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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra liouv.spad}
\author{Manuel Bronstein}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package LF LiouvillianFunction}
<<package LF LiouvillianFunction>>=
)abbrev package LF LiouvillianFunction
++ Author: Manuel Bronstein
++ Date Created: 1987
++ Date Last Updated: 10 August 1994
++ Keywords: liouvillian, function, primitive, exponential.
++ Examples: )r LF INPUT
++ Description:
++ This package provides liouvillian functions over an integral domain.
LiouvillianFunction(R, F): Exports == Implementation where
R:Join(OrderedSet, IntegralDomain)
F:Join(FunctionSpace R,RadicalCategory,TranscendentalFunctionCategory)
OP ==> BasicOperator
PR ==> Polynomial R
K ==> Kernel F
SE ==> Symbol
O ==> OutputForm
INP ==> InputForm
INV ==> error "Invalid argument"
SPECIALDIFF ==> "%specialDiff"
SPECIALDISP ==> "%specialDisp"
SPECIALINPUT ==> "%specialInput"
SPECIALEQUAL ==> "%specialEqual"
Exports ==> with
belong? : OP -> Boolean
++ belong?(op) checks if op is Liouvillian
operator: OP -> OP
++ operator(op) returns the Liouvillian operator based on op
Ei : F -> F
++ Ei(f) denotes the exponential integral
Si : F -> F
++ Si(f) denotes the sine integral
Ci : F -> F
++ Ci(f) denotes the cosine integral
li : F -> F
++ li(f) denotes the logarithmic integral
erf : F -> F
++ erf(f) denotes the error function
dilog : F -> F
++ dilog(f) denotes the dilogarithm
integral: (F, SE) -> F
++ integral(f,x) indefinite integral of f with respect to x.
integral: (F, SegmentBinding F) -> F
++ integral(f,x = a..b) denotes the definite integral of f with
++ respect to x from \spad{a} to b.
Implementation ==> add
iei : F -> F
isi : F -> F
ici : F -> F
ierf : F -> F
ili : F -> F
ili2 : F -> F
iint : List F -> F
eqint : (K,K) -> Boolean
dvint : (List F, SE) -> F
dvdint : (List F, SE) -> F
ddint : List F -> O
integrand : List F -> F
dummy := new()$SE :: F
opint := operator('integral)$CommonOperators
opdint := operator('%defint)$CommonOperators
opei := operator('Ei)$CommonOperators
opli := operator('li)$CommonOperators
opsi := operator('Si)$CommonOperators
opci := operator('Ci)$CommonOperators
opli2 := operator('dilog)$CommonOperators
operf := operator('erf)$CommonOperators
Si x == opsi x
Ci x == opci x
Ei x == opei x
erf x == operf x
li x == opli x
dilog x == opli2 x
belong? op == has?(op, 'prim)
isi x == kernel(opsi, x)
ici x == kernel(opci, x)
ierf x == (zero? x => 0; kernel(operf, x))
ili2 x == (one? x => INV; kernel(opli2, x))
integrand l == eval(first l, retract(second l)@K, third l)
integral(f:F, x:SE) == opint [eval(f, k:=kernel(x)$K, dummy), dummy, k::F]
iint l ==
zero? first l => 0
kernel(opint, l)
ddint l ==
int(integrand(l)::O * hconcat('d::O, third(l)::O),
third(rest l)::O, third(rest rest l)::O)
eqint(k1,k2) ==
a1:=argument k1
a2:=argument k2
res:=operator k1 = operator k2
if not res then return res
res:= a1 = a2
if res then return res
res:= (a1.3 = a2.3) and (subst(a1.1,[retract(a1.2)@K],[a2.2]) = a2.1)
dvint(l, x) ==
k := retract(second l)@K
differentiate(third l, x) * integrand l
+ opint [differentiate(first l, x), second l, third l]
dvdint(l, x) ==
x = retract(y := third l)@SE => 0
k := retract(d := second l)@K
differentiate(h := third rest rest l,x) * eval(f := first l, k, h)
- differentiate(g := third rest l, x) * eval(f, k, g)
+ opdint [differentiate(f, x), d, y, g, h]
integral(f:F, s: SegmentBinding F) ==
x := kernel(variable s)$K
opdint [eval(f,x,dummy), dummy, x::F, lo segment s, hi segment s]
ili x ==
x = 1 => INV
is?(x,'exp) => Ei first argument(retract(x)@K)
kernel(opli, x)
iei x ==
x = 0 => INV
is?(x,'log) => li first argument(retract(x)@K)
kernel(opei, x)
operator op ==
is?(op,'integral) => opint
is?(op,'%defint) => opdint
is?(op,'Ei) => opei
is?(op,'Si) => opsi
is?(op,'Ci) => opci
is?(op,'li) => opli
is?(op,'erf) => operf
is?(op,'dilog) => opli2
error "Not a Liouvillian operator"
evaluate(opei, iei)$BasicOperatorFunctions1(F)
evaluate(opli, ili)
evaluate(opsi, isi)
evaluate(opci, ici)
evaluate(operf, ierf)
evaluate(opli2, ili2)
evaluate(opint, iint)
derivative(opsi, sin(#1) / #1)
derivative(opci, cos(#1) / #1)
derivative(opei, exp(#1) / #1)
derivative(opli, inv log(#1))
derivative(operf, 2 * exp(-(#1**2)) / sqrt(pi()))
derivative(opli2, log(#1) / (1 - #1))
setProperty(opint,SPECIALEQUAL,eqint@((K,K) -> Boolean) pretend None)
setProperty(opint,SPECIALDIFF,dvint@((List F,SE) -> F) pretend None)
setProperty(opdint,SPECIALDIFF,dvdint@((List F,SE)->F) pretend None)
setProperty(opdint, SPECIALDISP, ddint@(List F -> O) pretend None)
if R has ConvertibleTo INP then
inint : List F -> INP
indint: List F -> INP
pint : List INP -> INP
pint l == convert concat(convert('integral)@INP, l)
inint l ==
r2:= convert([convert("::"::SE)@INP,convert(third l)@INP,convert("Symbol"::SE)@INP]@List INP)@INP
pint [convert(integrand l)@INP, r2]
indint l ==
pint [convert(integrand l)@INP,
convert concat(convert("="::SE)@INP,
[convert(third l)@INP,
convert concat(convert("SEGMENT"::SE)@INP,
[convert(third rest l)@INP,
convert(third rest rest l)@INP])])]
setProperty(opint, SPECIALINPUT, inint@(List F -> INP) pretend None)
setProperty(opdint, SPECIALINPUT, indint@(List F -> INP) pretend None)
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--Copyright (C) 2007-2009, Gabriel Dos Reis.
--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>>
-- SPAD files for the functional world should be compiled in the
-- following order:
--
-- op kl fspace algfunc elemntry LIOUV expr
<<package LF LiouvillianFunction>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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