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 Topics FrontPage SandBox SandBoxCombfunc <-- You are here. Add conjugate fricas)library BOP BOP1 BasicOperator is now explicitly exposed in frame initial BasicOperator will be automatically loaded when needed from /var/aw/var/LatexWiki/BOP.NRLIB/BOP BasicOperatorFunctions1 is now explicitly exposed in frame initial BasicOperatorFunctions1 will be automatically loaded when needed from /var/aw/var/LatexWiki/BOP1.NRLIB/BOP1 spad)abbrev category COMBOPC CombinatorialOpsCategory ++ Category for summations and products ++ Author: Manuel Bronstein ++ Date Created: ??? ++ Date Last Updated: 22 February 1993 (JHD/BMT) ++ Description: ++ CombinatorialOpsCategory is the category obtaining by adjoining ++ summations and products to the usual combinatorial operations; CombinatorialOpsCategory() : Category == CombinatorialFunctionCategory with factorials : % -> % ++ factorials(f) rewrites the permutations and binomials in f ++ in terms of factorials; factorials : (%, Symbol) -> % ++ factorials(f, x) rewrites the permutations and binomials in f ++ involving x in terms of factorials; summation : (%, Symbol) -> % ++ summation(f(n), n) returns the formal sum S(n) which verifies ++ S(n+1) - S(n) = f(n); summation : (%, SegmentBinding %) -> % ++ summation(f(n), n = a..b) returns f(a) + ... + f(b) as a ++ formal sum; product : (%, Symbol) -> % ++ product(f(n), n) returns the formal product P(n) which verifies ++ P(n+1)/P(n) = f(n); product : (%, SegmentBinding %) -> % ++ product(f(n), n = a..b) returns f(a) * ... * f(b) as a ++ formal product; )abbrev package COMBF CombinatorialFunction ++ Provides the usual combinatorial functions ++ Author: Manuel Bronstein, Martin Rubey ++ Date Created: 2 Aug 1988 ++ Date Last Updated: 30 October 2005 ++ Description: ++ Provides combinatorial functions over an integral domain. ++ Keywords: combinatorial, function, factorial. ++ Examples: )r COMBF INPUT CombinatorialFunction(R, F) : Exports == Implementation where R : Join(Comparable, IntegralDomain) F : FunctionSpace R OP ==> BasicOperator K ==> Kernel F SE ==> Symbol O ==> OutputForm SMP ==> SparseMultivariatePolynomial(R, K) Z ==> Integer POWER ==> '%power OPEXP ==> 'exp SPECIALDIFF ==> '%specialDiff SPECIALDISP ==> '%specialDisp SPECIALEQUAL ==> '%specialEqual Exports ==> with belong? : OP -> Boolean ++ belong?(op) is true if op is a combinatorial operator; operator : OP -> OP ++ operator(op) returns a copy of op with the domain-dependent ++ properties appropriate for F; ++ error if op is not a combinatorial operator; "^" : (F, F) -> F ++ a ^ b is the formal exponential a^b; binomial : (F, F) -> F ++ binomial(n, r) returns the number of subsets of r objects ++ taken among n objects, i.e. n!/(r! * (n-r)!); permutation : (F, F) -> F ++ permutation(n, r) returns the number of permutations of ++ n objects taken r at a time, i.e. n!/(n-r)!; factorial : F -> F ++ factorial(n) returns the factorial of n, i.e. n!; factorials : F -> F ++ factorials(f) rewrites the permutations and binomials in f ++ in terms of factorials; factorials : (F, SE) -> F ++ factorials(f, x) rewrites the permutations and binomials in f ++ involving x in terms of factorials; summation : (F, SE) -> F ++ summation(f(n), n) returns the formal sum S(n) which verifies ++ S(n+1) - S(n) = f(n); summation : (F, SegmentBinding F) -> F ++ summation(f(n), n = a..b) returns f(a) + ... + f(b) as a ++ formal sum; product : (F, SE) -> F ++ product(f(n), n) returns the formal product P(n) which verifies ++ P(n+1)/P(n) = f(n); product : (F, SegmentBinding F) -> F ++ product(f(n), n = a..b) returns f(a) * ... * f(b) as a ++ formal product; iifact : F -> F ++ iifact(x) should be local but conditional; iibinom : List F -> F ++ iibinom(l) should be local but conditional; iiperm : List F -> F ++ iiperm(l) should be local but conditional; iipow : List F -> F ++ iipow(l) should be local but conditional; iidsum : List F -> F ++ iidsum(l) should be local but conditional; iidprod : List F -> F ++ iidprod(l) should be local but conditional; ipow : List F -> F ++ ipow(l) should be local but conditional; Implementation ==> add COMB := 'comb ifact : F -> F iiipow : List F -> F iperm : List F -> F ibinom : List F -> F isum : List F -> F idsum : List F -> F iprod : List F -> F idprod : List F -> F dsum : List F -> O ddsum : List F -> O dprod : List F -> O ddprod : List F -> O equalsumprod : (K, K) -> Boolean equaldsumprod : (K, K) -> Boolean fourth : List F -> F dvpow1 : List F -> F dvpow2 : List F -> F summand : List F -> F dvsum : (List F, SE) -> F dvdsum : (List F, SE) -> F dvprod : (List F, SE) -> F dvdprod : (List F, SE) -> F facts : (F, List SE) -> F K2fact : (K, List SE) -> F smpfact : (SMP, List SE) -> F -- Must be a macro to produce fresh symbol for each use dummy ==> new()$SE :: F opfact := operator('factorial)$CommonOperators opperm := operator('permutation)$CommonOperators opbinom := operator('binomial)$CommonOperators opsum := operator('summation)$CommonOperators opdsum := operator('%defsum)$CommonOperators opprod := operator('product)$CommonOperators opdprod := operator('%defprod)$CommonOperators oppow := operator(POWER::Symbol)$CommonOperators factorial x == opfact x binomial(x, y) == opbinom [x, y] permutation(x, y) == opperm [x, y] import from F import from Kernel F number?(x : F) : Boolean == if R has RetractableTo(Z) then ground?(x) or ((retractIfCan(x)@Union(Fraction(Z),"failed")) case Fraction(Z)) else ground?(x) x ^ y == -- Do some basic simplifications is?(x, POWER) => args : List F := argument first kernels x not(#args = 2) => error "Too many arguments to ^" number?(first args) and number?(y) => oppow [first(args)^y, second args] oppow [first args, (second args)* y] -- Generic case exp : Union(Record(val:F,exponent:Z),"failed") := isPower x exp case Record(val : F, exponent : Z) => expr := exp::Record(val : F, exponent : Z) oppow [expr.val, (expr.exponent)*y] oppow [x, y] belong? op == has?(op, COMB) fourth l == third rest l dvpow1 l == second(l) * first(l) ^ (second l - 1) factorials x == facts(x, variables x) factorials(x, v) == facts(x, [v]) facts(x, l) == smpfact(numer x, l) / smpfact(denom x, l) summand l == eval(first l, retract(second l)@K, third l) product(x : F, i : SE) == dm := dummy opprod [eval(x, k := kernel(i)$K, dm), dm, k::F] summation(x : F, i : SE) == dm := dummy opsum [eval(x, k := kernel(i)$K, dm), dm, k::F] dvsum(l, x) == opsum [differentiate(first l, x), second l, third l] dvdsum(l, x) == x = retract(y := third l)@SE => 0 if member?(x, variables(h := third rest rest l)) or member?(x, variables(g := third rest l)) then error "a sum cannot be differentiated with respect to a bound" else opdsum [differentiate(first l, x), second l, y, g, h] dvprod(l, x) == dm := retract(dummy)@SE f := eval(first l, retract(second l)@K, dm::F) p := product(f, dm) opsum [differentiate(first l, x)/first l * p, second l, third l] dvdprod(l, x) == x = retract(y := third l)@SE => 0 if member?