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 Topics FrontPage SandBox SandBoxElemntry <-- You are here. Add conjugate spad)abbrev package EF ElementaryFunction ++ Author: Manuel Bronstein ++ Date Created: 1987 ++ Date Last Updated: 10 April 1995 ++ Keywords: elementary, function, logarithm, exponential. ++ Examples: )r EF INPUT ++ Description: Provides elementary functions over an integral domain. ElementaryFunction(R, F) : Exports == Implementation where R : Join(Comparable, IntegralDomain) F : Join(FunctionSpace R, RadicalCategory) B ==> Boolean L ==> List Z ==> Integer OP ==> BasicOperator K ==> Kernel F INV ==> error "Invalid argument" FSF ==> FunctionalSpecialFunction(R, F) Exports ==> with exp : F -> F ++ exp(x) applies the exponential operator to x log : F -> F ++ log(x) applies the logarithm operator to x sin : F -> F ++ sin(x) applies the sine operator to x cos : F -> F ++ cos(x) applies the cosine operator to x tan : F -> F ++ tan(x) applies the tangent operator to x cot : F -> F ++ cot(x) applies the cotangent operator to x sec : F -> F ++ sec(x) applies the secant operator to x csc : F -> F ++ csc(x) applies the cosecant operator to x asin : F -> F ++ asin(x) applies the inverse sine operator to x acos : F -> F ++ acos(x) applies the inverse cosine operator to x atan : F -> F ++ atan(x) applies the inverse tangent operator to x acot : F -> F ++ acot(x) applies the inverse cotangent operator to x asec : F -> F ++ asec(x) applies the inverse secant operator to x acsc : F -> F ++ acsc(x) applies the inverse cosecant operator to x sinh : F -> F ++ sinh(x) applies the hyperbolic sine operator to x cosh : F -> F ++ cosh(x) applies the hyperbolic cosine operator to x tanh : F -> F ++ tanh(x) applies the hyperbolic tangent operator to x coth : F -> F ++ coth(x) applies the hyperbolic cotangent operator to x sech : F -> F ++ sech(x) applies the hyperbolic secant operator to x csch : F -> F ++ csch(x) applies the hyperbolic cosecant operator to x asinh : F -> F ++ asinh(x) applies the inverse hyperbolic sine operator to x acosh : F -> F ++ acosh(x) applies the inverse hyperbolic cosine operator to x atanh : F -> F ++ atanh(x) applies the inverse hyperbolic tangent operator to x acoth : F -> F ++ acoth(x) applies the inverse hyperbolic cotangent operator to x asech : F -> F ++ asech(x) applies the inverse hyperbolic secant operator to x acsch : F -> F ++ acsch(x) applies the inverse hyperbolic cosecant operator to x pi : () -> F ++ pi() returns the pi operator belong? : OP -> Boolean ++ belong?(p) returns true if operator p is elementary operator : OP -> OP ++ operator(p) returns an elementary operator with the same symbol as p -- the following should be local, but are conditional iisqrt2 : () -> F ++ iisqrt2() should be local but conditional iisqrt3 : () -> F ++ iisqrt3() should be local but conditional iiexp : F -> F ++ iiexp(x) should be local but conditional iilog : F -> F ++ iilog(x) should be local but conditional iisin : F -> F ++ iisin(x) should be local but conditional iicos : F -> F ++ iicos(x) should be local but conditional iitan : F -> F ++ iitan(x) should be local but conditional iicot : F -> F ++ iicot(x) should be local but conditional iisec : F -> F ++ iisec(x) should be local but conditional iicsc : F -> F ++ iicsc(x) should be local but conditional iiasin : F -> F ++ iiasin(x) should be local but conditional iiacos : F -> F ++ iiacos(x) should be local but conditional iiatan : F -> F ++ iiatan(x) should be local but conditional iiacot : F -> F ++ iiacot(x) should be local but conditional iiasec : F -> F ++ iiasec(x) should be local but conditional iiacsc : F -> F ++ iiacsc(x) should be local but conditional iisinh : F -> F ++ iisinh(x) should be local but conditional iicosh : F -> F ++ iicosh(x) should be local but conditional iitanh : F -> F ++ iitanh(x) should be local but conditional iicoth : F -> F ++ iicoth(x) should be local but conditional iisech : F -> F ++ iisech(x) should be local but conditional iicsch : F -> F ++ iicsch(x) should be local but conditional iiasinh : F -> F ++ iiasinh(x) should be local but conditional iiacosh : F -> F ++ iiacosh(x) should be local but conditional iiatanh : F -> F ++ iiatanh(x) should be local but conditional iiacoth : F -> F ++ iiacoth(x) should be local but conditional iiasech : F -> F ++ iiasech(x) should be local but conditional iiacsch : F -> F ++ iiacsch(x) should be local but conditional specialTrigs:(F, L Record(func:F,pole:B)) -> Union(F, "failed") ++ specialTrigs(x, l) should be local but conditional localReal? : F -> Boolean ++ localReal?