/usr/share/axiom-20170501/src/algebra/EFULS.spad is in axiom-source 20170501-3.
This file is owned by root:root, with mode 0o644.
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++ Author: Clifton J. Williamson
++ Date Created: 6 February 1990
++ Date Last Updated: 25 February 1990
++ Description:
++ This domain provides elementary functions on any Laurent series
++ domain over a field which was constructed from a Taylor series
++ domain. These functions are implemented by calling the
++ corresponding functions on the Taylor series domain. We also
++ provide 'partial functions' which compute transcendental
++ functions of Laurent series when possible and return "failed"
++ when this is not possible.
ElementaryFunctionsUnivariateLaurentSeries(Coef,UTS,ULS) : SIG == CODE where
Coef : Algebra Fraction Integer
UTS : UnivariateTaylorSeriesCategory Coef
ULS : UnivariateLaurentSeriesConstructorCategory(Coef,UTS)
I ==> Integer
NNI ==> NonNegativeInteger
RN ==> Fraction Integer
S ==> String
STTF ==> StreamTranscendentalFunctions(Coef)
SIG ==> PartialTranscendentalFunctions(ULS) with
if Coef has Field then
"**" : (ULS,RN) -> ULS
++ s ** r raises a Laurent series s to a rational power r
--% Exponentials and Logarithms
exp : ULS -> ULS
++ exp(z) returns the exponential of Laurent series z.
log : ULS -> ULS
++ log(z) returns the logarithm of Laurent series z.
--% TrigonometricFunctionCategory
sin : ULS -> ULS
++ sin(z) returns the sine of Laurent series z.
cos : ULS -> ULS
++ cos(z) returns the cosine of Laurent series z.
tan : ULS -> ULS
++ tan(z) returns the tangent of Laurent series z.
cot : ULS -> ULS
++ cot(z) returns the cotangent of Laurent series z.
sec : ULS -> ULS
++ sec(z) returns the secant of Laurent series z.
csc : ULS -> ULS
++ csc(z) returns the cosecant of Laurent series z.
--% ArcTrigonometricFunctionCategory
asin : ULS -> ULS
++ asin(z) returns the arc-sine of Laurent series z.
acos : ULS -> ULS
++ acos(z) returns the arc-cosine of Laurent series z.
atan : ULS -> ULS
++ atan(z) returns the arc-tangent of Laurent series z.
acot : ULS -> ULS
++ acot(z) returns the arc-cotangent of Laurent series z.
asec : ULS -> ULS
++ asec(z) returns the arc-secant of Laurent series z.
acsc : ULS -> ULS
++ acsc(z) returns the arc-cosecant of Laurent series z.
--% HyperbolicFunctionCategory
sinh : ULS -> ULS
++ sinh(z) returns the hyperbolic sine of Laurent series z.
cosh : ULS -> ULS
++ cosh(z) returns the hyperbolic cosine of Laurent series z.
tanh : ULS -> ULS
++ tanh(z) returns the hyperbolic tangent of Laurent series z.
coth : ULS -> ULS
++ coth(z) returns the hyperbolic cotangent of Laurent series z.
sech : ULS -> ULS
++ sech(z) returns the hyperbolic secant of Laurent series z.
csch : ULS -> ULS
++ csch(z) returns the hyperbolic cosecant of Laurent series z.
--% ArcHyperbolicFunctionCategory
asinh : ULS -> ULS
++ asinh(z) returns the inverse hyperbolic sine of Laurent series z.
acosh : ULS -> ULS
++ acosh(z) returns the inverse hyperbolic cosine of Laurent series z.
atanh : ULS -> ULS
++ atanh(z) returns the inverse hyperbolic tangent of Laurent series z.
acoth : ULS -> ULS
++ acoth(z) returns the inverse hyperbolic cotangent of Laurent series z.
asech : ULS -> ULS
++ asech(z) returns the inverse hyperbolic secant of Laurent series z.
acsch : ULS -> ULS
++ acsch(z) returns the inverse hyperbolic cosecant of Laurent series z.