(x, variables(h := third rest rest l)) or member?(x, variables(g := third rest l)) then error "a product cannot be differentiated with respect to a bound" else opdsum cons(differentiate(first l, x)/first l, rest l) * opdprod l dprod l == prod(summand(l)::O, third(l)::O) ddprod l == prod(summand(l)::O, third(l)::O = fourth(l)::O, fourth(rest l)::O) dsum l == sum(summand(l)::O, third(l)::O) ddsum l == sum(summand(l)::O, third(l)::O = fourth(l)::O, fourth(rest l)::O) equalsumprod(s1, s2) == l1 := argument s1 l2 := argument s2 (eval(first l1, retract(second l1)@K, second l2) = first l2) equaldsumprod(s1, s2) == l1 := argument s1 l2 := argument s2 ((third rest l1 = third rest l2) and (third rest rest l1 = third rest rest l2) and (eval(first l1, retract(second l1)@K, second l2) = first l2)) product(x : F, s : SegmentBinding F) == k := kernel(variable s)$K dm := dummy opdprod [eval(x, k, dm), dm, k::F, lo segment s, hi segment s] summation(x : F, s : SegmentBinding F) == k := kernel(variable s)$K dm := dummy opdsum [eval(x, k, dm), dm, k::F, lo segment s, hi segment s] smpfact(p, l) == map(x +-> K2fact(x, l), x +-> x::F, p)$PolynomialCategoryLifting( IndexedExponents K, K, R, SMP, F) K2fact(k, l) == empty? [v for v in variables(kf := k::F) | member?(v, l)] => kf empty?(args : List F := [facts(a, l) for a in argument k]) => kf is?(k, opperm) => factorial(n := first args) / factorial(n - second args) is?(k, opbinom) => n := first args p := second args factorial(n) / (factorial(p) * factorial(n-p)) (operator k) args operator op == is?(op, 'factorial) => opfact is?(op, 'permutation) => opperm is?(op, 'binomial) => opbinom is?(op, 'summation) => opsum is?(op, '%defsum) => opdsum is?(op, 'product) => opprod is?(op, '%defprod) => opdprod is?(op, POWER) => oppow error "Not a combinatorial operator" iprod l == zero? first l => 0 -- one? first l => 1 (first l = 1) => 1 kernel(opprod, l) isum l == zero? first l => 0 kernel(opsum, l) idprod l == member?(retract(second l)@SE, variables first l) => kernel(opdprod, l) first(l) ^ (fourth rest l - fourth l + 1) idsum l == member?(retract(second l)@SE, variables first l) => kernel(opdsum, l) first(l) * (fourth rest l - fourth l + 1) ifact x == -- zero? x or one? x => 1 zero? x or (x = 1) => 1 kernel(opfact, x) ibinom l == n := first l ((p := second l) = 0) or (p = n) => 1 -- one? p or (p = n - 1) => n (p = 1) or (p = n - 1) => n kernel(opbinom, l) iperm l == zero? second l => 1 kernel(opperm, l) if R has RetractableTo Z then iidsum l == (r1 := retractIfCan(fourth l)@Union(Z,"failed")) case "failed" or (r2 := retractIfCan(fourth rest l)@Union(Z,"failed")) case "failed" or (k := retractIfCan(second l)@Union(K,"failed")) case "failed" => idsum l +/[eval(first l, k::K, i::F) for i in r1::Z .. r2::Z] iidprod l == (r1 := retractIfCan(fourth l)@Union(Z,"failed")) case "failed" or (r2 := retractIfCan(fourth rest l)@Union(Z,"failed")) case "failed" or (k := retractIfCan(second l)@Union(K,"failed")) case "failed" => idprod l */[eval(first l, k::K, i::F) for i in r1::Z .. r2::Z] iiipow l == (u := isExpt(x := first l, OPEXP)) case "failed" => kernel(oppow, l) rec := u::Record(var : K, exponent : Z) y := first argument(rec.var) (r := retractIfCan(y)@Union(Fraction Z, "failed")) case "failed" => kernel(oppow, l) (operator(rec.var)) (rec.exponent * y * second l) if F has RadicalCategory then ipow l == (r := retractIfCan(second l)@Union(Fraction Z,"failed")) case "failed" => iiipow l first(l) ^ (r::Fraction(Z)) else ipow l == (r := retractIfCan(second l)@Union(Z, "failed")) case "failed" => iiipow l first(l) ^ (r::Z) else ipow l == zero?