F;F*FenF*FVF*F6F*F=**F;F*
)F(FWF*FKF*F6F*F**(F:F*FTF*F?F*F***F;F*F'F*FKF*F?F*F***F5F*F'F*F<F*F6F
*F=*(FenF*FHF*F?F*F=**F-F*F?F*F.F*F(F*F=**F-F*F?F*F<F*F4F*F**(F+F*F'F*
F/F*F**(\"#7F*FEF*F<F*F=*(F5F*FEF*F6F*F**(F+F*FEF*FBF*F**(FaoF*F'F*FFF
*F=*(FaoF*FenF*F.F*F**(F+F*FhnF*F<F*F=*(F5F*F?F*FFF*F**(F5F*F'F*F<F*F*
*(F+F*FhnF*F.F*F=*&FEF*FTF*F**(F;F*F'F*FVF*F=*(F;F*FenF*FKF*F**&FhnF*F
HF*F=*(F+F*FEF*)F/\"#5F*F**(FaoF*F'F*FBF*F=*(FaoF*FenF*FFF*F**(F`pF*FE
F*FHF*F=**F5F*F/F*F:F*F6F*F**(FTF*F4F*F6F*F***F;F*FVF*F:F*F6F*F=**F;F*
FKF*FEF*F6F*F***F5F*FFF*F6F*F4F*F***F-F*F.F*F6F*F:F*F=**F5F*F_pF*F:F*F
6F*F***F-F*FBF*FEF*F6F*F=**F-F*FFF*F'F*F6F*F***F5F*F.F*FenF*F6F*F=**\"
#BF*FHF*F:F*F6F*F=**F)F*F/F*FEF*F6F*F*F***FHF*F2F*),&*$F<F*F*F(F=F;F*F
:F*F=F**&,&*&*&&%\"cG6#F*F*,&F(F5F6F*F*F*F2F=F**&**F5F*&Fjq6#F5F*F/F*F
\\rF*F*F2F=F*F**&F<F*)FcqF5F*F=F=*&*(F5F*,(&Fjq6#\"\"!F**&FiqF*F/F*F**
&F_rF*F<F*F*F*F\\rF*F**(FHF*FbrF*F2F*F=F**&*(F5F*FerF*,&*&*&F/F*F\\rF*
F*F2F=F5F*F=F*F**&F<F*FbqF*F=F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 44 "numer(normal(UseVariables(%)*d^pole_order));" }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#,`t*()%&ThetaG\"\"&\"\"\")%\"xG\"\"#F(%'Theta1GF(\"
\"'*(\"\"%F(&%\"cG6#\"\"!F()F*F'F(!\"\"*&F/F(F4F(F(*(F+F()F&F-F()F,F+F
(F(**\"#7F(F0F()F*F/F()F&F+F(F(**F+F(F0F(F<F(F,F(F5*(F+F(F<F(F,F(F(*(F
;F(F<F(F=F(F5*,F-F(F0F()F*\"\"$F(F,F(F=F(F(**F/F(FBF(F,F(F=F(F5*,F+F(F
&F(F0F(FBF(F,F(F5*(F8F(F,F(FBF(F5**FCF()F&F/F(F<F(F,F(F(**F-F(F=F(F<F(
F,F(F(*(F+F(F=F(F4F(F(*(F4F()F&FCF(F,F(F5**F-F(F<F(F8F(F,F(F5**F+F(FBF
()F&\"\")F(F,F(F(**F+F(FBF(F&F(F9F(F(**F+F(FBF(F9F(F=F(F5**F-F(F4F(FHF
(F,F(F(**FCF(F9F(F%F(F*F(F(**\"\"*F(F9F(FLF(F)F(F5*(F*F(F9F(F8F(F5**FC
F(F)F(F9F(FHF(F(*(F4F()F&FVF(F,F(F(**FCF(F<F()F&\"\"(F(F9F(F5*()F*FPF(
FLF(F,F(F5**F+F()F*F-F(F=F(F,F(F5**FCF(F[oF(FfnF(F,F(F5**FCF()F*FgnF(F
%F(F,F(F(*(FBF(FZF(F9F(F(**FCF(F4F(F%F(F9F(F(**F+F(F4F(F=F(F,F(F5*(F[o
F(FLF(F9F(F5**F-F(F9F(FHF(F*F(F5**F-F(F9F(F=F(F)F(F(*(F/F(F4F(F&F(F(*(
F/F(F<F(FOF(F(*(F;F(F4F(F8F(F5*(F;F(F[oF(FHF(F(*(F/F(F^oF(F=F(F5*(F/F(
F^oF(FHF(F5*&F<F(FZF(F(*(FCF(F4F(FfnF(F5*(FCF(F[oF(F%F(F(*&F^oF(FLF(F5
*(F/F(F<F()F&\"#5F(F(*(F;F(F4F(FOF(F5*(F;F(F[oF(F8F(F(*(FapF(F<F(FLF(F
5**FapF(FLF(F<F(&F16#F(F(F(**F+F(F&F(F4F(FfpF(F5**FPF(FHF(F<F(&F16#F+F
(F(**F+F(F&F(F4F(F0F(F5**F+F(F=F(F4F(FfpF(F5**F+F(FLF(F4F(FjpF(F5*,F'F
(FLF(FBF(F,F(FfpF(F(**F&F(F<F(F,F(FfpF(F5*,F/F(FHF(FBF(F,F(FjpF(F(*,F+
F(F=F(FBF(FfpF(F,F(F5*,F+F(FLF(FBF(FjpF(F,F(F5**F+F(F&F(FBF(F,F(F(*(FZ
