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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/algebra trigcat.spad}
+\author{Manuel Bronstein}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{category ELEMFUN ElementaryFunctionCategory}
+<<category ELEMFUN ElementaryFunctionCategory>>=
+)abbrev category ELEMFUN ElementaryFunctionCategory
+++ Category for the elementary functions
+++ Author: Manuel Bronstein
+++ Date Created: ???
+++ Date Last Updated: 14 May 1991
+++ Description: Category for the elementary functions;
+ElementaryFunctionCategory(): Category == with
+    log : $ -> $       ++ log(x) returns the natural logarithm of x.
+    exp : $ -> $       ++ exp(x) returns %e to the power x.
+    "**": ($, $) -> $  ++ x**y returns x to the power y.
+ add
+   if $ has Monoid then
+     x ** y == exp(y * log x)
+
+@
+\section{category AHYP ArcHyperbolicFunctionCategory}
+<<category AHYP ArcHyperbolicFunctionCategory>>=
+)abbrev category AHYP ArcHyperbolicFunctionCategory
+++ Category for the inverse hyperbolic trigonometric functions
+++ Author: ???
+++ Date Created: ???
+++ Date Last Updated: 14 May 1991
+++ Description:
+++ Category for the inverse hyperbolic trigonometric functions;
+ArcHyperbolicFunctionCategory(): Category == with
+    acosh: $ -> $ ++ acosh(x) returns the hyperbolic arc-cosine of x.
+    acoth: $ -> $ ++ acoth(x) returns the hyperbolic arc-cotangent of x.
+    acsch: $ -> $ ++ acsch(x) returns the hyperbolic arc-cosecant of x.
+    asech: $ -> $ ++ asech(x) returns the hyperbolic arc-secant of x.
+    asinh: $ -> $ ++ asinh(x) returns the hyperbolic arc-sine of x.
+    atanh: $ -> $ ++ atanh(x) returns the hyperbolic arc-tangent of x.
+
+@
+\section{category ATRIG ArcTrigonometricFunctionCategory}
+The [[asec]] and [[acsc]] functions were modified to include an
+intermediate test to check that the argument has a reciprocal values.
+<<category ATRIG ArcTrigonometricFunctionCategory>>=
+)abbrev category ATRIG ArcTrigonometricFunctionCategory
+++ Category for the inverse trigonometric functions
+++ Author: ???
+++ Date Created: ???
+++ Date Last Updated: 14 May 1991
+++ Description: Category for the inverse trigonometric functions;
+ArcTrigonometricFunctionCategory(): Category == with
+    acos: $ -> $       ++ acos(x) returns the arc-cosine of x.
+    acot: $ -> $       ++ acot(x) returns the arc-cotangent of x.
+    acsc: $ -> $       ++ acsc(x) returns the arc-cosecant of x.
+    asec: $ -> $       ++ asec(x) returns the arc-secant of x.
+    asin: $ -> $       ++ asin(x) returns the arc-sine of x.
+    atan: $ -> $       ++ atan(x) returns the arc-tangent of x.
+ add
+    if $ has Ring then
+       asec(x) ==
+         (a := recip x) case "failed" => error "asec: no reciprocal"
+         acos(a::$)
+       acsc(x) == 
+         (a := recip x) case "failed" => error "acsc: no reciprocal"
+         asin(a::$)
+
+@
+\section{category HYPCAT HyperbolicFunctionCategory}
+The [[csch]] and [[sech]] functions were modified to include an
+intermediate test to check that the argument has a reciprocal values.
+<<category HYPCAT HyperbolicFunctionCategory>>=
+)abbrev category HYPCAT HyperbolicFunctionCategory
+++ Category for the hyperbolic trigonometric functions
+++ Author: ???
+++ Date Created: ???
+++ Date Last Updated: 14 May 1991
+++ Description: Category for the hyperbolic trigonometric functions;
+HyperbolicFunctionCategory(): Category == with
+    cosh: $ -> $       ++ cosh(x) returns the hyperbolic cosine of x.
+    coth: $ -> $       ++ coth(x) returns the hyperbolic cotangent of x.
+    csch: $ -> $       ++ csch(x) returns the hyperbolic cosecant of x.
+    sech: $ -> $       ++ sech(x) returns the hyperbolic secant of x.
+    sinh: $ -> $       ++ sinh(x) returns the hyperbolic sine of x.
+    tanh: $ -> $       ++ tanh(x) returns the hyperbolic tangent of x.
