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--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
)abbrev package STTF StreamTranscendentalFunctions
++ Author: William Burge, Clifton J. Williamson
++ Date Created: 1986
++ Date Last Updated: 6 March 1995
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: Taylor series, elementary function, transcendental function
++ Examples:
++ References:
++ Description:
++ StreamTranscendentalFunctions implements transcendental functions on
++ Taylor series, where a Taylor series is represented by a stream of
++ its coefficients.
StreamTranscendentalFunctions(Coef): Exports == Implementation where
Coef : Algebra Fraction Integer
L ==> List
I ==> Integer
RN ==> Fraction Integer
SG ==> String
ST ==> Stream Coef
STT ==> StreamTaylorSeriesOperations Coef
YS ==> Y$ParadoxicalCombinatorsForStreams(Coef)
Exports ==> with
--% Exponentials and Logarithms
exp : ST -> ST
++ exp(st) computes the exponential of a power series st.
log : ST -> ST
++ log(st) computes the log of a power series.
** : (ST,ST) -> ST
++ st1 ** st2 computes the power of a power series st1 by another
++ power series st2.
--% TrigonometricFunctionCategory
sincos : ST -> Record(sin:ST, cos:ST)
++ sincos(st) returns a record containing the sine and cosine
++ of a power series st.
sin : ST -> ST
++ sin(st) computes sine of a power series st.
cos : ST -> ST
++ cos(st) computes cosine of a power series st.
tan : ST -> ST
++ tan(st) computes tangent of a power series st.
cot : ST -> ST
++ cot(st) computes cotangent of a power series st.
sec : ST -> ST
++ sec(st) computes secant of a power series st.
csc : ST -> ST
++ csc(st) computes cosecant of a power series st.
asin : ST -> ST
++ asin(st) computes arcsine of a power series st.
acos : ST -> ST
++ acos(st) computes arccosine of a power series st.
atan : ST -> ST
++ atan(st) computes arctangent of a power series st.
acot : ST -> ST
++ acot(st) computes arccotangent of a power series st.
asec : ST -> ST
++ asec(st) computes arcsecant of a power series st.
acsc : ST -> ST
++ acsc(st) computes arccosecant of a power series st.
--% HyperbloicTrigonometricFunctionCategory
sinhcosh: ST -> Record(sinh:ST, cosh:ST)
++ sinhcosh(st) returns a record containing
++ the hyperbolic sine and cosine
++ of a power series st.
sinh : ST -> ST
++ sinh(st) computes the hyperbolic sine of a power series st.
cosh : ST -> ST
++ cosh(st) computes the hyperbolic cosine of a power series st.
tanh : ST -> ST
++ tanh(st) computes the hyperbolic tangent of a power series st.
coth : ST -> ST
++ coth(st) computes the hyperbolic cotangent of a power series st.
sech : ST -> ST
++ sech(st) computes the hyperbolic secant of a power series st.
csch : ST -> ST
++ csch(st) computes the hyperbolic cosecant of a power series st.
asinh : ST -> ST
++ asinh(st) computes the inverse hyperbolic sine of a power series st.
acosh : ST -> ST
++ acosh(st) computes the inverse hyperbolic cosine
++ of a power series st.
atanh : ST -> ST
++ atanh(st) computes the inverse hyperbolic tangent
++ of a power series st.
acoth : ST -> ST
++ acoth(st) computes the inverse hyperbolic
++ cotangent of a power series st.
asech : ST -> ST
++ asech(st) computes the inverse hyperbolic secant of a
++ power series st.
acsch : ST -> ST
++ acsch(st) computes the inverse hyperbolic
++ cosecant of a power series st.