F(F)F(F,F(F(**FCF(FfnF(FBF(F,F(F5**FCF(F%F(F<F(F,F(F(**F+F(F8F(F,F(F)F
(F(**F-F(FHF(F,F(FBF(F5**F+F(F`pF(FBF(F,F(F(**F-F(FOF(F<F(F,F(F5**F-F(
F8F(F4F(F,F(F(**F+F(FHF(F[oF(F,F(F5**\"#BF(FLF(FBF(F,F(F5**F'F(F&F(F<F
(F,F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "This should be divisib
le by d, in order to make the pole order of f-diff(A2,x) smaller." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rem(%,d,Theta);" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#,.*&,8*&&%\"cG6#\"\"!\"\"\")%\"xG\"\"%F+\"#7*(F
.F+%'Theta1GF+)F-\"\"$F+!\"\"**\"\"'F+F'F+F2F+F1F+F+*(\"\")F+)F-\"\"&F
+&F(6#\"\"#F+F+*&F/F+F,F+F4**F.F+F,F+F1F+F;F+F+*(F=F+)F1F=F+)F-F=F+F+*
(F=F+F9F+&F(6#F+F+F4*(F=F+F,F+F1F+F+*&F=F+F9F+F+**F=F+F2F+F1F+FDF+F4F+
)%&ThetaGF=F+F+*&,6*&F1F+F2F+F=**F.F+F,F+F1F+FDF+F+*(F8F+F9F+FDF+F+*(F
.F+F2F+FAF+F4**F=F+F'F+F2F+F1F+F4*(F=F+)F-F6F+F;F+F4*(F=F+F'F+F9F+F4*&
F6F+F9F+F4*(F/F+F,F+F1F+F4**F=F+F,F+F1F+F;F+F4F+FJF+F+*(F.F+F'F+F9F+F4
*&F.F+F9F+F+**F=F+F'F+F,F+F1F+F4*(F=F+F,F+F1F+F+" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 16 "coeffs(%,Theta);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6%,**&&%\"cG6#\"\"!\"\"\")%\"xG\"\"&F)!\"%*&\"\"%F)F*F)F)
**\"\"#F)F%F))F+F/F)%'Theta1GF)!\"\"*(F1F)F2F)F3F)F),6*&F3F))F+\"\"$F)
F1**F/F)F2F)F3F)&F&6#F)F)F)*(\"\")F)F*F)F;F)F)*(F/F)F8F))F3F1F)F4**F1F
)F%F)F8F)F3F)F4*(F1F))F+\"\"'F)&F&6#F1F)F4*(F1F)F%F)F*F)F4*&FDF)F*F)F4
*(\"#7F)F2F)F3F)F4**F1F)F2F)F3F)FEF)F4,8*&F%F)F2F)FJ*(F/F)F3F)F8F)F4**
FDF)F%F)F8F)F3F)F)*(F>F)F*F)FEF)F)*&FJF)F2F)F4**F/F)F2F)F3F)FEF)F)*(F1
F)F@F))F+F1F)F)*(F1F)F*F)F;F)F4*(F1F)F2F)F3F)F)*&F1F)F*F)F)**F1F)F8F)F
3F)F;F)F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "solve(\{%\},\{
seq(c[i],i=0..degree(d,Theta)-1)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#<%/&%\"cG6#\"\"!\"\"\"/&F&6#F)*&,&%\"xGF)%'Theta1GF)F)F/!\"\"/&F&6#
\"\"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "A2:=UseFunctions
(subs(%,A2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G*&,&\"\"\"F'*&*
&,&%\"xGF'-%$expG6#F+F'F'-%#lnG6#,&*$)F+\"\"#F'F'F,F'F'F'F+!\"\"F'F'*&
)F/F5F'),&*$F8F'F'F+F6F5F'F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 59 "new_f:=normal(new_f-diff(A2,x)); # pole orders should drop." }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&new_fG*&,do*&)%\"xG\"\"'\"\"\"-%#ln
G6#,&*$)F)\"\"#F+F+-%$expG6#F)F+F+\"\"%*(F2F+)F,\"\"$F+)F3F2F+F+*&F(F+
)F,F2F+F+*(F6F+F(F+F8F+F+**F2F+F8F+F1F+F3F+F+*(F<F+)F)\"\"(F+F3F+F+*(F