+ add
+    if $ has Ring then
+       csch x == 
+         (a := recip(sinh x)) case "failed" => error "csch: no reciprocal"
+         a::$
+       sech x == 
+         (a := recip(cosh x)) case "failed" => error "sech: no reciprocal"
+         a::$
+       tanh x == sinh x * sech x
+       coth x == cosh x * csch x
+       if $ has ElementaryFunctionCategory then
+         cosh x ==
+           e := exp x
+           (e + recip(e)::$) * recip(2::$)::$
+         sinh(x):$ ==
+           e := exp x
+           (e - recip(e)::$) * recip(2::$)::$
+
+@
+\section{category TRANFUN TranscendentalFunctionCategory}
+The [[acsch]], [[asech]], and [[acoth]] functions were modified to
+include an intermediate test to check that the argument has a
+reciprocal values.
+<<category TRANFUN TranscendentalFunctionCategory>>=
+)abbrev category TRANFUN TranscendentalFunctionCategory
+++ Category for the transcendental elementary functions
+++ Author: Manuel Bronstein
+++ Date Created: ???
+++ Date Last Updated: 14 May 1991
+++ Description: Category for the transcendental elementary functions;
+TranscendentalFunctionCategory(): Category ==
+    Join(TrigonometricFunctionCategory,ArcTrigonometricFunctionCategory,
+         HyperbolicFunctionCategory,ArcHyperbolicFunctionCategory,
+         ElementaryFunctionCategory) with
+           pi : () -> $        ++ pi() returns the constant pi.
+   add
+     if $ has Ring then
+       pi()   == 2*asin(1)
+       acsch x == 
+         (a := recip x) case "failed" => error "acsch: no reciprocal"
+         asinh(a::$)
+       asech x == 
+         (a := recip x) case "failed" => error "asech: no reciprocal"
+         acosh(a::$)
+       acoth x == 
+         (a := recip x) case "failed" => error "acoth: no reciprocal"
+         atanh(a::$)
+     if $ has Field and $ has sqrt: $ -> $ then
+       asin x == atan(x/sqrt(1-x**2))
+       acos x == pi()/2::$ - asin x
+       acot x == pi()/2::$ - atan x
+       asinh x == log(x + sqrt(x**2 + 1))
+       acosh x == 2*log(sqrt((x+1)/2::$) + sqrt((x-1)/2::$))
+       atanh x == (log(1+x)-log(1-x))/2::$
+
+@
+\section{category TRIGCAT TrigonometricFunctionCategory}
+The [[csc]] and [[sec]] functions were modified to include an
+intermediate test to check that the argument has a reciprocal values.
+<<category TRIGCAT TrigonometricFunctionCategory>>=
+)abbrev category TRIGCAT TrigonometricFunctionCategory
+++ Category for the trigonometric functions
+++ Author: ???
+++ Date Created: ???
+++ Date Last Updated: 14 May 1991
+++ Description: Category for the trigonometric functions;
+TrigonometricFunctionCategory(): Category == with
+    cos: $ -> $        ++ cos(x) returns the cosine of x.
+    cot: $ -> $        ++ cot(x) returns the cotangent of x.
+    csc: $ -> $        ++ csc(x) returns the cosecant of x.
+    sec: $ -> $        ++ sec(x) returns the secant of x.
+    sin: $ -> $        ++ sin(x) returns the sine of x.
+    tan: $ -> $        ++ tan(x) returns the tangent of x.
+ add
+    if $ has Ring then
+       csc x == 
+         (a := recip(sin x)) case "failed" => error "csc: no reciprocal"
+         a::$
+       sec x == 
+         (a := recip(cos x)) case "failed" => error "sec: no reciprocal"
+         a::$
+       tan x == sin x * sec x
+       cot x == cos x * csc x
+
+@
+\section{category PRIMCAT PrimitiveFunctionCategory}
+<<category PRIMCAT PrimitiveFunctionCategory>>=
+)abbrev category PRIMCAT PrimitiveFunctionCategory
+++ Category for the integral functions
+++ Author: Manuel Bronstein
+++ Date Created: ???
+++ Date Last Updated: 14 May 1991
+++ Description: Category for the functions defined by integrals;
+PrimitiveFunctionCategory(): Category == with
+    integral: ($, Symbol) -> $
+      ++ integral(f, x) returns the formal integral of f dx.
+    integral: ($, SegmentBinding $) -> $
+      ++ integral(f, x = a..b) returns the formal definite integral
+      ++ of f dx for x between \spad{a} and b.
+
+@
+\section{category LFCAT LiouvillianFunctionCategory}
+<<category LFCAT LiouvillianFunctionCategory>>=
+)abbrev category LFCAT LiouvillianFunctionCategory
+++ Category for the transcendental Liouvillian functions
+++ Author: Manuel Bronstein
+++ Date Created: ???
+++ Date Last Updated: 14 May 1991
+++ Description: Category for the transcendental Liouvillian functions;
+LiouvillianFunctionCategory(): Category ==
+  Join(PrimitiveFunctionCategory, TranscendentalFunctionCategory) with
+    Ei      : $  -> $
+      ++ Ei(x) returns the exponential integral of x, i.e.