(x) should be local but conditional Implementation ==> add ELEM := 'elem ipi : List F -> F iexp : F -> F ilog : F -> F iiilog : F -> F isin : F -> F icos : F -> F itan : F -> F icot : F -> F isec : F -> F icsc : F -> F iasin : F -> F iacos : F -> F iatan : F -> F iacot : F -> F iasec : F -> F iacsc : F -> F isinh : F -> F icosh : F -> F itanh : F -> F icoth : F -> F isech : F -> F icsch : F -> F iasinh : F -> F iacosh : F -> F iatanh : F -> F iacoth : F -> F iasech : F -> F iacsch : F -> F dropfun : F -> F kernel : F -> K posrem : (Z, Z) -> Z iisqrt1 : () -> F valueOrPole : Record(func : F, pole : B) -> F oppi := operator('pi)$CommonOperators oplog := operator('log)$CommonOperators opexp := operator('exp)$CommonOperators opsin := operator('sin)$CommonOperators opcos := operator('cos)$CommonOperators optan := operator('tan)$CommonOperators opcot := operator('cot)$CommonOperators opsec := operator('sec)$CommonOperators opcsc := operator('csc)$CommonOperators opasin := operator('asin)$CommonOperators opacos := operator('acos)$CommonOperators opatan := operator('atan)$CommonOperators opacot := operator('acot)$CommonOperators opasec := operator('asec)$CommonOperators opacsc := operator('acsc)$CommonOperators opsinh := operator('sinh)$CommonOperators opcosh := operator('cosh)$CommonOperators optanh := operator('tanh)$CommonOperators opcoth := operator('coth)$CommonOperators opsech := operator('sech)$CommonOperators opcsch := operator('csch)$CommonOperators opasinh := operator('asinh)$CommonOperators opacosh := operator('acosh)$CommonOperators opatanh := operator('atanh)$CommonOperators opacoth := operator('acoth)$CommonOperators opasech := operator('asech)$CommonOperators opacsch := operator('acsch)$CommonOperators -- Pi is a domain... Pie, isqrt1, isqrt2, isqrt3 : F -- following code is conditionalized on arbitraryPrecision to recompute in -- case user changes the precision if R has TranscendentalFunctionCategory then Pie := pi()$R :: F else Pie := kernel(oppi, nil()$List(F)) if R has TranscendentalFunctionCategory and R has arbitraryPrecision then pi() == pi()$R :: F else pi() == Pie if R has imaginary : () -> R then isqrt1 := imaginary()$R :: F else isqrt1 := sqrt(-1::F) if R has RadicalCategory then isqrt2 := sqrt(2::R)::F isqrt3 := sqrt(3::R)::F else isqrt2 := sqrt(2::F) isqrt3 := sqrt(3::F) iisqrt1() == isqrt1 if R has RadicalCategory and R has arbitraryPrecision then iisqrt2() == sqrt(2::R)::F iisqrt3() == sqrt(3::R)::F else iisqrt2() == isqrt2 iisqrt3() == isqrt3 ipi l == pi() log x == oplog x exp x == opexp x sin x == opsin x cos x == opcos x tan x == optan x cot x == opcot x sec x == opsec x csc x == opcsc x asin x == opasin x acos x == opacos x atan x == opatan x acot x == opacot x asec x == opasec x acsc x == opacsc x sinh x == opsinh x cosh x == opcosh x tanh x == optanh x coth x == opcoth x sech x == opsech x csch x == opcsch x asinh x == opasinh x acosh x == opacosh x atanh x == opatanh x acoth x == opacoth x asech x == opasech x acsch x == opacsch x kernel x == retract(x)@K posrem(n, m) == ((r := n rem m) < 0 => r + m; r) valueOrPole rec == (rec.pole => INV; rec.func) belong? op == has?(op, ELEM) operator op == is?(op, 'pi) => oppi is?(op, 'log) => oplog is?(op, 'exp) => opexp is?(op, 'sin) => opsin is?(op, 'cos) => opcos is?(op, 'tan) => optan is?(op, 'cot) => opcot is?(op, 'sec) => opsec is?(op, 'csc) => opcsc is?(op, 'asin) => opasin is?(op, 'acos) => opacos is?(op, 'atan) => opatan is?(op, 'acot) => opacot is?(op, 'asec) => opasec is?(op, 'acsc) => opacsc is?(op, 'sinh) => opsinh is?