<F+)F)\"\"&F+F:F+F+**F2F+FCF+F,F+F3F+F+**F2F+FCF+F8F+F3F+F+*()F)F6F+F3
F+F<F+F+**F9F+F)F+F:F+F<F+F+**F9F+F,F+)F)F9F+F3F+!\"\"**F9F+F,F+F)F+F:
F+FL**F2F+)F,FDF+FKF+F3F+F+**F6F+F8F+FHF+F3F+FL**F2F+)F,FAF+FKF+F3F+F+
**F6F+FOF+FHF+F3F+FL*()F,F*F+F3F+F1F+F+**F2F+)F,F6F+F3F+FKF+FL*(FUF+FC
F+F3F+F+**F2F+FWF+F(F+F3F+FL*(FUF+FKF+F:F+F+**F2F+FWF+FHF+F:F+FL*&F:F+
F1F+FL*(F6F+FOF+FHF+F+*(\"\")F+F8F+FCF+FL*(F6F+FRF+FHF+F+*(FinF+FOF+FC
F+FL*&FUF+FHF+F+*(F2F+FWF+FCF+FL*(F2F+F3F+FKF+FL**F*F+F3F+F1F+F<F+F+*(
F8F+FKF+F3F+FL*(F,F+FHF+F3F+F+*(F8F+F)F+F:F+FL*(F,F+F1F+F:F+F+F+**F<F+
FKF+),&*$F<F+F+F)FLF2F+F/F+FL" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 27 "denom(UseVariables(new_f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*
*)%&ThetaG\"\"#\"\"\")%\"xG\"\"$F'),&*$F$F'!\"\"F)F'F&F',&*$)F)F&F'F'%
'Theta1GF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sqrfree(%,T
heta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*&)%\"xG\"\"$\"\"\",&*$)F&
\"\"#F(F(%'Theta1GF(F(7$7$%&ThetaGF,7$,&*$)F0F,F(F(F&!\"\"F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "%[2];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$7$%&ThetaG\"\"#7$,&*$)F%F&\"\"\"F+%\"xG!\"\"F&" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 163 "We don't care about the first par
t of sqrfree(%,Theta), we only care about the second part (%[2]) that \+
contains Theta, not the part that contains only Theta1 or x." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "d:=Theta*(Theta^2-x); # loca
tion of the poles of order 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG
*&%&ThetaG\"\"\",&*$)F&\"\"#F'F'%\"xG!\"\"F'" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "pole_order:=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%+pole_orderG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "A3
:=add(c[i]*Theta^i,i=0..degree(d,Theta)-1)/d^(pole_order-1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#A3G*&,(&%\"cG6#\"\"!\"\"\"*&&F(6#F+F+%&Th
etaGF+F+*&&F(6#\"\"#F+)F/F3F+F+F+*&F/F+,&*$F4F+F+%\"xG!\"\"F+F9" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "new_f - diff( UseFunctions(A
3) ,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,**&,do*&)%\"xG\"\"'\"\"\"-
%#lnG6#,&*$)F(\"\"#F*F*-%$expG6#F(F*F*\"\"%*(F1F*)F+\"\"$F*)F2F1F*F**&
F'F*)F+F1F*F**(F5F*F'F*F7F*F***F1F*F7F*F0F*F2F*F**(F;F*)F(\"\"(F*F2F*F
**(F;F*)F(\"\"&F*F9F*F***F1F*FBF*F+F*F2F*F***F1F*FBF*F7F*F2F*F**()F(F5
F*F2F*F;F*F***F8F*F(F*F9F*F;F*F***F8F*F+F*)F(F8F*F2F*!\"\"**F8F*F+F*F(