CODE ==> add
--% roots
RATPOWERS : Boolean := Coef has "**":(Coef,RN) -> Coef
TRANSFCN : Boolean := Coef has TranscendentalFunctionCategory
RATS : Boolean := Coef has retractIfCan: Coef -> Union(RN,"failed")
nthRootUTS:(UTS,I) -> Union(UTS,"failed")
nthRootUTS(uts,n) ==
-- assumed: n > 1, uts has non-zero constant term
coefficient(uts,0) = 1 => uts ** inv(n::RN)
RATPOWERS => uts ** inv(n::RN)
"failed"
nthRootIfCan(uls,nn) ==
(n := nn :: I) < 1 => error "nthRootIfCan: n must be positive"
n = 1 => uls
deg := degree uls
if zero? (coef := coefficient(uls,deg)) then
uls := removeZeroes(1000,uls); deg := degree uls
zero? (coef := coefficient(uls,deg)) =>
error "root of series with many leading zero coefficients"
(k := deg exquo n) case "failed" => "failed"
uts := taylor(uls * monomial(1,-deg))
(root := nthRootUTS(uts,n)) case "failed" => "failed"
monomial(1,k :: I) * (root :: UTS :: ULS)
if Coef has Field then
(uls:ULS) ** (r:RN) ==
num := numer r; den := denom r
den = 1 => uls ** num
deg := degree uls
if zero? (coef := coefficient(uls,deg)) then
uls := removeZeroes(1000,uls); deg := degree uls
zero? (coef := coefficient(uls,deg)) =>
error "power of series with many leading zero coefficients"
(k := deg exquo den) case "failed" =>
error "**: rational power does not exist"
uts := taylor(uls * monomial(1,-deg)) ** r
monomial(1,(k :: I) * num) * (uts :: ULS)
--% transcendental functions
applyIfCan: (UTS -> UTS,ULS) -> Union(ULS,"failed")
applyIfCan(fcn,uls) ==
uts := taylorIfCan uls
uts case "failed" => "failed"
fcn(uts :: UTS) :: ULS
expIfCan uls == applyIfCan(exp,uls)
sinIfCan uls == applyIfCan(sin,uls)
cosIfCan uls == applyIfCan(cos,uls)
asinIfCan uls == applyIfCan(asin,uls)
acosIfCan uls == applyIfCan(acos,uls)
asecIfCan uls == applyIfCan(asec,uls)
acscIfCan uls == applyIfCan(acsc,uls)
sinhIfCan uls == applyIfCan(sinh,uls)
coshIfCan uls == applyIfCan(cosh,uls)
asinhIfCan uls == applyIfCan(asinh,uls)
acoshIfCan uls == applyIfCan(acosh,uls)
atanhIfCan uls == applyIfCan(atanh,uls)
acothIfCan uls == applyIfCan(acoth,uls)
asechIfCan uls == applyIfCan(asech,uls)
acschIfCan uls == applyIfCan(acsch,uls)
logIfCan uls ==
uts := taylorIfCan uls
uts case "failed" => "failed"
zero? coefficient(ts := uts :: UTS,0) => "failed"
log(ts) :: ULS
tanIfCan uls ==
-- don't call 'tan' on a UTS (tan(uls) may have a singularity)
uts := taylorIfCan uls
uts case "failed" => "failed"
sc := sincos(coefficients(uts :: UTS))$STTF
(cosInv := recip(series(sc.cos) :: ULS)) case "failed" => "failed"
(series(sc.sin) :: ULS) * (cosInv :: ULS)
cotIfCan uls ==
-- don't call 'cot' on a UTS (cot(uls) may have a singularity)
uts := taylorIfCan uls
uts case "failed" => "failed"
sc := sincos(coefficients(uts :: UTS))$STTF
(sinInv := recip(series(sc.sin) :: ULS)) case "failed" => "failed"
(series(sc.cos) :: ULS) * (sinInv :: ULS)
secIfCan uls ==
cos := cosIfCan uls
cos case "failed" => "failed"
(cosInv := recip(cos :: ULS)) case "failed" => "failed"
cosInv :: ULS
cscIfCan uls ==
sin := sinIfCan uls
sin case "failed" => "failed"
(sinInv := recip(sin :: ULS)) case "failed" => "failed"
sinInv :: ULS
atanIfCan uls ==
coef := coefficient(uls,0)
(ord := order(uls,0)) = 0 and coef * coef = -1 => "failed"
cc : Coef :=
ord < 0 =>
TRANSFCN =>
RATS =>
lc := coefficient(uls,ord)
(rat := retractIfCan(lc)@Union(RN,"failed")) case "failed" =>
(1/2) * pi()
(rat :: RN) > 0 => (1/2) * pi()
(-1/2) * pi()
(1/2) * pi()
return "failed"
coef = 0 => 0
TRANSFCN => atan coef
return "failed"
(z := recip(1 + uls*uls)) case "failed" => "failed"
(cc :: ULS) + integrate(differentiate(uls) * (z :: ULS))
acotIfCan uls ==
coef := coefficient(uls,0)
(ord := order(uls,0)) = 0 and coef * coef = -1 => "failed"
cc : Coef :=
ord < 0 =>
RATS =>
lc := coefficient(uls,ord)
(rat := retractIfCan(lc)@Union(RN,"failed")) case "failed" => 0
(rat :: RN) > 0 => 0