(op, 'cosh) => opcosh is?(op, 'tanh) => optanh is?(op, 'coth) => opcoth is?(op, 'sech) => opsech is?(op, 'csch) => opcsch is?(op, 'asinh) => opasinh is?(op, 'acosh) => opacosh is?(op, 'atanh) => opatanh is?(op, 'acoth) => opacoth is?(op, 'asech) => opasech is?(op, 'acsch) => opacsch error "Not an elementary operator" dropfun x == ((k := retractIfCan(x)@Union(K, "failed")) case "failed") or empty?(argument(k::K)) => 0 first argument(k::K) if R has RetractableTo Z then specialTrigs(x, values) == (r := retractIfCan(y := x/pi())@Union(Fraction Z, "failed")) case "failed" => "failed" q := r::Fraction(Integer) m := minIndex values (n := retractIfCan(q)@Union(Z, "failed")) case Z => even?(n::Z) => valueOrPole(values.m) valueOrPole(values.(m+1)) (n := retractIfCan(2*q)@Union(Z, "failed")) case Z => -- one?(s := posrem(n::Z, 4)) => valueOrPole(values.(m+2)) (s := posrem(n::Z, 4)) = 1 => valueOrPole(values.(m+2)) valueOrPole(values.(m+3)) (n := retractIfCan(3*q)@Union(Z, "failed")) case Z => -- one?(s := posrem(n::Z, 6)) => valueOrPole(values.(m+4)) (s := posrem(n::Z, 6)) = 1 => valueOrPole(values.(m+4)) s = 2 => valueOrPole(values.(m+5)) s = 4 => valueOrPole(values.(m+6)) valueOrPole(values.(m+7)) (n := retractIfCan(4*q)@Union(Z, "failed")) case Z => -- one?(s := posrem(n::Z, 8)) => valueOrPole(values.(m+8)) (s := posrem(n::Z, 8)) = 1 => valueOrPole(values.(m+8)) s = 3 => valueOrPole(values.(m+9)) s = 5 => valueOrPole(values.(m+10)) valueOrPole(values.(m+11)) (n := retractIfCan(6*q)@Union(Z, "failed")) case Z => -- one?(s := posrem(n::Z, 12)) => valueOrPole(values.(m+12)) (s := posrem(n::Z, 12)) = 1 => valueOrPole(values.(m+12)) s = 5 => valueOrPole(values.(m+13)) s = 7 => valueOrPole(values.(m+14)) valueOrPole(values.(m+15)) "failed" else specialTrigs(x, values) == "failed" isin x == zero? x => 0 y := dropfun x is?(x, opasin) => y is?(x, opacos) => sqrt(1 - y^2) is?(x, opatan) => y / sqrt(1 + y^2) is?(x, opacot) => inv sqrt(1 + y^2) is?(x, opasec) => sqrt(y^2 - 1) / y is?(x, opacsc) => inv y h := inv(2::F) s2 := h * iisqrt2() s3 := h * iisqrt3() u := specialTrigs(x, [[0, false], [0, false], [1, false], [-1, false], [s3, false], [s3, false], [-s3, false], [-s3, false], [s2, false], [s2, false], [-s2, false], [-s2, false], [h, false], [h, false], [-h, false], [-h, false]]) u case F => u :: F kernel(opsin, x) icos x == zero? x => 1 y := dropfun x is?(x, opasin) => sqrt(1 - y^2) is?(x, opacos) => y is?(x, opatan) => inv sqrt(1 + y^2) is?(x, opacot) => y / sqrt(1 + y^2) is?(x, opasec) => inv y is?(x, opacsc) => sqrt(y^2 - 1) / y h := inv(2::F) s2 := h * iisqrt2() s3 := h * iisqrt3() u := specialTrigs(x, [[1, false], [-1, false], [0, false], [0, false], [h, false], [-h, false], [-h, false], [h, false], [s2, false], [-s2, false], [-s2, false], [s2, false], [s3, false], [-s3, false], [-s3, false], [s3, false]]) u case F => u :: F kernel(opcos, x) itan x == zero? x => 0 y := dropfun x is?(x, opasin) => y / sqrt(1 - y^2) is?(x, opacos) => sqrt(1 - y^2) / y is?(x, opatan) => y is?(x, opacot) => inv y is?(x, opasec) => sqrt(y^2 - 1) is?(x, opacsc) => inv sqrt(y^2 - 1) s33 := (s3 := iisqrt3()) / (3::F) u := specialTrigs(x, [[0, false], [0, false], [0, true], [0, true], [s3, false], [-s3, false], [s3, false], [-s3, false], [1, false], [-1, false], [1, false], [-1, false], [s33, false], [-s33, false], [s33, false], [-s33, false]]) u case F => u :: F kernel(optan, x) icot x == zero? x => INV y := dropfun x is?(x, opasin) => sqrt(1 - y^2) / y is?(x, opacos) => y / sqrt(1 - y^2) is?(x, opatan) => inv y is?(x, opacot) => y is?(x, opasec) => inv sqrt(y^2 - 1) is?(x, opacsc) => sqrt(y^2 - 1) s33 := (s3 := iisqrt3()) / (3::F) u := specialTrigs(x, [[0, true], [0, true], [0, false], [0, false], [s33, false], [-s33, false], [s33, false], [-s33, false], [1, false], [-1, false], [1, false], [-1, false], [s3, false], [-s3, false], [s3, false], [-s3, false]]) u case F => u :: F kernel(opcot, x) isec x == zero? x => 1 y := dropfun x is?(x, opasin) => inv sqrt(1 - y^2) is?(x, opacos) => inv y is?(x, opatan) => sqrt(1 + y^2) is?