F*F9F*FK**F1F*)F+FCF*FJF*F2F*F***F5F*F7F*FGF*F2F*FK**F1F*)F+F@F*FJF*F2
F*F***F5F*FNF*FGF*F2F*FK*()F+F)F*F2F*F0F*F***F1F*)F+F5F*F2F*FJF*FK*(FT
F*FBF*F2F*F***F1F*FVF*F'F*F2F*FK*(FTF*FJF*F9F*F***F1F*FVF*FGF*F9F*FK*&
F9F*F0F*FK*(F5F*FNF*FGF*F**(\"\")F*F7F*FBF*FK*(F5F*FQF*FGF*F**(FhnF*FN
F*FBF*FK*&FTF*FGF*F**(F1F*FVF*FBF*FK*(F1F*F2F*FJF*FK**F)F*F2F*F0F*F;F*
F**(F7F*FJF*F2F*FK*(F+F*FGF*F2F*F**(F7F*F(F*F9F*FK*(F+F*F0F*F9F*F*F***
F;F*FJF*),&*$F;F*F*F(FKF1F*F.F*FKF**&,&*&*&&%\"cG6#F*F*,&F(F1F2F*F*F*F
.FKF**&**F1F*&F\\p6#F1F*F+F*F^pF*F*F.FKF*F**&F+F*FeoF*FKFK*&*&,(&F\\p6
#\"\"!F**&F[pF*F+F*F**&FapF*F;F*F*F*F^pF*F**(F;F*FeoF*F.F*FKF**&*&FfpF
*,&*&*&F+F*F^pF*F*F.FKF1F*FKF*F**&F+F*FdoF*FKF*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 24 "normal(UseVariables(%));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#,$*&,dq*&&%\"cG6#\"\"!\"\"\")%\"xG\"\"&F+\"\"#*(F'F+)F-
\"\"%F+%'Theta1GF+F+**\"\"$F+%&ThetaGF+)F-F5F+F3F+F+**F/F+)F6F2F+F1F+)
F3F/F+F+**\"\"'F+F3F+)F-F/F+)F6F/F+!\"\"*()F6F5F+F7F+F3F+F+*(F6F+F1F+F
3F+F?*(FAF+F-F+F:F+F+*(F6F+F=F+F:F+F?**F5F+F-F+F:F+F>F+F?*(F1F+F3F+F>F
+F?**F/F+F,F+FAF+F3F+F?**F/F+F,F+F6F+F3F+F?*(F>F+F,F+F:F+F?*(F>F+)F-\"
\"(F+F3F+F?**F5F+F6F+F-F+F:F+F+**F/F+)F6F.F+F7F+F3F+F?**F2F+FAF+F1F+F3
F+F+**F/F+)F6FLF+F7F+F3F+F?**F2F+FOF+F1F+F3F+F+*()F6F<F+F3F+F=F+F?**F/
F+F9F+F3F+F7F+F+*(FUF+F,F+F3F+F?**F/F+F9F+)F-F<F+F3F+F+*(FUF+F7F+F:F+F
?**F/F+FAF+F=F+F3F+F?**F2F+FAF+F1F+&F(6#F+F+F?**F6F+F7F+F'F+F3F+F+**F>
F+F7F+FgnF+F3F+F+**FAF+F7F+&F(6#F/F+F3F+F+*(F6F+F,F+F'F+F+*(F>F+F,F+Fg
nF+F+*(FAF+F,F+F\\oF+F+*,F5F+F7F+F'F+F3F+F>F+F?**F<F+F1F+F'F+F>F+F?**F
/F+F>F+F,F+F\\oF+F?**F/F+F9F+F1F+F\\oF+F?*,F/F+FAF+F7F+F3F+FgnF+F?**F9
F+F7F+F3F+F\\oF+F?**F>F+F1F+F3F+F\\oF+F?*(F/F+FAF+F:F+F?*(F/F+F9F+F,F+
F+*(\"\")F+FOF+F,F+F+*&FUF+F1F+F?*(F[pF+FAF+F,F+F+*(F2F+FRF+F1F+F?*(F2
F+FOF+F1F+F?*(F2F+FYF+FAF+F?*&FYF+F>F+F?*(F2F+FYF+F6F+F?*&F:F+F=F+F+*(
F/F+F3F+F7F+F+F+**F>F+F7F+),&*$F>F+F?F-F+F/F+,&*$F=F+F+F3F+F+F?F?" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "normal(%*d^pole_order);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&,dq*&&%\"cG6#\"\"!\"\"\")%\"xG\"\"
&F+\"\"#*(F'F+)F-\"\"%F+%'Theta1GF+F+**\"\"$F+%&ThetaGF+)F-F5F+F3F+F+*
*F/F+)F6F2F+F1F+)F3F/F+F+**\"\"'F+F3F+)F-F/F+)F6F/F+!\"\"*()F6F5F+F7F+
F3F+F+*(F6F+F1F+F3F+F?*(FAF+F-F+F:F+F+*(F6F+F=F+F:F+F?**F5F+F-F+F:F+F>
F+F?*(F1F+F3F+F>F+F?**F/F+F,F+FAF+F3F+F?**F/F+F,F+F6F+F3F+F?*(F>F+F,F+
