/usr/share/axiom-20170501/src/algebra/EXPR2UPS.spad is in axiom-source 20170501-3.
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++ Author: Clifton J. Williamson
++ Date Created: 9 May 1989
++ Date Last Updated: 20 September 1993
++ Description:
++ This package provides functions to convert functional expressions
++ to power series.
ExpressionToUnivariatePowerSeries(R,FE) : SIG == CODE where
R : Join(GcdDomain,OrderedSet,RetractableTo Integer,_
LinearlyExplicitRingOver Integer)
FE : Join(AlgebraicallyClosedField,TranscendentalFunctionCategory,_
FunctionSpace R)
EQ ==> Equation
I ==> Integer
NNI ==> NonNegativeInteger
RN ==> Fraction Integer
SY ==> Symbol
UTS ==> UnivariateTaylorSeries
ULS ==> UnivariateLaurentSeries
UPXS ==> UnivariatePuiseuxSeries
GSER ==> GeneralUnivariatePowerSeries
EFULS ==> ElementaryFunctionsUnivariateLaurentSeries
EFUPXS ==> ElementaryFunctionsUnivariatePuiseuxSeries
FS2UPS ==> FunctionSpaceToUnivariatePowerSeries
Prob ==> Record(func:String,prob:String)
ANY1 ==> AnyFunctions1
SIG ==> with
taylor : SY -> Any
++ \spad{taylor(x)} returns x viewed as a Taylor series.
taylor : FE -> Any
++ \spad{taylor(f)} returns a Taylor expansion of the expression f.
++ Note that f should have only one variable; the series will be
++ expanded in powers of that variable.
taylor : (FE,NNI) -> Any
++ \spad{taylor(f,n)} returns a Taylor expansion of the expression f.
++ Note that f should have only one variable; the series will be
++ expanded in powers of that variable and terms will be computed
++ up to order at least n.
taylor : (FE,EQ FE) -> Any
++ \spad{taylor(f,x = a)} expands the expression f as a Taylor series
++ in powers of \spad{(x - a)}.
taylor : (FE,EQ FE,NNI) -> Any
++ \spad{taylor(f,x = a)} expands the expression f as a Taylor series
++ in powers of \spad{(x - a)}; terms will be computed up to order
++ at least n.
laurent : SY -> Any
++ \spad{laurent(x)} returns x viewed as a Laurent series.
laurent : FE -> Any
++ \spad{laurent(f)} returns a Laurent expansion of the expression f.
++ Note that f should have only one variable; the series will be
++ expanded in powers of that variable.
laurent : (FE,I) -> Any
++ \spad{laurent(f,n)} returns a Laurent expansion of the expression f.
++ Note that f should have only one variable; the series will be
++ expanded in powers of that variable and terms will be computed
++ up to order at least n.
laurent : (FE,EQ FE) -> Any
++ \spad{laurent(f,x = a)} expands the expression f as a Laurent series
++ in powers of \spad{(x - a)}.
laurent : (FE,EQ FE,I) -> Any
++ \spad{laurent(f,x = a,n)} expands the expression f as a Laurent
++ series in powers of \spad{(x - a)}; terms will be computed up to order
++ at least n.
puiseux : SY -> Any
++ \spad{puiseux(x)} returns x viewed as a Puiseux series.
puiseux : FE -> Any
++ \spad{puiseux(f)} returns a Puiseux expansion of the expression f.
++ Note that f should have only one variable; the series will be
++ expanded in powers of that variable.
puiseux : (FE,RN) -> Any
++ \spad{puiseux(f,n)} returns a Puiseux expansion of the expression f.
++ Note that f should have only one variable; the series will be
++ expanded in powers of that variable and terms will be computed
++ up to order at least n.
puiseux : (FE,EQ FE) -> Any
++ \spad{puiseux(f,x = a)} expands the expression f as a Puiseux series
++ in powers of \spad{(x - a)}.
puiseux : (FE,EQ FE,RN) -> Any
++ \spad{puiseux(f,x = a,n)} expands the expression f as a Puiseux
++ series in powers of \spad{(x - a)}; terms will be computed up to order
++ at least n.
series : SY -> Any
++ \spad{series(x)} returns x viewed as a series.
series : FE -> Any
++ \spad{series(f)} returns a series expansion of the expression f.
++ Note that f should have only one variable; the series will be
++ expanded in powers of that variable.
series : (FE,RN) -> Any
++ \spad{series(f,n)} returns a series expansion of the expression f.
++ Note that f should have only one variable; the series will be
++ expanded in powers of that variable and terms will be computed
++ up to order at least n.
series : (FE,EQ FE) -> Any
++ \spad{series(f,x = a)} expands the expression f as a series
++ in powers of (x - a).
series : (FE,EQ FE,RN) -> Any
++ \spad{series(f,x = a,n)} expands the expression f as a series
++ in powers of (x - a); terms will be computed up to order
++ at least n.