(x, opacot) => sqrt(1 + y^2) / y is?(x, opasec) => y is?(x, opacsc) => y / sqrt(y^2 - 1) s2 := iisqrt2() s3 := 2 * iisqrt3() / (3::F) h := 2::F u := specialTrigs(x, [[1, false], [-1, false], [0, true], [0, true], [h, false], [-h, false], [-h, false], [h, false], [s2, false], [-s2, false], [-s2, false], [s2, false], [s3, false], [-s3, false], [-s3, false], [s3, false]]) u case F => u :: F kernel(opsec, x) icsc x == zero? x => INV y := dropfun x is?(x, opasin) => inv y is?(x, opacos) => inv sqrt(1 - y^2) is?(x, opatan) => sqrt(1 + y^2) / y is?(x, opacot) => sqrt(1 + y^2) is?(x, opasec) => y / sqrt(y^2 - 1) is?(x, opacsc) => y s2 := iisqrt2() s3 := 2 * iisqrt3() / (3::F) h := 2::F u := specialTrigs(x, [[0, true], [0, true], [1, false], [-1, false], [s3, false], [s3, false], [-s3, false], [-s3, false], [s2, false], [s2, false], [-s2, false], [-s2, false], [h, false], [h, false], [-h, false], [-h, false]]) u case F => u :: F kernel(opcsc, x) iasin x == zero? x => 0 -- one? x => pi() / (2::F) (x = 1) => pi() / (2::F) x = -1 => - pi() / (2::F) -- y := dropfun x -- is?(x, opsin) => y -- is?(x, opcos) => pi() / (2::F) - y kernel(opasin, x) iacos x == zero? x => pi() / (2::F) -- one? x => 0 (x = 1) => 0 x = -1 => pi() -- y := dropfun x -- is?(x, opsin) => pi() / (2::F) - y -- is?(x, opcos) => y kernel(opacos, x) iatan x == zero? x => 0 -- one? x => pi() / (4::F) (x = 1) => pi() / (4::F) x = -1 => - pi() / (4::F) x = (r3 := iisqrt3()) => pi() / (3::F) -- one?(x*r3) => pi() / (6::F) (x*r3) = 1 => pi() / (6::F) -- y := dropfun x -- is?(x, optan) => y -- is?(x, opcot) => pi() / (2::F) - y kernel(opatan, x) iacot x == zero? x => pi() / (2::F) -- one? x => pi() / (4::F) (x = 1) => pi() / (4::F) x = -1 => 3 * pi() / (4::F) x = (r3 := iisqrt3()) => pi() / (6::F) x = -r3 => 5 * pi() / (6::F) -- one?(xx := x*r3) => pi() / (3::F) (xx := x*r3) = 1 => pi() / (3::F) xx = -1 => 2* pi() / (3::F) -- y := dropfun x -- is?(x, optan) => pi() / (2::F) - y -- is?(x, opcot) => y kernel(opacot, x) iasec x == zero? x => INV -- one? x => 0 (x = 1) => 0 x = -1 => pi() -- y := dropfun x -- is?(x, opsec) => y -- is?(x, opcsc) => pi() / (2::F) - y kernel(opasec, x) iacsc x == zero? x => INV -- one? x => pi() / (2::F) (x = 1) => pi() / (2::F) x = -1 => - pi() / (2::F) -- y := dropfun x -- is?(x, opsec) => pi() / (2::F) - y -- is?(x, opcsc) => y kernel(opacsc, x) isinh x == zero? x => 0 y := dropfun x is?(x, opasinh) => y is?(x, opacosh) => sqrt(y^2 - 1) is?(x, opatanh) => y / sqrt(1 - y^2) is?(x, opacoth) => - inv sqrt(y^2 - 1) is?(x, opasech) => sqrt(1 - y^2) / y is?(x, opacsch) => inv y kernel(opsinh, x) icosh x == zero? x => 1 y := dropfun x is?(x, opasinh) => sqrt(y^2 + 1) is?(x, opacosh) => y is?(x, opatanh) => inv sqrt(1 - y^2) is?(x, opacoth) => y / sqrt(y^2 - 1) is?(x, opasech) => inv y is?(x, opacsch) => sqrt(y^2 + 1) / y kernel(opcosh, x) itanh x == zero? x => 0 y := dropfun x is?(x, opasinh) => y / sqrt(y^2 + 1) is?(x, opacosh) => sqrt(y^2 - 1) / y is?(x, opatanh) => y is?(x, opacoth) => inv y is?(x, opasech) => sqrt(1 - y^2) is?(x, opacsch) => inv sqrt(y^2 + 1) kernel(optanh, x) icoth x == zero? x => INV y := dropfun x is?(x, opasinh) => sqrt(y^2 + 1) / y is?(x, opacosh) => y / sqrt(y^2 - 1) is?(x, opatanh) => inv y is?(x, opacoth) => y is?(x, opasech) => inv sqrt(1 - y^2) is?(x, opacsch) => sqrt(y^2 + 1) kernel(opcoth, x) isech x == zero? x => 1 y := dropfun x is?(x, opasinh) => inv sqrt(y^2 + 1) is?(x, opacosh) => inv y is?(x, opatanh) => sqrt(1 - y^2) is?(x, opacoth) => sqrt(y^2 - 1) / y is?(x, opasech) => y is?(x, opacsch) => y / sqrt(y^2 + 1) kernel(opsech, x) icsch x == zero? x => INV y := dropfun x is?(x, opasinh) => inv y is?(x, opacosh) => inv sqrt(y^2 - 1) is?(x, opatanh) => sqrt(1 - y^2) / y is?(x, opacoth) => - sqrt(y^2 - 1) is?(x, opasech) => y / sqrt(1 - y^2) is?(x, opacsch) => y kernel(opcsch, x) iasinh x == -- is?(x, opsinh) => first argument kernel x kernel(opasinh, x) iacosh x == -- is?(x, opcosh) => first argument kernel x kernel(opacosh, x) iatanh x == -- is?(x, optanh) => first argument kernel x kernel(opatanh, x) iacoth x == zero? x => INV -- is?