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diff --git a/src/algebra/sttf.spad.pamphlet b/src/algebra/sttf.spad.pamphlet new file mode 100644 index 00000000..1cbc7038 --- /dev/null +++ b/src/algebra/sttf.spad.pamphlet @@ -0,0 +1,743 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/algebra sttf.spad} +\author{William Burge, Clifton J. Williamson} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +\section{package STTF StreamTranscendentalFunctions} +<<package STTF StreamTranscendentalFunctions>>= +)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 :: 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 :: ST,1 :: 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 :: 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 :: 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 :: 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 :: 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 :: ST + (coef := frst z) = 0 => + integrate(0,powern(-1/2,(1 :: ST) - z*z) * (deriv z)) + TRANSFCN => + coef = 1 or coef = -1 => + x := (1 :: 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 :: 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 :: 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 :: ST) - z*z) * (deriv z)) + error concat("acos: ",TRCONST) + + atan z == + empty? z => 0 :: ST + (coef := frst z) = 0 => + integrate(0,(recip((1 :: ST) + z*z) :: ST) * (deriv z)) + TRANSFCN => + (y := recip((1 :: 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 :: 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 :: 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::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 :: 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::ST))*(deriv z)) / z) + error concat("acsc: ",TRCONST) + +--% Hyperbolic Trigonometric Functions + + sinhcosh z == + empty? z => [0 :: ST,1 :: 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 :: 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 :: ST + (coef := frst z) = 0 => log(z + powern(1/2,(1 :: ST) + z*z)) + TRANSFCN => + x := (1 :: 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 :: 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 :: ST))) + error concat("acosh: ",TRCONST) + + atanh z == + empty? z => 0 :: ST + (coef := frst z) = 0 => + (inv(2::RN)::Coef) * log(((1 :: ST) + z)/((1 :: ST) - z)) + TRANSFCN => + coef = 1 or coef = -1 => error concat("atanh: ",LOGS) + (inv(2::RN)::Coef) * log(((1 :: ST) + z)/((1 :: 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 :: ST))/(z - (1 :: 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 :: 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 :: ST) + powern(1/2,x))/z) + log(((1 :: ST) + powern(1/2,(1 :: 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 :: 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 :: ST) + powern(1/2,x))/z) + error concat("acsch: ",TRCONST) + +@ +\section{package STTFNC StreamTranscendentalFunctionsNonCommutative} +<<package STTFNC StreamTranscendentalFunctionsNonCommutative>>= +)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 :: 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))$STTF + compose(log1PlusX,z - monom(1,0)) + error concat("log: ",ONE) + + (z1:ST) ** (z2:ST) == exp(log(z1) * z2) + +--% Trigonometric Functions + + sin z == + empty? z => 0 :: ST + (frst z) = 0 => + sinx := sin(monom(1,1))$STTF + compose(sinx,z) + error concat("sin: ",ZERO) + + cos z == + empty? z => 1 :: ST + (frst z) = 0 => + cosx := cos(monom(1,1))$STTF + compose(cosx,z) + error concat("cos: ",ZERO) + + tan z == + empty? z => 0 :: ST + (frst z) = 0 => + tanx := tan(monom(1,1))$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 :: ST + (frst z) = 0 => + secx := sec(monom(1,1))$STTF + compose(secx,z) + error concat("sec: ",ZERO) + + csc z == + empty? z => error "csc: csc(0) is undefined" + (frst z) = 0 => error concat("csc: ",NPOWERS) + error concat("csc: ",ZERO) + + asin z == + empty? z => 0 :: ST + (frst z) = 0 => + asinx := asin(monom(1,1))$STTF + compose(asinx,z) + error concat("asin: ",ZERO) + + atan z == + empty? z => 0 :: ST + (frst z) = 0 => + atanx := atan(monom(1,1))$STTF + compose(atanx,z) + error concat("atan: ",ZERO) + + acos z == error "acos: acos undefined on this coefficient domain" + acot z == error "acot: acot undefined on this coefficient domain" + asec z == error "asec: asec undefined on this coefficient domain" + acsc z == error "acsc: acsc undefined on this coefficient domain" + +--% Hyperbolic Trigonometric Functions + + sinh z == + empty? z => 0 :: ST + (frst z) = 0 => + sinhx := sinh(monom(1,1))$STTF + compose(sinhx,z) + error concat("sinh: ",ZERO) + + cosh z == + empty? z => 1 :: ST + (frst z) = 0 => + coshx := cosh(monom(1,1))$STTF + compose(coshx,z) + error concat("cosh: ",ZERO) + + tanh z == + empty? z => 0 :: ST + (frst z) = 0 => + tanhx := tanh(monom(1,1))$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 :: ST + (frst z) = 0 => + sechx := sech(monom(1,1))$STTF + compose(sechx,z) + error concat("sech: ",ZERO) + + csch z == + empty? z => error "csch: csch(0) is undefined" + (frst z) = 0 => error concat("csch: ",NPOWERS) + error concat("csch: ",ZERO) + + asinh z == + empty? z => 0 :: ST + (frst z) = 0 => + asinhx := asinh(monom(1,1))$STTF + compose(asinhx,z) + error concat("asinh: ",ZERO) + + atanh z == + empty? z => 0 :: 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" + +@ +\section{License} +<<license>>= +--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +--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. +@ +<<*>>= +<<license>> + +<<package STTF StreamTranscendentalFunctions>> +<<package STTFNC StreamTranscendentalFunctionsNonCommutative>> +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} |