F:F+F?*(F>F+)F-\"\"(F+F3F+F?**F5F+F6F+F-F+F:F+F+**F/F+)F6F.F+F7F+F3F+F
?**F2F+FAF+F1F+F3F+F+**F/F+)F6FLF+F7F+F3F+F?**F2F+FOF+F1F+F3F+F+*()F6F
<F+F3F+F=F+F?**F/F+F9F+F3F+F7F+F+*(FUF+F,F+F3F+F?**F/F+F9F+)F-F<F+F3F+
F+*(FUF+F7F+F:F+F?**F/F+FAF+F=F+F3F+F?**F2F+FAF+F1F+&F(6#F+F+F?**F6F+F
7F+F'F+F3F+F+**F>F+F7F+FgnF+F3F+F+**FAF+F7F+&F(6#F/F+F3F+F+*(F6F+F,F+F
'F+F+*(F>F+F,F+FgnF+F+*(FAF+F,F+F\\oF+F+*,F5F+F7F+F'F+F3F+F>F+F?**F<F+
F1F+F'F+F>F+F?**F/F+F>F+F,F+F\\oF+F?**F/F+F9F+F1F+F\\oF+F?*,F/F+FAF+F7
F+F3F+FgnF+F?**F9F+F7F+F3F+F\\oF+F?**F>F+F1F+F3F+F\\oF+F?*(F/F+FAF+F:F
+F?*(F/F+F9F+F,F+F+*(\"\")F+FOF+F,F+F+*&FUF+F1F+F?*(F[pF+FAF+F,F+F+*(F
2F+FRF+F1F+F?*(F2F+FOF+F1F+F?*(F2F+FYF+FAF+F?*&FYF+F>F+F?*(F2F+FYF+F6F
+F?*&F:F+F=F+F+*(F/F+F3F+F7F+F+F+*&F7F+,&*$F=F+F+F3F+F+F?F?" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "numer(%);" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#,dq*&&%\"cG6#\"\"!\"\"\")%\"xG\"\"&F)!\"#*(F%F))F+\"
\"%F)%'Theta1GF)!\"\"**\"\"$F)%&ThetaGF))F+F4F)F1F)F2**\"\"#F))F5F0F)F
/F))F1F8F)F2**\"\"'F)F1F))F+F8F))F5F8F)F)*()F5F4F)F6F)F1F)F2*(F5F)F/F)
F1F)F)*(F@F)F+F)F:F)F2*(F5F)F=F)F:F)F)**F4F)F+F)F:F)F>F)F)*(F/F)F1F)F>
F)F)**F8F)F*F)F@F)F1F)F)**F8F)F*F)F5F)F1F)F)*(F>F)F*F)F:F)F)*(F>F))F+
\"\"(F)F1F)F)**F4F)F5F)F+F)F:F)F2**F8F))F5F,F)F6F)F1F)F)**F0F)F@F)F/F)
F1F)F2**F8F))F5FKF)F6F)F1F)F)**F0F)FNF)F/F)F1F)F2*()F5F<F)F1F)F=F)F)**
F8F)F9F)F1F)F6F)F2*(FTF)F*F)F1F)F)**F8F)F9F))F+F<F)F1F)F2*(FTF)F6F)F:F
)F)**F8F)F@F)F=F)F1F)F)**F0F)F@F)F/F)&F&6#F)F)F)**F5F)F6F)F%F)F1F)F2**
F>F)F6F)FfnF)F1F)F2**F@F)F6F)&F&6#F8F)F1F)F2*(F5F)F*F)F%F)F2*(F>F)F*F)
FfnF)F2*(F@F)F*F)F[oF)F2*,F4F)F6F)F%F)F1F)F>F)F)**F<F)F/F)F%F)F>F)F)**
F8F)F>F)F*F)F[oF)F)**F8F)F9F)F/F)F[oF)F)*,F8F)F@F)F6F)F1F)FfnF)F)**F9F
)F6F)F1F)F[oF)F)**F>F)F/F)F1F)F[oF)F)*(F8F)F@F)F:F)F)*(F8F)F9F)F*F)F2*
(\"\")F)FNF)F*F)F2*&FTF)F/F)F)*(FjoF)F@F)F*F)F2*(F0F)FQF)F/F)F)*(F0F)F
NF)F/F)F)*(F0F)FXF)F@F)F)*&FXF)F>F)F)*(F0F)FXF)F5F)F)*&F:F)F=F)F2*(F8F
)F1F)F6F)F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rem(%,d,Thet
a);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,.*&,2*()%\"xG\"\"$\"\"\"%'Thet
a1GF*&%\"cG6#F*F*!\"\"*&)F(\"\"&F*F,F*F/*(\"\"'F*&F-6#\"\"!F*)F(\"\"%F
*F**(F9F*F1F*&F-6#\"\"#F*F***F)F*F5F*F'F*F+F*F***F=F*F8F*F+F*F;F*F**(F
4F*F+F*)F(F=F*F**(F)F*F(F*)F+F=F*F*F*)%&ThetaGF=F*F**&,2*(F8F*F+F*F;F*
F/*&F+F*F'F*F/*&)F(F4F*F;F*F/*&F(F*FCF*F/*(F5F*F'F*F+F*F/*&F5F*F1F*F/*
*F=F*F8F*F+F*F,F*F**(F9F*F1F*F,F*F*F*FEF*F**(F=F*F5F*F1F*F/*(F5F*F8F*F