(x, opcoth) => first argument kernel x kernel(opacoth, x) iasech x == zero? x => INV -- is?(x, opsech) => first argument kernel x kernel(opasech, x) iacsch x == zero? x => INV -- is?(x, opcsch) => first argument kernel x kernel(opacsch, x) iexp x == zero? x => 1 is?(x, oplog) => first argument kernel x smaller?(x, 0) and empty? variables x => inv iexp(-x) R has RetractableTo Z => i := iisqrt1() xi := x / i y := xi / pi() -- this test saves us a lot of effort in the common case -- when no trigonometic simplifiaction is possible retractIfCan(y)@Union(Fraction Z, "failed") case "failed" => kernel(opexp, x) h := inv(2::F) s2 := h * iisqrt2() s3 := h * iisqrt3() u := specialTrigs(xi, [[1, false], [-1, false], [i, false], [-i, false], [h + i * s3, false], [-h + i * s3, false], [-h - i * s3, false], [h - i * s3, false], [s2 + i * s2, false], [-s2 + i * s2, false], [-s2 - i * s2, false], [s2 - i * s2, false], [s3 + i * h, false], [-s3 + i * h, false], [-s3 - i * h, false], [s3 - i * h, false]]) u case F => u :: F kernel(opexp, x) kernel(opexp, x) -- THIS DETERMINES WHEN TO PERFORM THE log exp f -> f SIMPLIFICATION -- CURRENT BEHAVIOR: --- IF R IS COMPLEX(S) THEN ONLY ELEMENTS WHICH ARE RETRACTABLE TO R --- AND EQUAL TO THEIR CONJUGATES ARE DEEMED REAL (OVERRESTRICTIVE FOR NOW) --- OTHERWISE (e.g. R = INT OR FRAC INT), ALL THE ELEMENTS ARE DEEMED REAL --+ ELEMENTS WHICH ARE NOT EQUAL TO THEIR CONJUGATES ARE DEEMED NOT REAL --+ ALL OTHER ELEMENTS ARE DEEMED REAL if (R has imaginary : () -> R) and (R has conjugate : R -> R) then --+ if (F has conjugate : F -> F) then --+ localReal? x == x = conjugate(x) --+ else if (R has imaginary : () -> R) and (R has conjugate : R -> R) then localReal? x == (u := retractIfCan(x)@Union(R, "failed")) case R and (u::R) = conjugate(u::R) else localReal? x == true iiilog x == zero? x => INV -- one? x => 0 (x = 1) => 0 (u := isExpt(x, opexp)) case Record(var : K, exponent : Integer) => rec := u::Record(var : K, exponent : Integer) arg := first argument(rec.var) localReal? arg => rec.exponent * first argument(rec.var) ilog x ilog x ilog x == -- ((num1 := one?(num := numer x)) or num = -1) and (den := denom x) ~= 1 ((num1 := ((num := numer x) = 1)) or num = -1) and (den := denom x) ~= 1 and empty? variables x => - kernel(oplog, (num1 => den; -den)::F) kernel(oplog, x) if R has ElementaryFunctionCategory then iilog x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iiilog x log(r::R)::F iiexp x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iexp x exp(r::R)::F else iilog x == iiilog x iiexp x == iexp x if R has TrigonometricFunctionCategory then iisin x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => isin x sin(r::R)::F iicos x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icos x cos(r::R)::F iitan x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => itan x tan(r::R)::F iicot x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icot x cot(r::R)::F iisec x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => isec x sec(r::R)::F iicsc x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icsc x csc(r::R)::F else iisin x == isin x iicos x == icos x iitan x == itan x iicot x == icot x iisec x == isec x iicsc x == icsc x if R has ArcTrigonometricFunctionCategory then iiasin x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iasin x asin(r::R)::F iiacos x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacos x acos(r::R)::F iiatan x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iatan x atan(r::R)::F iiacot x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacot x acot(r::R)::F iiasec x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iasec x asec(r::R)::F iiacsc x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacsc x acsc(r::R)::F else iiasin x == iasin x iiacos x == iacos x iiatan x == iatan x iiacot x == iacot x iiasec x == iasec x iiacsc x == iacsc x if R has HyperbolicFunctionCategory then iisinh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => isinh x sinh(r::R)::F iicosh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icosh