Implementation ==> add
import StreamTaylorSeriesOperations Coef
TRANSFCN : Boolean := Coef has TranscendentalFunctionCategory
--% Error Reporting
TRCONST : SG := "series expansion involves transcendental constants"
NPOWERS : SG := "series expansion has terms of negative degree"
FPOWERS : SG := "series expansion has terms of fractional degree"
MAYFPOW : SG := "series expansion may have terms of fractional degree"
LOGS : SG := "series expansion has logarithmic term"
NPOWLOG : SG :=
"series expansion has terms of negative degree or logarithmic term"
FPOWLOG : SG :=
"series expansion has terms of fractional degree or logarithmic term"
NOTINV : SG := "leading coefficient not invertible"
--% Exponentials and Logarithms
expre:(Coef,ST,ST) -> ST
expre(r,e,dx) == lazyIntegrate(r,e*dx)
exp z ==
empty? z => 1@Coef :: ST
(coef := frst z) = 0 => YS expre(1,#1,deriv z)
TRANSFCN => YS expre(exp coef,#1,deriv z)
error concat("exp: ",TRCONST)
log z ==
empty? z => error "log: constant coefficient should not be 0"
(coef := frst z) = 0 => error "log: constant coefficient should not be 0"
coef = 1 => lazyIntegrate(0,deriv z/z)
TRANSFCN => lazyIntegrate(log coef,deriv z/z)
error concat("log: ",TRCONST)
z1:ST ** z2:ST == exp(z2 * log z1)
--% Trigonometric Functions
sincosre:(Coef,Coef,L ST,ST,Coef) -> L ST
sincosre(rs,rc,sc,dx,sign) ==
[lazyIntegrate(rs,(second sc)*dx),lazyIntegrate(rc,sign*(first sc)*dx)]
-- When the compiler had difficulties with the above definition,
-- I did the following to help it:
-- sincosre:(Coef,Coef,L ST,ST,Coef) -> L ST
-- sincosre(rs,rc,sc,dx,sign) ==
-- st1 : ST := (second sc) * dx
-- st2 : ST := (first sc) * dx
-- st2 := sign * st2
-- [lazyIntegrate(rs,st1),lazyIntegrate(rc,st2)]
sincos z ==
empty? z => [0@Coef :: ST,1@Coef :: ST]
l :=
(coef := frst z) = 0 => YS(sincosre(0,1,#1,deriv z,-1),2)
TRANSFCN => YS(sincosre(sin coef,cos coef,#1,deriv z,-1),2)
error concat("sincos: ",TRCONST)
[first l,second l]
sin z == sincos(z).sin
cos z == sincos(z).cos
tanre:(Coef,ST,ST,Coef) -> ST
tanre(r,t,dx,sign) == lazyIntegrate(r,((1@Coef :: ST) + sign*t*t)*dx)
-- When the compiler had difficulties with the above definition,
-- I did the following to help it:
-- tanre:(Coef,ST,ST,Coef) -> ST
-- tanre(r,t,dx,sign) ==
-- st1 : ST := t * t
-- st1 := sign * st1
-- st2 : ST := 1 :: ST
-- st1 := st2 + st1
-- st1 := st1 * dx
-- lazyIntegrate(r,st1)
tan z ==
empty? z => 0@Coef :: ST
(coef := frst z) = 0 => YS tanre(0,#1,deriv z,1)
TRANSFCN => YS tanre(tan coef,#1,deriv z,1)
error concat("tan: ",TRCONST)
cotre:(Coef,ST,ST) -> ST
cotre(r,t,dx) == lazyIntegrate(r,-((1@Coef :: ST) + t*t)*dx)
-- When the compiler had difficulties with the above definition,
-- I did the following to help it:
-- cotre:(Coef,ST,ST) -> ST
-- cotre(r,t,dx) ==
-- st1 : ST := t * t
-- st2 : ST := 1 :: ST
-- st1 := st2 + st1
-- st1 := st1 * dx
-- st1 := -st1
-- lazyIntegrate(r,st1)
cot z ==
empty? z => error "cot: cot(0) is undefined"
(coef := frst z) = 0 => error concat("cot: ",NPOWERS)
TRANSFCN => YS cotre(cot coef,#1,deriv z)
error concat("cot: ",TRCONST)
sec z ==
empty? z => 1@Coef :: ST
frst z = 0 => recip(cos z) :: ST
TRANSFCN =>
cosz := cos z
first cosz = 0 => error concat("sec: ",NPOWERS)
recip(cosz) :: ST
error concat("sec: ",TRCONST)
csc z ==
empty? z => error "csc: csc(0) is undefined"