CODE ==> add
performSubstitution: (FE,SY,FE) -> FE
performSubstitution(fcn,x,a) ==
zero? a => fcn
xFE := x :: FE
eval(fcn,xFE = xFE + a)
iTaylor: (FE,SY,FE) -> Any
iTaylor(fcn,x,a) ==
pack := FS2UPS(R,FE,I,ULS(FE,x,a),_
EFULS(FE,UTS(FE,x,a),ULS(FE,x,a)),x)
ans := exprToUPS(fcn,false,"just do it")$pack
ans case %problem =>
ans.%problem.prob = "essential singularity" =>
error "No Taylor expansion: essential singularity"
ans.%problem.func = "log" =>
error "No Taylor expansion: logarithmic singularity"
ans.%problem.func = "nth root" =>
error "No Taylor expansion: fractional powers in expansion"
error "No Taylor expansion"
uls := ans.%series
(uts := taylorIfCan uls) case "failed" =>
error "No Taylor expansion: pole"
any1 := ANY1(UTS(FE,x,a))
coerce(uts :: UTS(FE,x,a))$any1
taylor(x:SY) ==
uts := UTS(FE,x,0$FE); any1 := ANY1(uts)
coerce(monomial(1,1)$uts)$any1
taylor(fcn:FE) ==
null(vars := variables fcn) =>
error "taylor: expression has no variables"
not null rest vars =>
error "taylor: expression has more than one variable"
taylor(fcn,(first(vars) :: FE) = 0)
taylor(fcn:FE,n:NNI) ==
null(vars := variables fcn) =>
error "taylor: expression has no variables"
not null rest vars =>
error "taylor: expression has more than one variable"
x := first vars
uts := UTS(FE,x,0$FE); any1 := ANY1(uts)
series := retract(taylor(fcn,(x :: FE) = 0))$any1
coerce(extend(series,n))$any1
taylor(fcn:FE,eq:EQ FE) ==
(xx := retractIfCan(lhs eq)@Union(SY,"failed")) case "failed" =>
error "taylor: left hand side must be a variable"
x := xx :: SY; a := rhs eq
iTaylor(performSubstitution(fcn,x,a),x,a)
taylor(fcn,eq,n) ==
(xx := retractIfCan(lhs eq)@Union(SY,"failed")) case "failed" =>
error "taylor: left hand side must be a variable"
x := xx :: SY; a := rhs eq
any1 := ANY1(UTS(FE,x,a))
series := retract(iTaylor(performSubstitution(fcn,x,a),x,a))$any1
coerce(extend(series,n))$any1
iLaurent: (FE,SY,FE) -> Any
iLaurent(fcn,x,a) ==
pack := FS2UPS(R,FE,I,ULS(FE,x,a),_
EFULS(FE,UTS(FE,x,a),ULS(FE,x,a)),x)
ans := exprToUPS(fcn,false,"just do it")$pack
ans case %problem =>
ans.%problem.prob = "essential singularity" =>
error "No Laurent expansion: essential singularity"
ans.%problem.func = "log" =>
error "No Laurent expansion: logarithmic singularity"
ans.%problem.func = "nth root" =>
error "No Laurent expansion: fractional powers in expansion"
error "No Laurent expansion"
any1 := ANY1(ULS(FE,x,a))
coerce(ans.%series)$any1
laurent(x:SY) ==
uls := ULS(FE,x,0$FE); any1 := ANY1(uls)
coerce(monomial(1,1)$uls)$any1
laurent(fcn:FE) ==
null(vars := variables fcn) =>
error "laurent: expression has no variables"
not null rest vars =>
error "laurent: expression has more than one variable"
laurent(fcn,(first(vars) :: FE) = 0)
laurent(fcn:FE,n:I) ==
null(vars := variables fcn) =>
error "laurent: expression has no variables"
not null rest vars =>
error "laurent: expression has more than one variable"
x := first vars
uls := ULS(FE,x,0$FE); any1 := ANY1(uls)
series := retract(laurent(fcn,(x :: FE) = 0))$any1
coerce(extend(series,n))$any1
laurent(fcn:FE,eq:EQ FE) ==
(xx := retractIfCan(lhs eq)@Union(SY,"failed")) case "failed" =>
error "taylor: left hand side must be a variable"
x := xx :: SY; a := rhs eq
iLaurent(performSubstitution(fcn,x,a),x,a)
laurent(fcn,eq,n) ==
(xx := retractIfCan(lhs eq)@Union(SY,"failed")) case "failed" =>
error "taylor: left hand side must be a variable"
x := xx :: SY; a := rhs eq
any1 := ANY1(ULS(FE,x,a))