+F*F/*(F=F*F+F*F'F*F/*&FCF*FAF*F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "coeffs(%,Theta);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%,
**&&%\"cG6#\"\"!\"\"\")%\"xG\"\"&F)!\"#*(F%F))F+\"\"%F)%'Theta1GF)!\"
\"*(\"\"#F)F1F))F+\"\"$F)F2*&)F1F4F))F+F4F)F2,2*(F/F)F1F)&F&6#F4F)F2*&
F1F)F5F)F2*&)F+\"\"'F)F<F)F2*&F+F)F8F)F2*(F%F)F5F)F1F)F2F$F2**F4F)F/F)
F1F)&F&6#F)F)F)*(F0F)F*F)FEF)F),2*(F5F)F1F)FEF)F2*&F*F)FEF)F2*(FAF)F%F
)F/F)F)*(F0F)F*F)F<F)F)**F6F)F%F)F5F)F1F)F)**F4F)F/F)F1F)F<F)F)*(FAF)F
1F)F9F)F)*(F6F)F+F)F8F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
46 "solve(\{%\},\{seq(c[i],i=0..degree(d,Theta)-1)\});" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#<%/&%\"cG6#\"\"!,$*&%'Theta1G\"\"\"*$)%\"xG\"\"#F,
!\"\"F1/&F&6#F,F(/&F&6#F0F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "A3:=UseFunctions(subs(%,A3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#A3G,$*&-%$expG6#%\"xG\"\"\"*()F*\"\"#F+-%#lnG6#,&*$F-F+F+F'F+F+,&*$)
F/F.F+F+F*!\"\"F+F7F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ne
w_f:=normal(new_f - diff(A3,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
&new_fG*&,2*(-%$expG6#%\"xG\"\"\"F+F,)-%#lnG6#,&*$)F+\"\"#F,F,F(F,F4F,
F4*(\"\"%F,F-F,F3F,F,*(F.F,F+F,)F(F4F,F,*&F.F,F3F,F,*&F(F,F.F,F,*(F.F,
)F+\"\"$F,F(F,F,*(F4F,F+F,F(F,F,*&F6F,F3F,F,F,*(F.F,F+F,F1F,!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fv:=UseVariables(new_f);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fvG*&,2*(%'Theta1G\"\"\"%\"xGF))%&T
hetaG\"\"#F)F-*(\"\"%F)F+F))F*F-F)F)*(F,F)F*F))F(F-F)F)*&F,F)F0F)F)*&F
,F)F(F)F)*(F,F))F*\"\"$F)F(F)F)*(F-F)F(F)F*F)F)*&F/F)F0F)F)F)*(F,F)F*F
),&*$F0F)F)F(F)F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "s
qrfree(denom(fv),Theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$*&%\"xG
\"\"\",&*$)F%\"\"#F&F&%'Theta1GF&F&7#7$%&ThetaGF&" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 5 "%[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$
%&ThetaG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "d:=Theta;
 pole_order:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG%&ThetaG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+pole_orderG\"\"\"" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "a,b:=numer(new_f), denom(new_f);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"aG%\"bG6$,2*(-%$expG6#%\"xG\"\"
\"F-F.)-%#lnG6#,&*$)F-\"\"#F.F.F*F.F6F.F6*(\"\"%F.F/F.F5F.F.*(F0F.F-F.