(x := first l) => zero? second l => error "0 ^ 0" 0 -- one? x or zero?(n := second l) => 1 (x = 1) or zero?(n : F := second l) => 1 -- one? n => x (n = 1) => x (u := isExpt(x, OPEXP)) case "failed" => kernel(oppow, l) rec := u::Record(var : K, exponent : Z) -- one?(y := first argument(rec.var)) or y = -1 => ((y := first argument(rec.var))=1) or y = -1 => (operator(rec.var)) (rec.exponent * y * n) kernel(oppow, l) if R has CombinatorialFunctionCategory then iifact x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => ifact x factorial(r::R)::F iiperm l == (r1 := retractIfCan(first l)@Union(R,"failed")) case "failed" or (r2 := retractIfCan(second l)@Union(R,"failed")) case "failed" => iperm l permutation(r1::R, r2::R)::F if R has RetractableTo(Z) and F has Algebra(Fraction(Z)) then -- Try to explicitly evaluete binomial coefficient -- We currently simplify binomial coefficients for non-negative -- integral second argument, using the formula -- $$\binom{n}{k}=\frac{1}{k!}\prod_{i=0..k-1} (n-i),$$ iibinom l == (s := retractIfCan(second l)@Union(R,"failed")) case R and (t := retractIfCan(s)@Union(Z,"failed")) case Z and t>0 => ans := 1::F for i in 0..t-1 repeat ans := ans*(first l - i::R::F) (1/factorial t) * ans (s := retractIfCan(first l-second l)@Union(R,"failed")) case R and (t := retractIfCan(s)@Union(Z,"failed")) case Z and t>0 => ans := 1::F for i in 1..t repeat ans := ans*(second l+i::R::F) (1/factorial t) * ans (r1 := retractIfCan(first l)@Union(R,"failed")) case "failed" or (r2 := retractIfCan(second l)@Union(R,"failed")) case "failed" => ibinom l binomial(r1::R, r2::R)::F else iibinom l == (r1 := retractIfCan(first l)@Union(R,"failed")) case "failed" or (r2 := retractIfCan(second l)@Union(R,"failed")) case "failed" => ibinom l binomial(r1::R, r2::R)::F else iifact x == ifact x iibinom l == ibinom l iiperm l == iperm l if R has ElementaryFunctionCategory then iipow l == (r1 := retractIfCan(first l)@Union(R,"failed")) case "failed" or (r2 := retractIfCan(second l)@Union(R,"failed")) case "failed" => ipow l (r1::R ^ r2::R)::F else iipow l == ipow l if F has ElementaryFunctionCategory then dvpow2 l == if zero?(first l) then 0 else log(first l) * first(l) ^ second(l) evaluate(opfact, iifact)$BasicOperatorFunctions1(F) evaluate(oppow, iipow) evaluate(opperm, iiperm) evaluate(opbinom, iibinom) evaluate(opsum, isum) evaluate(opdsum, iidsum) evaluate(opprod, iprod) evaluate(opdprod, iidprod) derivative(oppow, [dvpow1, dvpow2]) setProperty(opsum, SPECIALDIFF, dvsum@((List F, SE) -> F) pretend None) setProperty(opdsum, SPECIALDIFF, dvdsum@((List F, SE)->F) pretend None) setProperty(opprod, SPECIALDIFF, dvprod@((List F, SE)->F) pretend None) setProperty(opdprod, SPECIALDIFF, dvdprod@((List F, SE)->F) pretend None) setProperty(opsum, SPECIALDISP, dsum@(List F -> O) pretend None) setProperty(opdsum, SPECIALDISP, ddsum@(List F -> O) pretend None) setProperty(opprod, SPECIALDISP, dprod@(List F -> O) pretend None) setProperty(opdprod, SPECIALDISP, ddprod@(List F -> O) pretend None) setProperty(opsum, SPECIALEQUAL, equalsumprod@((K, K) -> Boolean) pretend None) setProperty(opdsum, SPECIALEQUAL, equaldsumprod@((K, K) -> Boolean) pretend None) setProperty(opprod, SPECIALEQUAL, equalsumprod@((K, K) -> Boolean) pretend None) setProperty(opdprod, SPECIALEQUAL, equaldsumprod@((K, K) -> Boolean) pretend