\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} <>= )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()$SE :: F opint := operator("integral"::Symbol)$CommonOperators opdint := operator("%defint"::Symbol)$CommonOperators opei := operator("Ei"::Symbol)$CommonOperators opli := operator("li"::Symbol)$CommonOperators opsi := operator("Si"::Symbol)$CommonOperators opci := operator("Ci"::Symbol)$CommonOperators opli2 := operator("dilog"::Symbol)$CommonOperators operf := operator("erf"::Symbol)$CommonOperators Si x == opsi x Ci x == opci x Ei x == opei x erf x == operf x li x == opli x dilog x == opli2 x belong? op == has?(op, "prim") isi x == kernel(opsi, x) ici x == kernel(opci, x) ierf x == (zero? x => 0; kernel(operf, x)) -- ili2 x == (one? x => INV; kernel(opli2, x)) ili2 x == ((x = 1) => INV; kernel(opli2, x)) integrand l == eval(first l, retract(second l)@K, third l) integral(f:F, x:SE) == opint [eval(f, k:=kernel(x)$K, dummy), dummy, k::F] iint l == zero? first l => 0 kernel(opint, l) ddint l == int(integrand(l)::O * hconcat("d"::SE::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)$K opdint [eval(f,x,dummy), dummy, x::F, lo segment s, hi segment s] ili x == x = 1 => INV is?(x, "exp"::Symbol) => Ei first argument(retract(x)@K) kernel(opli, x) iei x == x = 0 => INV is?(x, "log"::Symbol) => li first argument(retract(x)@K) kernel(opei, x) operator op == is?(op, "integral"::Symbol) => opint is?(op, "%defint"::Symbol) => opdint is?(op, "Ei"::Symbol) => opei is?(op, "Si"::Symbol) => opsi is?(op, "Ci"::Symbol) => opci is?(op, "li"::Symbol) => opli is?(op, "erf"::Symbol) => operf is?(op, "dilog"::Symbol) => opli2 error "Not a Liouvillian operator" evaluate(opei, iei)$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"::SE)@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} <>= --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. @ <<*>>= <> -- SPAD files for the functional world should be compiled in the -- following order: -- -- op kl fspace algfunc elemntry LIOUV expr <> @ \eject \begin{thebibliography}{99} \bibitem{1} nothing \end{thebibliography} \end{document}