+      ++ the integral of \spad{exp(x)/x dx}.
+    Si      : $  -> $
+      ++ Si(x) returns the sine integral of x, i.e.
+      ++ the integral of \spad{sin(x) / x dx}.
+    Ci      : $  -> $
+      ++ Ci(x) returns the cosine integral of x, i.e.
+      ++ the integral of \spad{cos(x) / x dx}.
+    li      : $  -> $
+      ++ li(x) returns the logarithmic integral of x, i.e.
+      ++ the integral of \spad{dx / log(x)}.
+    dilog   : $  -> $
+      ++ dilog(x) returns the dilogarithm of x, i.e.
+      ++ the integral of \spad{log(x) / (1 - x) dx}.
+    erf     : $  -> $
+      ++ erf(x) returns the error function of x, i.e.
+      ++ \spad{2 / sqrt(%pi)} times the integral of \spad{exp(-x**2) dx}.
+
+@
+\section{category CFCAT CombinatorialFunctionCategory}
+<<category CFCAT CombinatorialFunctionCategory>>=
+)abbrev category CFCAT CombinatorialFunctionCategory
+++ Category for the usual combinatorial functions
+++ Author: Manuel Bronstein
+++ Date Created: ???
+++ Date Last Updated: 14 May 1991
+++ Description: Category for the usual combinatorial functions;
+CombinatorialFunctionCategory(): Category == with
+    binomial   : ($, $) -> $
+      ++ binomial(n,r) returns the \spad{(n,r)} binomial coefficient
+      ++ (often denoted in the literature by \spad{C(n,r)}).
+      ++ Note: \spad{C(n,r) = n!/(r!(n-r)!)} where \spad{n >= r >= 0}.
+    factorial  : $ -> $
+      ++ factorial(n) computes the factorial of n
+      ++ (denoted in the literature by \spad{n!})
+      ++ Note: \spad{n! = n (n-1)! when n > 0}; also, \spad{0! = 1}.
+    permutation: ($, $) -> $
+      ++ permutation(n, m) returns the number of
+      ++ permutations of n objects taken m at a time.
+      ++ Note: \spad{permutation(n,m) = n!/(n-m)!}.
+
+@
+\section{category SPFCAT SpecialFunctionCategory}
+<<category SPFCAT SpecialFunctionCategory>>=
+)abbrev category SPFCAT SpecialFunctionCategory
+++ Category for the other special functions
+++ Author: Manuel Bronstein
+++ Date Created: ???
+++ Date Last Updated: 11 May 1993
+++ Description: Category for the other special functions;
+SpecialFunctionCategory(): Category == with
+    abs :      $ -> $
+        ++ abs(x) returns the absolute value of x.
+    Gamma:     $ -> $
+        ++ Gamma(x) is the Euler Gamma function.
+    Beta:      ($,$)->$
+        ++ Beta(x,y) is \spad{Gamma(x) * Gamma(y)/Gamma(x+y)}.
+    digamma:   $ -> $
+        ++ digamma(x) is the logarithmic derivative of \spad{Gamma(x)}
+        ++ (often written \spad{psi(x)} in the literature).
+    polygamma: ($, $) -> $
+        ++ polygamma(k,x) is the \spad{k-th} derivative of \spad{digamma(x)},
+        ++ (often written \spad{psi(k,x)} in the literature).
+    Gamma:     ($, $) -> $
+        ++ Gamma(a,x) is the incomplete Gamma function.
+    besselJ:   ($,$) -> $
+        ++ besselJ(v,z) is the Bessel function of the first kind.
+    besselY:   ($,$) -> $
+        ++ besselY(v,z) is the Bessel function of the second kind.
+    besselI:   ($,$) -> $
+        ++ besselI(v,z) is the modified Bessel function of the first kind.
+    besselK:   ($,$) -> $
+        ++ besselK(v,z) is the modified Bessel function of the second kind.
+    airyAi:    $ -> $
+        ++ airyAi(x) is the Airy function \spad{Ai(x)}.
+    airyBi:    $ -> $
+        ++ airyBi(x) is the Airy function \spad{Bi(x)}.
+
+@
+\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>>
+
+<<category ELEMFUN ElementaryFunctionCategory>>
+<<category TRIGCAT TrigonometricFunctionCategory>>
+<<category ATRIG ArcTrigonometricFunctionCategory>>
+<<category HYPCAT HyperbolicFunctionCategory>>
+<<category AHYP ArcHyperbolicFunctionCategory>>
+<<category TRANFUN TranscendentalFunctionCategory>>
+<<category PRIMCAT PrimitiveFunctionCategory>>
+<<category LFCAT LiouvillianFunctionCategory>>
+<<category CFCAT CombinatorialFunctionCategory>>
+<<category SPFCAT SpecialFunctionCategory>>
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}