None) )abbrev package FSPECF FunctionalSpecialFunction )boot$tryRecompileArguments := nil ++ Provides the special functions ++ Author: Manuel Bronstein ++ Date Created: 18 Apr 1989 ++ Description: Provides some special functions over an integral domain. ++ Keywords: special, function. ++ Date Last Updated: 2 October 2014 ++ Added: 'conjugate' ++ Modied: 'abs' FunctionalSpecialFunction(R, F) : Exports == Implementation where R : Join(Comparable, IntegralDomain) F : FunctionSpace R OP ==> BasicOperator K ==> Kernel F SE ==> Symbol SPECIALDIFF ==> '%specialDiff Exports ==> with belong? : OP -> Boolean ++ belong?(op) is true if op is a special function operator; operator : OP -> OP ++ operator(op) returns a copy of op with the domain-dependent ++ properties appropriate for F; ++ error if op is not a special function operator abs : F -> F ++ abs(f) returns the absolute value operator applied to f conjugate: F -> F ++ conjugate(f) returns the conjugate operator applied to f Gamma : F -> F ++ Gamma(f) returns the formal Gamma function applied to f Gamma : (F, F) -> F ++ Gamma(a, x) returns the incomplete Gamma function applied to a and x Beta : (F, F) -> F ++ Beta(x, y) returns the beta function applied to x and y digamma : F->F ++ digamma(x) returns the digamma function applied to x polygamma : (F, F) ->F ++ polygamma(x, y) returns the polygamma function applied to x and y besselJ : (F, F) -> F ++ besselJ(x, y) returns the besselj function applied to x and y besselY : (F, F) -> F ++ besselY(x, y) returns the bessely function applied to x and y besselI : (F, F) -> F ++ besselI(x, y) returns the besseli function applied to x and y besselK : (F, F) -> F ++ besselK(x, y) returns the besselk function applied to x and y airyAi : F -> F ++ airyAi(x) returns the Airy Ai function applied to x airyAiPrime : F -> F ++ airyAiPrime(x) returns the derivative of Airy Ai function applied to x airyBi : F -> F ++ airyBi(x) returns the Airy Bi function applied to x airyBiPrime : F -> F ++ airyBiPrime(x) returns the derivative of Airy Bi function applied to x lambertW : F -> F ++ lambertW(x) is the Lambert W function at x polylog : (F, F) -> F ++ polylog(s, x) is the polylogarithm of order s at x weierstrassP : (F, F, F) -> F ++ weierstrassP(g2, g3, x) weierstrassPPrime : (F, F, F) -> F ++ weierstrassPPrime(g2, g3, x) weierstrassSigma : (F, F, F) -> F ++ weierstrassSigma(g2, g3, x) weierstrassZeta : (F, F, F) -> F ++ weierstrassZeta(g2, g3, x) -- weierstrassPInverse : (F, F, F) -> F -- ++ weierstrassPInverse(g2, g3, z) is the inverse of Weierstrass -- ++ P function, defined by the formula -- ++ \spad{weierstrassP(g2, g3, weierstrassPInverse(g2, g3, z)) = z} whittakerM : (F, F, F) -> F ++ whittakerM(k, m, z) is the Whittaker M function whittakerW : (F, F, F) -> F ++ whittakerW(k, m, z) is the Whittaker W function angerJ : (F, F) -> F ++ angerJ(v, z) is the Anger J function weberE : (F, F) -> F ++ weberE(v, z) is the Weber E function struveH : (F, F) -> F ++ struveH(v, z) is the Struve H function struveL : (F, F) -> F ++ struveL(v, z) is the Struve L function defined by the formula ++ \spad{struveL(v, z) = -%i^exp(-v*%pi*%i/2)*struveH(v, %i*z)} hankelH1 : (F, F) -> F ++ hankelH1(v, z) is first Hankel function (Bessel function of ++ the third kind) hankelH2 : (F, F) -> F ++ hankelH2(v, z) is the second Hankel function (Bessel function of ++ the third kind) lommelS1 : (F, F, F) -> F ++ lommelS1(mu, nu, z) is