x cosh(r::R)::F iitanh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => itanh x tanh(r::R)::F iicoth x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icoth x coth(r::R)::F iisech x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => isech x sech(r::R)::F iicsch x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => icsch x csch(r::R)::F else iisinh x == isinh x iicosh x == icosh x iitanh x == itanh x iicoth x == icoth x iisech x == isech x iicsch x == icsch x if R has ArcHyperbolicFunctionCategory then iiasinh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iasinh x asinh(r::R)::F iiacosh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacosh x acosh(r::R)::F iiatanh x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iatanh x atanh(r::R)::F iiacoth x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacoth x acoth(r::R)::F iiasech x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iasech x asech(r::R)::F iiacsch x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iacsch x acsch(r::R)::F else iiasinh x == iasinh x iiacosh x == iacosh x iiatanh x == iatanh x iiacoth x == iacoth x iiasech x == iasech x iiacsch x == iacsch x import from BasicOperatorFunctions1(F) evaluate(oppi, ipi) evaluate(oplog, iilog) -- holomorhic derivative(evaluate(conjugate(evaluate(opexp, iiexp)),(x:F):F+->iiexp(conjugate(x)$FSF)),(x:F):F+->0) derivative(evaluate(conjugate(evaluate(opsin, iisin)),(x:F):F +-> iisin(conjugate(x)$FSF)),(x:F):F+->0) derivative(evaluate(conjugate(evaluate(opcos, iicos)),(x:F):F +-> iicos(conjugate(x)$FSF)),(x:F):F+->0) derivative(evaluate(conjugate(evaluate(optan, iitan)),(x:F):F +-> iitan(conjugate(x)$FSF)),(x:F):F+->0) derivative(evaluate(conjugate(evaluate(opcot, iicot)),(x:F):F +-> iicot(conjugate(x)$FSF)),(x:F):F+->0) derivative(evaluate(conjugate(evaluate(opsec, iisec)),(x:F):F +-> iisec(conjugate(x)$FSF)),(x:F):F+->0) derivative(evaluate(conjugate(evaluate(opcsc, iicsc)),(x:F):F +-> iicsc(conjugate(x)$FSF)),(x:F):F+->0) evaluate(opasin, iiasin) evaluate(opacos, iiacos) evaluate(opatan, iiatan) evaluate(opacot, iiacot) evaluate(opasec, iiasec) evaluate(opacsc, iiacsc) evaluate(opsinh, iisinh) evaluate(opcosh, iicosh) evaluate(optanh, iitanh) evaluate(opcoth, iicoth) evaluate(opsech, iisech) evaluate(opcsch, iicsch) evaluate(opasinh, iiasinh) evaluate(opacosh, iiacosh) evaluate(opatanh, iiatanh) evaluate(opacoth, iiacoth) evaluate(opasech, iiasech) evaluate(opacsch, iiacsch) derivative(opexp, exp) derivative(oplog, inv) derivative(opsin, cos) derivative(opcos, (x : F) : F +-> - sin x) derivative(optan, (x : F) : F +-> 1 + tan(x)^2) derivative(opcot, (x : F) : F +-> - 1 - cot(x)^2) derivative(opsec, (x : F) : F +-> tan(x) * sec(x)) derivative(opcsc, (x : F) : F +-> - cot(x) * csc(x)) derivative(opasin, (x : F) : F +-> inv sqrt(1 - x^2)) derivative(opacos, (x : F) : F +-> - inv sqrt(1 - x^2)) derivative(opatan, (x : F) : F +-> inv(1 + x^2)) derivative(opacot, (x : F) : F +-> - inv(1 + x^2)) derivative(opasec, (x : F) : F +-> inv(x^2 * sqrt(1 - inv(x^2)))) derivative(opacsc, (x : F) : F +-> - inv(x^2 * sqrt(1 - inv(x^2)))) derivative(opsinh, cosh) derivative(opcosh, sinh) derivative(optanh, (x : F) : F +-> 1 - tanh(x)^2) derivative(opcoth, (x : F) : F +-> 1 - coth(x)^2) derivative(opsech, (x : F) : F +-> - tanh(x) * sech(x)) derivative(opcsch, (x : F) : F +-> - coth(x) * csch(x)) derivative(opasinh, (x : F) : F +-> inv sqrt(1 + x^2)) derivative(opacosh, (x : F) : F +-> inv sqrt(x^2 - 1)) derivative(opatanh, (x : F) : F +-> inv(1 - x^2)) derivative(opacoth, (x : F) : F +-> inv(1 - x^2)) derivative(opasech, (x : F) : F +-> - inv(x^2 * sqrt(inv(x^2) - 1))) derivative(opacsch, (x : F) : F +-> - inv(x^2 * sqrt(inv(x^2) + 1))) --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 expr spad Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5788449648295288209-25px001.spad using old system compiler. EF abbreviates package ElementaryFunction ------------------------------------------------------------------------ initializing NRLIB EF for ElementaryFunction compiling into