TRANSFCN =>
sinz := sin z
first sinz = 0 => error concat("csc: ",NPOWERS)
recip(sinz) :: ST
error concat("csc: ",TRCONST)
orderOrFailed : ST -> Union(I,"failed")
orderOrFailed x ==
-- returns the order of x or "failed"
-- if -1 is returned, the series is identically zero
for n in 0..1000 repeat
empty? x => return -1
not zero? frst x => return n :: I
x := rst x
"failed"
asin z ==
empty? z => 0@Coef :: ST
(coef := frst z) = 0 =>
integrate(0,powern(-1/2,(1@Coef :: ST) - z*z) * (deriv z))
TRANSFCN =>
coef = 1 or coef = -1 =>
x := (1@Coef :: ST) - z*z
-- compute order of 'x'
(ord := orderOrFailed x) case "failed" =>
error concat("asin: ",MAYFPOW)
(order := ord :: I) = -1 => return asin(coef) :: ST
odd? order => error concat("asin: ",FPOWERS)
squirt := powern(1/2,x)
(quot := (deriv z) exquo squirt) case "failed" =>
error concat("asin: ",NOTINV)
integrate(asin coef,quot :: ST)
integrate(asin coef,powern(-1/2,(1@Coef :: ST) - z*z) * (deriv z))
error concat("asin: ",TRCONST)
acos z ==
empty? z =>
TRANSFCN => acos(0)$Coef :: ST
error concat("acos: ",TRCONST)
TRANSFCN =>
coef := frst z
coef = 1 or coef = -1 =>
x := (1@Coef :: ST) - z*z
-- compute order of 'x'
(ord := orderOrFailed x) case "failed" =>
error concat("acos: ",MAYFPOW)
(order := ord :: I) = -1 => return acos(coef) :: ST
odd? order => error concat("acos: ",FPOWERS)
squirt := powern(1/2,x)
(quot := (-deriv z) exquo squirt) case "failed" =>
error concat("acos: ",NOTINV)
integrate(acos coef,quot :: ST)
integrate(acos coef,-powern(-1/2,(1@Coef :: ST) - z*z) * (deriv z))
error concat("acos: ",TRCONST)
atan z ==
empty? z => 0@Coef :: ST
(coef := frst z) = 0 =>
integrate(0,(recip((1@Coef :: ST) + z*z) :: ST) * (deriv z))
TRANSFCN =>
(y := recip((1@Coef :: ST) + z*z)) case "failed" =>
error concat("atan: ",LOGS)
integrate(atan coef,(y :: ST) * (deriv z))
error concat("atan: ",TRCONST)
acot z ==
empty? z =>
TRANSFCN => acot(0)$Coef :: ST
error concat("acot: ",TRCONST)
TRANSFCN =>
(y := recip((1@Coef :: ST) + z*z)) case "failed" =>
error concat("acot: ",LOGS)
integrate(acot frst z,-(y :: ST) * (deriv z))
error concat("acot: ",TRCONST)
asec z ==
empty? z => error "asec: constant coefficient should not be 0"
TRANSFCN =>
(coef := frst z) = 0 =>
error "asec: constant coefficient should not be 0"
coef = 1 or coef = -1 =>
x := z*z - (1@Coef :: ST)
-- compute order of 'x'
(ord := orderOrFailed x) case "failed" =>
error concat("asec: ",MAYFPOW)
(order := ord :: I) = -1 => return asec(coef) :: ST
odd? order => error concat("asec: ",FPOWERS)
squirt := powern(1/2,x)
(quot := (deriv z) exquo squirt) case "failed" =>
error concat("asec: ",NOTINV)
(quot2 := (quot :: ST) exquo z) case "failed" =>
error concat("asec: ",NOTINV)
integrate(asec coef,quot2 :: ST)
integrate(asec coef,(powern(-1/2,z*z-(1@Coef::ST))*(deriv z)) / z)
error concat("asec: ",TRCONST)
acsc z ==
empty? z => error "acsc: constant coefficient should not be zero"
TRANSFCN =>
(coef := frst z) = 0 =>
error "acsc: constant coefficient should not be zero"
coef = 1 or coef = -1 =>
x := z*z - (1@Coef :: ST)
-- compute order of 'x'
(ord := orderOrFailed x) case "failed" =>
error concat("acsc: ",MAYFPOW)
(order := ord :: I) = -1 => return acsc(coef) :: ST
odd? order => error concat("acsc: ",FPOWERS)
squirt := powern(1/2,x)