the Lommel s function lommelS2 : (F, F, F) -> F ++ lommelS2(mu, nu, z) is the Lommel S function kummerM : (F, F, F) -> F ++ kummerM(a, b, z) is the Kummer M function kummerU : (F, F, F) -> F ++ kummerU(a, b, z) is the Kummer U function legendreP : (F, F, F) -> F ++ legendreP(nu, mu, z) is the Legendre P function legendreQ : (F, F, F) -> F ++ legendreQ(nu, mu, z) is the Legendre Q function kelvinBei : (F, F) -> F ++ kelvinBei(v, z) is the Kelvin bei function defined by equality ++ \spad{kelvinBei(v, z) = imag(besselJ(v, exp(3*%pi*%i/4)*z))} ++ for z and v real kelvinBer : (F, F) -> F ++ kelvinBer(v, z) is the Kelvin ber function defined by equality ++ \spad{kelvinBer(v, z) = real(besselJ(v, exp(3*%pi*%i/4)*z))} ++ for z and v real kelvinKei : (F, F) -> F ++ kelvinKei(v, z) is the Kelvin kei function defined by equality ++ \spad{kelvinKei(v, z) = ++ imag(exp(-v*%pi*%i/2)*besselK(v, exp(%pi*%i/4)*z))} ++ for z and v real kelvinKer : (F, F) -> F ++ kelvinKer(v, z) is the Kelvin kei function defined by equality ++ \spad{kelvinKer(v, z) = ++ real(exp(-v*%pi*%i/2)*besselK(v, exp(%pi*%i/4)*z))} ++ for z and v real ellipticK : F -> F ++ ellipticK(m) is the complete elliptic integral of the ++ first kind: \spad{ellipticK(m) = ++ integrate(1/sqrt((1-t^2)*(1-m*t^2)), t = 0..1)} ellipticE : F -> F ++ ellipticE(m) is the complete elliptic integral of the ++ second kind: \spad{ellipticE(m) = ++ integrate(sqrt(1-m*t^2)/sqrt(1-t^2), t = 0..1)} ellipticE : (F, F) -> F ++ ellipticE(z, m) is the incomplete elliptic integral of the ++ second kind: \spad{ellipticE(z, m) = ++ integrate(sqrt(1-m*t^2)/sqrt(1-t^2), t = 0..z)} ellipticF : (F, F) -> F ++ ellipticF(z, m) is the incomplete elliptic integral of the ++ first kind : \spad{ellipticF(z, m) = ++ integrate(1/sqrt((1-t^2)*(1-m*t^2)), t = 0..z)} ellipticPi : (F, F, F) -> F ++ ellipticPi(z, n, m) is the incomplete elliptic integral of ++ the third kind: \spad{ellipticPi(z, n, m) = ++ integrate(1/((1-n*t^2)*sqrt((1-t^2)*(1-m*t^2))), t = 0..z)} jacobiSn : (F, F) -> F ++ jacobiSn(z, m) is the Jacobi elliptic sn function, defined ++ by the formula \spad{jacobiSn(ellipticF(z, m), m) = z} jacobiCn : (F, F) -> F ++ jacobiCn(z, m) is the Jacobi elliptic cn function, defined ++ by \spad{jacobiCn(z, m)^2 + jacobiSn(z, m)^2 = 1} and ++ \spad{jacobiCn(0, m) = 1} jacobiDn : (F, F) -> F ++ jacobiDn(z, m) is the Jacobi elliptic dn function, defined ++ by \spad{jacobiDn(z, m)^2 + m*jacobiSn(z, m)^2 = 1} and ++ \spad{jacobiDn(0, m) = 1} jacobiZeta : (F, F) -> F ++ jacobiZeta(z, m) is the Jacobi elliptic zeta function, defined ++ by \spad{D(jacobiZeta(z, m), z) = ++ jacobiDn(z, m)^2 - ellipticE(m)/ellipticK(m)} and ++ \spad{jacobiZeta(0, m) = 0}. jacobiTheta : (F, F) -> F ++ jacobiTheta(q, z) is the third Jacobi Theta function lerchPhi : (F, F, F) -> F ++ lerchPhi(z, s, a) is the Lerch Phi function riemannZeta : F -> F ++ riemannZeta(z) is the Riemann Zeta function charlierC : (F, F, F) -> F ++ charlierC(n, a, z) is the Charlier polynomial hermiteH : (F, F) -> F ++ hermiteH(n, z) is the Hermite polynomial jacobiP : (F, F, F, F) -> F ++ jacobiP(n, a, b, z) is the Jacobi polynomial laguerreL: (F, F, F) -> F ++ laguerreL(n, a, z) is the Laguerre polynomial meixnerM : (F, F, F, F) -> F ++ meixnerM(n, b, c, z) is the Meixner polynomial if F has RetractableTo(Integer) then hypergeometricF : (List F, List F, F) -> F ++ hypergeometricF(la, lb, z) is the generalized hypergeometric ++ function meijerG : (List F, List