NRLIB EF ****** Domain: R already in scope ****** Domain: F already in scope ****** Domain: R already in scope augmenting R: (TranscendentalFunctionCategory) ****** Domain: R already in scope augmenting R: (TranscendentalFunctionCategory) ****** Domain: R already in scope augmenting R: (arbitraryPrecision) compiling exported pi : () -> F Time: 0.05 SEC. compiling exported pi : () -> F Time: 0 SEC. compiling exported pi : () -> F Time: 0 SEC. augmenting R: (SIGNATURE R imaginary (R)) ****** Domain: R already in scope augmenting R: (RadicalCategory) compiling local iisqrt1 : () -> F Time: 0.01 SEC. ****** Domain: R already in scope augmenting R: (RadicalCategory) ****** Domain: R already in scope augmenting R: (arbitraryPrecision) compiling exported iisqrt2 : () -> F Time: 0 SEC. compiling exported iisqrt3 : () -> F Time: 0.01 SEC. compiling exported iisqrt2 : () -> F Time: 0 SEC. compiling exported iisqrt3 : () -> F Time: 0 SEC. compiling exported iisqrt2 : () -> F Time: 0 SEC. compiling exported iisqrt3 : () -> F Time: 0 SEC. compiling local ipi : List F -> F Time: 0 SEC. compiling exported log : F -> F Time: 0 SEC. compiling exported exp : F -> F Time: 0 SEC. compiling exported sin : F -> F Time: 0 SEC. compiling exported cos : F -> F Time: 0 SEC. compiling exported tan : F -> F Time: 0 SEC. compiling exported cot : F -> F Time: 0 SEC. compiling exported sec : F -> F Time: 0 SEC. compiling exported csc : F -> F Time: 0 SEC. compiling exported asin : F -> F Time: 0 SEC. compiling exported acos : F -> F Time: 0 SEC. compiling exported atan : F -> F Time: 0 SEC. compiling exported acot : F -> F Time: 0 SEC. compiling exported asec : F -> F Time: 0 SEC. compiling exported acsc : F -> F Time: 0 SEC. compiling exported sinh : F -> F Time: 0 SEC. compiling exported cosh : F -> F Time: 0 SEC. compiling exported tanh : F -> F Time: 0.01 SEC. compiling exported coth : F -> F Time: 0 SEC. compiling exported sech : F -> F Time: 0 SEC. compiling exported csch : F -> F Time: 0 SEC. compiling exported asinh : F -> F Time: 0 SEC. compiling exported acosh : F -> F Time: 0 SEC. compiling exported atanh : F -> F Time: 0 SEC. compiling exported acoth : F -> F Time: 0 SEC. compiling exported asech : F -> F Time: 0 SEC. compiling exported acsch : F -> F Time: 0 SEC. compiling local kernel : F -> Kernel F Time: 0 SEC. compiling local posrem : (Integer,Integer) -> Integer Time: 0 SEC. compiling local valueOrPole : Record(func: F,pole: Boolean) -> F Time: 0 SEC. compiling exported belong? : BasicOperator -> Boolean Time: 0 SEC. compiling exported operator : BasicOperator -> BasicOperator Time: 0.01 SEC. compiling local dropfun : F -> F Time: 0.01 SEC. ****** Domain: R already in scope augmenting R: (RetractableTo (Integer)) compiling exported specialTrigs : (F,List Record(func: F,pole: Boolean)) -> Union(F,failed) Time: 0.05 SEC. compiling exported specialTrigs : (F,List Record(func: F,pole: Boolean)) -> Union(F,failed) EF;specialTrigs;FLU;45 is replaced by CONS1failed Time: 0 SEC. compiling local isin : F -> F Time: 0.02 SEC. compiling local icos : F -> F Time: 0.03 SEC. compiling local itan : F -> F Time: 0.02 SEC. compiling local icot : F -> F Time: 0.02 SEC. compiling local isec : F -> F Time: 0.02 SEC. compiling local icsc : F -> F Time: 0.03 SEC. compiling local iasin : F -> F Time: 0 SEC. compiling local iacos : F -> F Time: 0.01 SEC. compiling local iatan : F -> F Time: 0.01 SEC. compiling local iacot : F -> F Time: 0.02 SEC. compiling local iasec : F -> F Time: 0 SEC. compiling local iacsc : F -> F Time: 0.01 SEC. compiling local isinh : F -> F Time: 0.01 SEC. compiling local icosh : F -> F Time: 0.01 SEC. compiling local itanh : F -> F Time: 0.01 SEC. compiling local icoth : F -> F Time: 0.02 SEC. compiling local isech : F -> F Time: 0.01 SEC. compiling local icsch : F -> F Time: 0.01 SEC. compiling local iasinh : F -> F Time: 0 SEC. compiling local iacosh : F -> F Time: 0 SEC. compiling local iatanh : F -> F Time: 0 SEC. compiling local iacoth : F -> F Time: 0 SEC. compiling local iasech : F -> F Time: 0.01 SEC. compiling