)F*F6F.F.*&F0F.F5F.F.*&F*F.F0F.F.*(F0F.)F-\"\"$F.F*F.F.*(F6F.F-F.F*F.F
.*&F8F.F5F.F.*(F0F.F-F.F3F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "b_prime:=diff(b,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(b_primeG
,(*&,&%\"xG\"\"#-%$expG6#F(\"\"\"F-F(F-F-*&-%#lnG6#,&*$)F(F)F-F-F*F-F-
F2F-F-*(F/F-F(F-F'F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "r
esultant(a-z*b_prime,b,theta); # have to use variables, otherwise erro
r" }}{PARA 8 "" 1 "" {TEXT -1 40 "Error, (in resultant) invalid argume
nts\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "UseVariables([a,b,
b_prime]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,2*(%'Theta1G\"\"\"%\"
xGF')%&ThetaG\"\"#F'F+*(\"\"%F'F)F')F(F+F'F'*(F*F'F(F')F&F+F'F'*&F*F'F
.F'F'*&F*F'F&F'F'*(F*F')F(\"\"$F'F&F'F'*(F+F'F&F'F(F'F'*&F-F'F.F'F'*(F
*F'F(F',&*$F.F'F'F&F'F',(*&,&F(F+F&F'F'F(F'F'*&F*F'F9F'F'*(F*F'F(F'F=F
'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "op(%);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6%,2*(%'Theta1G\"\"\"%\"xGF&)%&ThetaG\"\"#F&F**(\"
\"%F&F(F&)F'F*F&F&*(F)F&F'F&)F%F*F&F&*&F)F&F-F&F&*&F)F&F%F&F&*(F)F&)F'
\"\"$F&F%F&F&*(F*F&F%F&F'F&F&*&F,F&F-F&F&*(F)F&F'F&,&*$F-F&F&F%F&F&,(*
&,&F'F*F%F&F&F'F&F&*&F)F&F8F&F&*(F)F&F'F&F<F&F&" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "av, bv, bv_prime:=%;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>6%%#avG%#bvG%)bv_primeG6%,2*(%'Theta1G\"\"\"%\"xGF,)%&
ThetaG\"\"#F,F0*(\"\"%F,F.F,)F-F0F,F,*(F/F,F-F,)F+F0F,F,*&F/F,F3F,F,*&
F/F,F+F,F,*(F/F,)F-\"\"$F,F+F,F,*(F0F,F+F,F-F,F,*&F2F,F3F,F,*(F/F,F-F,
,&*$F3F,F,F+F,F,,(*&,&F-F0F+F,F,F-F,F,*&F/F,F>F,F,*(F/F,F-F,FBF,F," }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "resultant(av - z*bv_prime, \+
bv, Theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,**&%'Theta1G\"\"\"%
\"xGF'\"\"#*&\"\"%F')F(F)F'F'*(F)F'%\"zGF'F,F'!\"\"*(F.F'F&F'F(F'F/F'F
,F'),&*$F,F'F'F&F'F)F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f
actor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**)%\"xG\"\"$\"\"\",&!
\"#F(%\"zGF(F(,&F&\"\"#%'Theta1GF(F(),&*$)F&F-F(F(F.F(F-F(!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "residues:=\{solve(%,z)\};" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)residuesG<#\"\"#" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 45 "add(r*log(gcd(av-r*bv_prime,bv)),r=residu
es);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6#%&ThetaG\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "A4:=UseFunctions(%); # Note
: we only find log(things with Theta in it), we won't find log(x) this
 way." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A4G,$-%#lnG6#-F'6#,&*$)%\"
xG\"\"#\"\"\"F0-%$expG6#F.F0F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 32 "new_f:=normal(new_f-diff(A4,x));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&new_fG*&,.*(-%$expG6#%\"xG\"\"\"-%#lnG6#,&*$)F+\"\"#F,F,F(F,F
,F+F,F3*(\"\"%F,F-F,F2F,F,*&)F(F3F,F+F,F,F1F,F(F,*&F(F,)F+\"\"$F,F,F,*
&F+F,F0F,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fv:=UseVa
riables(new_f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fvG*&,.*(%\"xG\"
\"\"%&ThetaGF)%'Theta1GF)\"\"#*(\"\"%F)F*F))F(F,F)F)*&F(F))F+F,F)F)*$F
/F)F)F+F)*&F+F))F(\"\"$F)F)F)*&F(F),&F2F)F+F)F)!\"\"" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 10 "denom(fv);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*&%\"xG\"\"\",&*$)F$\"\"#F%F%%'Theta1GF%F%" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 10 "indets(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#<$%\"xG%'Theta1G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Theta no lo
nger appears in the denominator. So fv is a polynomial in Theta, f is \+
a polynomial in theta." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "n
ew_f:=collect(new_f,theta,normal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%&new_fG,&*&*&-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F0-%$expG6#F.F0F0,&F.F/F1F
0F0F0F+!\"\"F/*&,&*&F.F0F1F0F0F0F0F0F.F5F0" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "d:=degree(new_f,theta);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"dG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "lcoeff(new_f,theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"x