(quot := (-deriv z) exquo squirt) case "failed" =>
error concat("acsc: ",NOTINV)
(quot2 := (quot :: ST) exquo z) case "failed" =>
error concat("acsc: ",NOTINV)
integrate(acsc coef,quot2 :: ST)
integrate(acsc coef,-(powern(-1/2,z*z-(1@Coef::ST))*(deriv z)) / z)
error concat("acsc: ",TRCONST)
--% Hyperbolic Trigonometric Functions
sinhcosh z ==
empty? z => [0@Coef :: ST,1@Coef :: ST]
l :=
(coef := frst z) = 0 => YS(sincosre(0,1,#1,deriv z,1),2)
TRANSFCN => YS(sincosre(sinh coef,cosh coef,#1,deriv z,1),2)
error concat("sinhcosh: ",TRCONST)
[first l,second l]
sinh z == sinhcosh(z).sinh
cosh z == sinhcosh(z).cosh
tanh z ==
empty? z => 0@Coef :: ST
(coef := frst z) = 0 => YS tanre(0,#1,deriv z,-1)
TRANSFCN => YS tanre(tanh coef,#1,deriv z,-1)
error concat("tanh: ",TRCONST)
coth z ==
tanhz := tanh z
empty? tanhz => error "coth: coth(0) is undefined"
(frst tanhz) = 0 => error concat("coth: ",NPOWERS)
recip(tanhz) :: ST
sech z ==
coshz := cosh z
(empty? coshz) or (frst coshz = 0) => error concat("sech: ",NPOWERS)
recip(coshz) :: ST
csch z ==
sinhz := sinh z
(empty? sinhz) or (frst sinhz = 0) => error concat("csch: ",NPOWERS)
recip(sinhz) :: ST
asinh z ==
empty? z => 0@Coef :: ST
(coef := frst z) = 0 => log(z + powern(1/2,(1@Coef :: ST) + z*z))
TRANSFCN =>
x := (1@Coef :: ST) + z*z
-- compute order of 'x', in case coefficient(z,0) = +- %i
(ord := orderOrFailed x) case "failed" =>
error concat("asinh: ",MAYFPOW)
(order := ord :: I) = -1 => return asinh(coef) :: ST
odd? order => error concat("asinh: ",FPOWERS)
-- the argument to 'log' must have a non-zero constant term
log(z + powern(1/2,x))
error concat("asinh: ",TRCONST)
acosh z ==
empty? z =>
TRANSFCN => acosh(0)$Coef :: ST
error concat("acosh: ",TRCONST)
TRANSFCN =>
coef := frst z
coef = 1 or coef = -1 =>
x := z*z - (1@Coef :: ST)
-- compute order of 'x'
(ord := orderOrFailed x) case "failed" =>
error concat("acosh: ",MAYFPOW)
(order := ord :: I) = -1 => return acosh(coef) :: ST
odd? order => error concat("acosh: ",FPOWERS)
-- the argument to 'log' must have a non-zero constant term
log(z + powern(1/2,x))
log(z + powern(1/2,z*z - (1@Coef :: ST)))
error concat("acosh: ",TRCONST)
atanh z ==
empty? z => 0@Coef :: ST
(coef := frst z) = 0 =>
(inv(2::RN)::Coef) * log(((1@Coef :: ST) + z)/((1@Coef :: ST) - z))
TRANSFCN =>
coef = 1 or coef = -1 => error concat("atanh: ",LOGS)
(inv(2::RN)::Coef) * log(((1@Coef :: ST) + z)/((1@Coef :: ST) - z))
error concat("atanh: ",TRCONST)
acoth z ==
empty? z =>
TRANSFCN => acoth(0)$Coef :: ST
error concat("acoth: ",TRCONST)
TRANSFCN =>
frst z = 1 or frst z = -1 => error concat("acoth: ",LOGS)
(inv(2::RN)::Coef) * log((z + (1@Coef :: ST))/(z - (1@Coef :: ST)))
error concat("acoth: ",TRCONST)
asech z ==
empty? z => error "asech: asech(0) is undefined"
TRANSFCN =>
(coef := frst z) = 0 => error concat("asech: ",NPOWLOG)
coef = 1 or coef = -1 =>
x := (1@Coef :: ST) - z*z
-- compute order of 'x'
(ord := orderOrFailed x) case "failed" =>
error concat("asech: ",MAYFPOW)
(order := ord :: I) = -1 => return asech(coef) :: ST
odd? order => error concat("asech: ",FPOWERS)
log(((1@Coef :: ST) + powern(1/2,x))/z)
log(((1@Coef :: ST) + powern(1/2,(1@Coef :: ST) - z*z))/z)
error concat("asech: ",TRCONST)
acsch z ==
empty? z => error "acsch: acsch(0) is undefined"
TRANSFCN =>