series := retract(iLaurent(performSubstitution(fcn,x,a),x,a))$any1
coerce(extend(series,n))$any1
iPuiseux: (FE,SY,FE) -> Any
iPuiseux(fcn,x,a) ==
pack := FS2UPS(R,FE,RN,UPXS(FE,x,a),_
EFUPXS(FE,ULS(FE,x,a),UPXS(FE,x,a),_
EFULS(FE,UTS(FE,x,a),ULS(FE,x,a))),x)
ans := exprToUPS(fcn,false,"just do it")$pack
ans case %problem =>
ans.%problem.prob = "essential singularity" =>
error "No Puiseux expansion: essential singularity"
ans.%problem.func = "log" =>
error "No Puiseux expansion: logarithmic singularity"
error "No Puiseux expansion"
any1 := ANY1(UPXS(FE,x,a))
coerce(ans.%series)$any1
puiseux(x:SY) ==
upxs := UPXS(FE,x,0$FE); any1 := ANY1(upxs)
coerce(monomial(1,1)$upxs)$any1
puiseux(fcn:FE) ==
null(vars := variables fcn) =>
error "puiseux: expression has no variables"
not null rest vars =>
error "puiseux: expression has more than one variable"
puiseux(fcn,(first(vars) :: FE) = 0)
puiseux(fcn:FE,n:RN) ==
null(vars := variables fcn) =>
error "puiseux: expression has no variables"
not null rest vars =>
error "puiseux: expression has more than one variable"
x := first vars
upxs := UPXS(FE,x,0$FE); any1 := ANY1(upxs)
series := retract(puiseux(fcn,(x :: FE) = 0))$any1
coerce(extend(series,n))$any1
puiseux(fcn:FE,eq:EQ FE) ==
(xx := retractIfCan(lhs eq)@Union(SY,"failed")) case "failed" =>
error "taylor: left hand side must be a variable"
x := xx :: SY; a := rhs eq
iPuiseux(performSubstitution(fcn,x,a),x,a)
puiseux(fcn,eq,n) ==
(xx := retractIfCan(lhs eq)@Union(SY,"failed")) case "failed" =>
error "taylor: left hand side must be a variable"
x := xx :: SY; a := rhs eq
any1 := ANY1(UPXS(FE,x,a))
series := retract(iPuiseux(performSubstitution(fcn,x,a),x,a))$any1
coerce(extend(series,n))$any1
iSeries: (FE,SY,FE) -> Any
iSeries(fcn,x,a) ==
pack := FS2UPS(R,FE,RN,UPXS(FE,x,a), _
EFUPXS(FE,ULS(FE,x,a),UPXS(FE,x,a), _
EFULS(FE,UTS(FE,x,a),ULS(FE,x,a))),x)
ans := exprToUPS(fcn,false,"just do it")$pack
ans case %problem =>
ansG := exprToGenUPS(fcn,false,"just do it")$pack
ansG case %problem =>
ansG.%problem.prob = "essential singularity" =>
error "No series expansion: essential singularity"
error "No series expansion"
anyone := ANY1(GSER(FE,x,a))
coerce((ansG.%series) :: GSER(FE,x,a))$anyone
any1 := ANY1(UPXS(FE,x,a))
coerce(ans.%series)$any1
series(x:SY) ==
upxs := UPXS(FE,x,0$FE); any1 := ANY1(upxs)
coerce(monomial(1,1)$upxs)$any1
series(fcn:FE) ==
null(vars := variables fcn) =>
error "series: expression has no variables"
not null rest vars =>
error "series: expression has more than one variable"
series(fcn,(first(vars) :: FE) = 0)
series(fcn:FE,n:RN) ==
null(vars := variables fcn) =>
error "series: expression has no variables"
not null rest vars =>
error "series: expression has more than one variable"
x := first vars
upxs := UPXS(FE,x,0$FE); any1 := ANY1(upxs)
series := retract(series(fcn,(x :: FE) = 0))$any1
coerce(extend(series,n))$any1
series(fcn:FE,eq:EQ FE) ==
(xx := retractIfCan(lhs eq)@Union(SY,"failed")) case "failed" =>
error "taylor: left hand side must be a variable"
x := xx :: SY; a := rhs eq
iSeries(performSubstitution(fcn,x,a),x,a)
series(fcn,eq,n) ==
(xx := retractIfCan(lhs eq)@Union(SY,"failed")) case "failed" =>
error "taylor: left hand side must be a variable"
x := xx :: SY; a := rhs eq
any1 := ANY1(UPXS(FE,x,a))
series := retract(iSeries(performSubstitution(fcn,x,a),x,a))$any1
coerce(extend(series,n))$any1
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