TRANSFCN => pi()
return "failed"
0
TRANSFCN => acot coef
return "failed"
(z := recip(1 + uls*uls)) case "failed" => "failed"
(cc :: ULS) - integrate(differentiate(uls) * (z :: ULS))
tanhIfCan uls ==
-- don't call 'tanh' on a UTS (tanh(uls) may have a singularity)
uts := taylorIfCan uls
uts case "failed" => "failed"
sc := sinhcosh(coefficients(uts :: UTS))$STTF
(coshInv := recip(series(sc.cosh) :: ULS)) case "failed" =>
"failed"
(series(sc.sinh) :: ULS) * (coshInv :: ULS)
cothIfCan uls ==
-- don't call 'coth' on a UTS (coth(uls) may have a singularity)
uts := taylorIfCan uls
uts case "failed" => "failed"
sc := sinhcosh(coefficients(uts :: UTS))$STTF
(sinhInv := recip(series(sc.sinh) :: ULS)) case "failed" =>
"failed"
(series(sc.cosh) :: ULS) * (sinhInv :: ULS)
sechIfCan uls ==
cosh := coshIfCan uls
cosh case "failed" => "failed"
(coshInv := recip(cosh :: ULS)) case "failed" => "failed"
coshInv :: ULS
cschIfCan uls ==
sinh := sinhIfCan uls
sinh case "failed" => "failed"
(sinhInv := recip(sinh :: ULS)) case "failed" => "failed"
sinhInv :: ULS
applyOrError:(ULS -> Union(ULS,"failed"),S,ULS) -> ULS
applyOrError(fcn,name,uls) ==
ans := fcn uls
ans case "failed" =>
error concat(name," of function with singularity")
ans :: ULS
exp uls == applyOrError(expIfCan,"exp",uls)
log uls == applyOrError(logIfCan,"log",uls)
sin uls == applyOrError(sinIfCan,"sin",uls)
cos uls == applyOrError(cosIfCan,"cos",uls)
tan uls == applyOrError(tanIfCan,"tan",uls)
cot uls == applyOrError(cotIfCan,"cot",uls)
sec uls == applyOrError(secIfCan,"sec",uls)
csc uls == applyOrError(cscIfCan,"csc",uls)
asin uls == applyOrError(asinIfCan,"asin",uls)
acos uls == applyOrError(acosIfCan,"acos",uls)
asec uls == applyOrError(asecIfCan,"asec",uls)
acsc uls == applyOrError(acscIfCan,"acsc",uls)
sinh uls == applyOrError(sinhIfCan,"sinh",uls)
cosh uls == applyOrError(coshIfCan,"cosh",uls)
tanh uls == applyOrError(tanhIfCan,"tanh",uls)
coth uls == applyOrError(cothIfCan,"coth",uls)
sech uls == applyOrError(sechIfCan,"sech",uls)
csch uls == applyOrError(cschIfCan,"csch",uls)
asinh uls == applyOrError(asinhIfCan,"asinh",uls)
acosh uls == applyOrError(acoshIfCan,"acosh",uls)
atanh uls == applyOrError(atanhIfCan,"atanh",uls)
acoth uls == applyOrError(acothIfCan,"acoth",uls)
asech uls == applyOrError(asechIfCan,"asech",uls)
acsch uls == applyOrError(acschIfCan,"acsch",uls)
atan uls ==
-- code is duplicated so that correct error messages will be returned
coef := coefficient(uls,0)
(ord := order(uls,0)) = 0 and coef * coef = -1 =>
error "atan: series expansion has logarithmic term"
cc : Coef :=
ord < 0 =>
TRANSFCN =>
RATS =>
lc := coefficient(uls,ord)
(rat := retractIfCan(lc)@Union(RN,"failed")) case "failed" =>
(1/2) * pi()
(rat :: RN) > 0 => (1/2) * pi()
(-1/2) * pi()
(1/2) * pi()
error "atan: series expansion involves transcendental constants"
coef = 0 => 0
TRANSFCN => atan coef
error "atan: series expansion involves transcendental constants"
(z := recip(1 + uls*uls)) case "failed" =>
error "atan: leading coefficient not invertible"
(cc :: ULS) + integrate(differentiate(uls) * (z :: ULS))
acot uls ==
-- code is duplicated so that correct error messages will be returned
coef := coefficient(uls,0)
(ord := order(uls,0)) = 0 and coef * coef = -1 =>
error "acot: series expansion has logarithmic term"
cc : Coef :=
ord < 0 =>
RATS =>
lc := coefficient(uls,ord)
(rat := retractIfCan(lc)@Union(RN,"failed")) case "failed" => 0
(rat :: RN) > 0 => 0
TRANSFCN => pi()
error "acot: series expansion involves transcendental constants"
0
TRANSFCN => acot coef
error "acot: series expansion involves transcendental constants"
(z := recip(1 + uls*uls)) case "failed" =>
error "acot: leading coefficient not invertible"
(cc :: ULS) - integrate(differentiate(uls) * (z :: ULS))
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