frst z = 0 => error concat("acsch: ",NPOWLOG)
x := z*z + (1@Coef :: ST)
-- compute order of 'x'
(ord := orderOrFailed x) case "failed" =>
error concat("acsc: ",MAYFPOW)
(order := ord :: I) = -1 => return acsch(frst z) :: ST
odd? order => error concat("acsch: ",FPOWERS)
log(((1@Coef :: ST) + powern(1/2,x))/z)
error concat("acsch: ",TRCONST)
)abbrev package STTFNC StreamTranscendentalFunctionsNonCommutative
++ Author: Clifton J. Williamson
++ Date Created: 26 May 1994
++ Date Last Updated: 26 May 1994
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: Taylor series, transcendental function, non-commutative
++ Examples:
++ References:
++ Description:
++ StreamTranscendentalFunctionsNonCommutative implements transcendental
++ functions on Taylor series over a non-commutative ring, where a Taylor
++ series is represented by a stream of its coefficients.
StreamTranscendentalFunctionsNonCommutative(Coef): _
Exports == Implementation where
Coef : Algebra Fraction Integer
I ==> Integer
SG ==> String
ST ==> Stream Coef
STTF ==> StreamTranscendentalFunctions Coef
Exports ==> with
--% Exponentials and Logarithms
exp : ST -> ST
++ exp(st) computes the exponential of a power series st.
log : ST -> ST
++ log(st) computes the log of a power series.
** : (ST,ST) -> ST
++ st1 ** st2 computes the power of a power series st1 by another
++ power series st2.
--% TrigonometricFunctionCategory
sin : ST -> ST
++ sin(st) computes sine of a power series st.
cos : ST -> ST
++ cos(st) computes cosine of a power series st.
tan : ST -> ST
++ tan(st) computes tangent of a power series st.
cot : ST -> ST
++ cot(st) computes cotangent of a power series st.
sec : ST -> ST
++ sec(st) computes secant of a power series st.
csc : ST -> ST
++ csc(st) computes cosecant of a power series st.
asin : ST -> ST
++ asin(st) computes arcsine of a power series st.
acos : ST -> ST
++ acos(st) computes arccosine of a power series st.
atan : ST -> ST
++ atan(st) computes arctangent of a power series st.
acot : ST -> ST
++ acot(st) computes arccotangent of a power series st.
asec : ST -> ST
++ asec(st) computes arcsecant of a power series st.
acsc : ST -> ST
++ acsc(st) computes arccosecant of a power series st.
--% HyperbloicTrigonometricFunctionCategory
sinh : ST -> ST
++ sinh(st) computes the hyperbolic sine of a power series st.
cosh : ST -> ST
++ cosh(st) computes the hyperbolic cosine of a power series st.
tanh : ST -> ST
++ tanh(st) computes the hyperbolic tangent of a power series st.
coth : ST -> ST
++ coth(st) computes the hyperbolic cotangent of a power series st.
sech : ST -> ST
++ sech(st) computes the hyperbolic secant of a power series st.
csch : ST -> ST
++ csch(st) computes the hyperbolic cosecant of a power series st.
asinh : ST -> ST
++ asinh(st) computes the inverse hyperbolic sine of a power series st.
acosh : ST -> ST
++ acosh(st) computes the inverse hyperbolic cosine
++ of a power series st.
atanh : ST -> ST
++ atanh(st) computes the inverse hyperbolic tangent
++ of a power series st.
acoth : ST -> ST
++ acoth(st) computes the inverse hyperbolic
++ cotangent of a power series st.
asech : ST -> ST
++ asech(st) computes the inverse hyperbolic secant of a
++ power series st.
acsch : ST -> ST
++ acsch(st) computes the inverse hyperbolic
++ cosecant of a power series st.