local iacsch : F -> F Time: 0 SEC. compiling local iexp : F -> F ****** Domain: R already in scope augmenting R: (RetractableTo (Integer)) Time: 0.02 SEC. augmenting R: (SIGNATURE R imaginary (R)) augmenting R: (SIGNATURE R conjugate (R R)) compiling exported localReal? : F -> Boolean Time: 0 SEC. compiling exported localReal? : F -> Boolean EF;localReal?;FB;72 is replaced by QUOTET Time: 0 SEC. compiling exported localReal? : F -> Boolean EF;localReal?;FB;73 is replaced by QUOTET Time: 0.01 SEC. compiling local iiilog : F -> F Time: 0 SEC. compiling local ilog : F -> F Time: 0.02 SEC. ****** Domain: R already in scope augmenting R: (ElementaryFunctionCategory) compiling exported iilog : F -> F Time: 0 SEC. compiling exported iiexp : F -> F Time: 0 SEC. compiling exported iilog : F -> F Time: 0 SEC. compiling exported iiexp : F -> F Time: 0 SEC. ****** Domain: R already in scope augmenting R: (TrigonometricFunctionCategory) compiling exported iisin : F -> F Time: 0 SEC. compiling exported iicos : F -> F Time: 0.01 SEC. compiling exported iitan : F -> F Time: 0 SEC. compiling exported iicot : F -> F Time: 0 SEC. compiling exported iisec : F -> F Time: 0 SEC. compiling exported iicsc : F -> F Time: 0 SEC. compiling exported iisin : F -> F Time: 0 SEC. compiling exported iicos : F -> F Time: 0 SEC. compiling exported iitan : F -> F Time: 0 SEC. compiling exported iicot : F -> F Time: 0 SEC. compiling exported iisec : F -> F Time: 0 SEC. compiling exported iicsc : F -> F Time: 0 SEC. ****** Domain: R already in scope augmenting R: (ArcTrigonometricFunctionCategory) compiling exported iiasin : F -> F Time: 0.01 SEC. compiling exported iiacos : F -> F Time: 0 SEC. compiling exported iiatan : F -> F Time: 0 SEC. compiling exported iiacot : F -> F Time: 0 SEC. compiling exported iiasec : F -> F Time: 0 SEC. compiling exported iiacsc : F -> F Time: 0 SEC. compiling exported iiasin : F -> F Time: 0 SEC. compiling exported iiacos : F -> F Time: 0 SEC. compiling exported iiatan : F -> F Time: 0 SEC. compiling exported iiacot : F -> F Time: 0 SEC. compiling exported iiasec : F -> F Time: 0 SEC. compiling exported iiacsc : F -> F Time: 0 SEC. ****** Domain: R already in scope augmenting R: (HyperbolicFunctionCategory) compiling exported iisinh : F -> F Time: 0.01 SEC. compiling exported iicosh : F -> F Time: 0 SEC. compiling exported iitanh : F -> F Time: 0 SEC. compiling exported iicoth : F -> F Time: 0 SEC. compiling exported iisech : F -> F Time: 0 SEC. compiling exported iicsch : F -> F Time: 0 SEC. compiling exported iisinh : F -> F Time: 0 SEC. compiling exported iicosh : F -> F Time: 0 SEC. compiling exported iitanh : F -> F Time: 0 SEC. compiling exported iicoth : F -> F Time: 0 SEC. compiling exported iisech : F -> F Time: 0 SEC. compiling exported iicsch : F -> F Time: 0 SEC. ****** Domain: R already in scope augmenting R: (ArcHyperbolicFunctionCategory) compiling exported iiasinh : F -> F Time: 0.01 SEC. compiling exported iiacosh : F -> F Time: 0 SEC. compiling exported iiatanh : F -> F Time: 0 SEC. compiling exported iiacoth : F -> F Time: 0 SEC. compiling exported iiasech : F -> F Time: 0 SEC. compiling exported iiacsch : F -> F Time: 0 SEC. compiling exported iiasinh : F -> F Time: 0 SEC. compiling exported iiacosh : F -> F Time: 0 SEC. compiling exported iiatanh : F -> F Time: 0 SEC. compiling exported iiacoth : F -> F Time: 0 SEC. compiling exported iiasech : F -> F Time: 0 SEC. compiling exported iiacsch : F -> F Time: 0 SEC. importing BasicOperatorFunctions1 F ****** comp fails at level 3 with expression: ****** (|derivative| (|evaluate| | << | (|conjugate| (|evaluate| |opexp| |iiexp|)) | >> | (+-> (|:| (|:| |x| F) F) (|iiexp| ((|Sel| (|FunctionalSpecialFunction| R F) |conjugate|) |x|)))) (+-> (|:| (|:| |x| F) F) 0)) ****** level 3 ****** $x:= (conjugate (evaluate opexp iiexp))$m:= $EmptyMode$f:= ((((|constantOpIfCan| #) (|constantOperator| #) (|derivative| #) (|evaluate| #) ...))) >> Apparent user error: no modemap for conjugate with 1 arguments

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