G\"\"#-%$expG6#F&\"\"\"F+,&*$)F&F'F+F+F(F+!\"\"F'" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 32 "int(%,x); # Use integration in K" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F,-%$expG6#F*
F,F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(theta=theta/(d
+1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&*$)%\"xG\"\"#\"\"
\"F+-%$expG6#F)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "A5:=%*
theta^d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A5G*$)-%#lnG6#,&*$)%\"x
G\"\"#\"\"\"F/-%$expG6#F-F/F.F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 49 "new_f:=collect(new_f - diff(A5,x), theta,normal);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%&new_fG*&,&*&%\"xG\"\"\"-%$expG6#F(F)F)F)F)F)
F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "indets(new_f);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%$expG6#%\"xGF'" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 16 "member(theta,%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "theta \+
no longer appears in new_f" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
36 "int(new_f,x); # Use integration in K" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&-%$expG6#%\"xG\"\"\"-%#lnGF&F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "result := %+A1+A2+A3+A4+A5;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'resultG,0-%$expG6#%\"xG\"\"\"-%#lnGF(F**&F&F**&)F)\"
\"#F*)-F,6#,&*$F/F*F*F&F*\"\"$F*!\"\"F**&,&F*F**&*&,&F)F*F&F*F*F2F*F*F
)F7F*F**&)F2F0F*),&*$F>F*F*F)F7F0F*F7F**&F&F**(F/F*F2F*F@F*F7F7*&F0F*-
F,6#F2F*F*FAF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "check := \+
normal(diff(result,x) - f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&chec
kG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 209 "As you can see, as lo
ng as f (or new_f) still has theta in the denominator, we're only usin
g things like normal, diff, sqrfree, solve, and such. Once new_f is in
 K[theta], only then do we use integration in K." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "How can integration fail?
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 166 "As l
ong as the pole_order is >1 at any finite point, we can always reduce \+
it. That never fails, we don't use any integration but only other comm
ands that always work." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 554 "When the pole_order=1, then we compute the residues
. Now if any of them is non-constant, then no elementary integral exis
ts because according to Liouvilles principle, if an elementary integra
l exists then f must have a very special form, in particular these res
idues must be constant. So if any residue is non-constant (contains an
 x) then the algorithm ends, no elementary integral exists. However, i
f all residues are constant, then the computation proceeds and f (new_
f) is reduced to a new_f with no theta in the denominator. So new_f is
 in K[theta]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 152 "When f (or new_f) is in K[theta], then we integrate lcoe
ff(f,theta), which is an element of K. If the degree d is >0, then we \+
distinguish several cases." }}{PARA 0 "" 0 "" {TEXT -1 112 "1) lcoeff(
f,theta) does not have an elementary anti-derivative. Then f has no el
ementary anti-derivative either." }}{PARA 0 "" 0 "" {TEXT -1 357 "2) l
coeff(f,theta) does have an elementary anti-derivative but it contains
 a logarithmic extension other than theta. Well, then we get theta^d*n
ew_logarithm, but that's not allowed in Liouvilles theorem, new logari
thms always come with a constant coefficient, not with something that \+
contains theta. So in case 2, there is again no elementary integral of
 f." }}{PARA 0 "" 0 "" {TEXT -1 324 "3) lcoeff(f,theta) has an element
ary integral without any new logarithms (except for perhaps theta). So
 then int(lcoeff(f,theta),x) must be of the form c*theta+a for some a \+
in K, and constant c (which could be 0). Then take A=(c/(d+1) * theta \+
+ a)*theta^d, and you see that f-diff(A,x) is a polynomial in theta of
 degree <d." }}{PARA 0 "" 0 "" {TEXT -1 286 "4) Now either we hit 1) o
r 2) at some point so that no elementary integral exists, or we must r
each degree d=0 at some point. At that point f has been reduced to an \+
element new_f in K. So in that situation, f has an elementary integral
 if and only this new_f has an elementary integral." }}}}{MARK "8 0 0
" 0 }{VIEWOPTS 1 1 0 1 1 1803 }