Implementation ==> add
import StreamTaylorSeriesOperations(Coef)
--% Error Reporting
ZERO : SG := "series must have constant coefficient zero"
ONE : SG := "series must have constant coefficient one"
NPOWERS : SG := "series expansion has terms of negative degree"
--% Exponentials and Logarithms
exp z ==
empty? z => 1@Coef :: ST
(frst z) = 0 =>
expx := exp(monom(1,1))$STTF
compose(expx,z)
error concat("exp: ",ZERO)
log z ==
empty? z => error concat("log: ",ONE)
(frst z) = 1 =>
log1PlusX := log(monom(1,0) + monom(1,1))$STTF
compose(log1PlusX,z - monom(1,0))
error concat("log: ",ONE)
(z1:ST) ** (z2:ST) == exp(log(z1) * z2)
--% Trigonometric Functions
sin z ==
empty? z => 0@Coef :: ST
(frst z) = 0 =>
sinx := sin(monom(1,1))$STTF
compose(sinx,z)
error concat("sin: ",ZERO)
cos z ==
empty? z => 1@Coef :: ST
(frst z) = 0 =>
cosx := cos(monom(1,1))$STTF
compose(cosx,z)
error concat("cos: ",ZERO)
tan z ==
empty? z => 0@Coef :: ST
(frst z) = 0 =>
tanx := tan(monom(1,1))$STTF
compose(tanx,z)
error concat("tan: ",ZERO)
cot z ==
empty? z => error "cot: cot(0) is undefined"
(frst z) = 0 => error concat("cot: ",NPOWERS)
error concat("cot: ",ZERO)
sec z ==
empty? z => 1@Coef :: ST
(frst z) = 0 =>
secx := sec(monom(1,1))$STTF
compose(secx,z)
error concat("sec: ",ZERO)
csc z ==
empty? z => error "csc: csc(0) is undefined"
(frst z) = 0 => error concat("csc: ",NPOWERS)
error concat("csc: ",ZERO)
asin z ==
empty? z => 0@Coef :: ST
(frst z) = 0 =>
asinx := asin(monom(1,1))$STTF
compose(asinx,z)
error concat("asin: ",ZERO)
atan z ==
empty? z => 0@Coef :: ST
(frst z) = 0 =>
atanx := atan(monom(1,1))$STTF
compose(atanx,z)
error concat("atan: ",ZERO)
acos z == error "acos: acos undefined on this coefficient domain"
acot z == error "acot: acot undefined on this coefficient domain"
asec z == error "asec: asec undefined on this coefficient domain"
acsc z == error "acsc: acsc undefined on this coefficient domain"
--% Hyperbolic Trigonometric Functions
sinh z ==
empty? z => 0@Coef :: ST
(frst z) = 0 =>
sinhx := sinh(monom(1,1))$STTF
compose(sinhx,z)
error concat("sinh: ",ZERO)
cosh z ==
empty? z => 1@Coef :: ST
(frst z) = 0 =>
coshx := cosh(monom(1,1))$STTF
compose(coshx,z)
error concat("cosh: ",ZERO)
tanh z ==
empty? z => 0@Coef :: ST
(frst z) = 0 =>
tanhx := tanh(monom(1,1))$STTF
compose(tanhx,z)
error concat("tanh: ",ZERO)
coth z ==
empty? z => error "coth: coth(0) is undefined"
(frst z) = 0 => error concat("coth: ",NPOWERS)
error concat("coth: ",ZERO)
sech z ==
empty? z => 1@Coef :: ST
(frst z) = 0 =>
sechx := sech(monom(1,1))$STTF
compose(sechx,z)
error concat("sech: ",ZERO)
csch z ==
empty? z => error "csch: csch(0) is undefined"
(frst z) = 0 => error concat("csch: ",NPOWERS)
error concat("csch: ",ZERO)
asinh z ==
empty? z => 0@Coef :: ST
(frst z) = 0 =>
asinhx := asinh(monom(1,1))$STTF
compose(asinhx,z)
error concat("asinh: ",ZERO)
atanh z ==
empty? z => 0@Coef :: ST
(frst z) = 0 =>
atanhx := atanh(monom(1,1))$STTF
compose(atanhx,z)
error concat("atanh: ",ZERO)
acosh z == error "acosh: acosh undefined on this coefficient domain"
acoth z == error "acoth: acoth undefined on this coefficient domain"
asech z == error "asech: asech undefined on this coefficient domain"
acsch z == error "acsch: acsch undefined on this coefficient domain"
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