F, List F, List F, F) -> F ++ meijerG(la, lb, lc, ld, z) is the meijerG function -- Functions below should be local but conditional iiGamma : F -> F ++ iiGamma(x) should be local but conditional; iiabs : F -> F ++ iiabs(x) should be local but conditional; iiconjugate: F -> F ++ iiconjugate(x) should be local but conditional; iiBeta : List F -> F ++ iiBeta(x) should be local but conditional; iidigamma : F -> F ++ iidigamma(x) should be local but conditional; iipolygamma : List F -> F ++ iipolygamma(x) should be local but conditional; iiBesselJ : List F -> F ++ iiBesselJ(x) should be local but conditional; iiBesselY : List F -> F ++ iiBesselY(x) should be local but conditional; iiBesselI : List F -> F ++ iiBesselI(x) should be local but conditional; iiBesselK : List F -> F ++ iiBesselK(x) should be local but conditional; iiAiryAi : F -> F ++ iiAiryAi(x) should be local but conditional; iiAiryAiPrime : F -> F ++ iiAiryAiPrime(x) should be local but conditional; iiAiryBi : F -> F ++ iiAiryBi(x) should be local but conditional; iiAiryBiPrime : F -> F ++ iiAiryBiPrime(x) should be local but conditional; iAiryAi : F -> F ++ iAiryAi(x) should be local but conditional; iAiryAiPrime : F -> F ++ iAiryAiPrime(x) should be local but conditional; iAiryBi : F -> F ++ iAiryBi(x) should be local but conditional; iAiryBiPrime : F -> F ++ iAiryBiPrime(x) should be local but conditional; iiHypergeometricF : List F -> F ++ iiHypergeometricF(l) should be local but conditional; iiPolylog : (F, F) -> F ++ iiPolylog(x, s) should be local but conditional; iLambertW : F -> F ++ iLambertW(x) should be local but conditional; Implementation ==> add SPECIAL := 'special INP ==> InputForm SPECIALINPUT ==> '%specialInput iabs : F -> F iGamma : F -> F iBeta : (F, F) -> F idigamma : F -> F iiipolygamma : (F, F) -> F iiiBesselJ : (F, F) -> F iiiBesselY : (F, F) -> F iiiBesselI : (F, F) -> F iiiBesselK : (F, F) -> F iPolylog : List F -> F iWeierstrassP : (F, F, F) -> F iWeierstrassPPrime : (F, F, F) -> F iWeierstrassSigma : (F, F, F) -> F iWeierstrassZeta : (F, F, F) -> F iiWeierstrassP : List F -> F iiWeierstrassPPrime : List F -> F iiWeierstrassSigma : List F -> F iiWeierstrassZeta : List F -> F iiMeijerG : List F -> F opabs := operator('abs)$CommonOperators opconjugate := operator('conjugate)$CommonOperators opsqrt := operator('sqrt)$CommonOperators opGamma := operator('Gamma)$CommonOperators opGamma2 := operator('Gamma2)$CommonOperators opBeta := operator('Beta)$CommonOperators opdigamma := operator('digamma)$CommonOperators oppolygamma := operator('polygamma)$CommonOperators opBesselJ := operator('besselJ)$CommonOperators opBesselY := operator('besselY)$CommonOperators opBesselI := operator('besselI)$CommonOperators opBesselK := operator('besselK)$CommonOperators opAiryAi := operator('airyAi)$CommonOperators opAiryAiPrime := operator('airyAiPrime)$CommonOperators opAiryBi := operator('airyBi)$CommonOperators opAiryBiPrime := operator('airyBiPrime)$CommonOperators opLambertW := operator('lambertW)$CommonOperators opPolylog := operator('polylog)$CommonOperators opWeierstrassP := operator('weierstrassP)$CommonOperators opWeierstrassPPrime := operator('weierstrassPPrime)$CommonOperators opWeierstrassSigma := operator('weierstrassSigma)$CommonOperators opWeierstrassZeta := operator('weierstrassZeta)$CommonOperators opHypergeometricF := operator('hypergeometricF)$CommonOperators opMeijerG := operator('meijerG)$CommonOperators opCharlierC := operator('charlierC)$CommonOperators opHermiteH := operator('hermiteH)$CommonOperators opJacobiP := operator('jacobiP)$CommonOperators opLaguerreL := operator('laguerreL)$CommonOperators opMeixnerM := operator('meixnerM)$CommonOperators op_log_gamma := operator('%logGamma)$CommonOperators op_eis := operator('%eis)$CommonOperators op_erfs := operator('%erfs)$CommonOperators op_erfis := operator('%erfis)$CommonOperators abs x == opabs x conjugate x == opconjugate x Gamma(x) == opGamma(x) Gamma(a, x) == opGamma2(a, x) Beta(x, y) == opBeta(x, y) digamma x == opdigamma(x) polygamma(k, x)== oppolygamma(k, x) besselJ(a, x) == opBesselJ(a, x) besselY(a, x) == opBesselY(a, x) besselI(a, x) == opBesselI(a, x) besselK(a, x) == opBesselK(a, x) airyAi(x) == opAiryAi(x) airyAiPrime(x) == opAiryAiPrime(x) airyBi(x) == opAiryBi(x) airyBiPrime(x) == opAiryBiPrime(x) lambertW(x) == opLambertW(x) polylog(s, x) == opPolylog(s, x) weierstrassP(g2, g3, x) == opWeierstrassP(g2, g3, x) weierstrassPPrime(g2, g3, x) == opWeierstrassPPrime(g2, g3, x) weierstrassSigma(g2, g3, x) == opWeierstrassSigma(g2, g3, x) weierstrassZeta(g2, g3, x) == opWeierstrassZeta(g2, g3, x) if F has RetractableTo(Integer) then hypergeometricF(a, b, z) == nai := #a nbi := #b z = 0 and nai <= nbi + 1 => 1 p := (#a)::F q := (#b)::F opHypergeometricF concat(concat(a, concat(b, [z])), [p, q]) meijerG(a, b, c, d, z) == n1 := (#a)::F n2 := (#b)::F m1 := (#c)::F m2 := (#d)::F opMeijerG concat(concat(a, concat(b, concat(c, concat(d, [z])))), [n1, n2, m1, m2]) import from List Kernel(F) opdiff := operator(operator('%diff)$CommonOperators)$F dummy ==> new()$SE :: F ahalf : F := recip(2::F)::F athird : F := recip(3::F)::F afourth : F := recip(4::F)::F asixth : F := recip(6::F)::F twothirds : F := 2*athird threehalfs : F := 3*ahalf -- Helpers for partially defined derivatives grad2(l : List F, t : SE, op : OP, d2 : (F, F) -> F ) : F == x1 := l(1) x2 := l(2) dm := dummy differentiate(x1, t)*kernel(opdiff, [op [dm, x2], dm, x1]) + differentiate(x2, t)*d2(x1, x2) grad3(l : List F, t : SE, op : OP, d3 : (F, F, F) -> F ) : F == x1 := l(1) x2 := l(2) x3 := l(3) dm1 := dummy dm2 := dummy differentiate(x1, t)*kernel(opdiff, [op [dm1, x2, x3], dm1, x1]) + differentiate(x2, t)*kernel(opdiff, [op [x1, dm2, x3], dm2, x2]) + differentiate(x3, t)*d3(x1, x2, x3) grad4(l : List F, t : SE, op : OP, d4 : (F, F, F, F) -> F ) : F == x1 := l(1) x2 := l(2) x3 := l(3) x4 := l(4) dm1 := dummy dm2 := dummy dm3 := dummy kd1 := kernel(opdiff, [op [dm1, x2, x3, x4], dm1, x1]) kd2 := kernel(opdiff, [op [x1, dm2, x3, x4], dm2, x2]) kd3 := kernel(opdiff, [op [x1, x2, dm3, x4], dm3, x3]) differentiate(x1, t)*kd1 + differentiate(x2, t)*kd2 + differentiate(x3, t)*kd3 + differentiate(x4, t)*d4(x1, x2, x3, x4) -- handle WeierstrassPInverse )if false opWeierstrassPInverse := operator('weierstrassPInverse)$CommonOperators weierstrassPInverse(g2, g3, z) == opWeierstrassPInverse(g2, g3, z) eWeierstrassPInverse(g2 : F, g3 : F, z : F) : F == kernel(opWeierstrassPInverse, [g2, g3, z]) elWeierstrassPInverse(l : List F) : F == eWeierstrassPInverse(l(1), l(2), l(3)) evaluate(opWeierstrassPInverse, elWeierstrassPInverse)$BasicOperatorFunctions1(F) if F has RadicalCategory then eWeierstrassPInverseGrad_g2(l : List F) : F == g2 := l(1) g3 := l(2) z := l(3) error "unimplemented" eWeierstrassPInverseGrad_g3(l : List F) : F == g2 := l(1) g3 := l(2) z := l(3) error "unimplemented" eWeierstrassPInverseGrad_z(l : List F) : F == g2 := l(1) g3 := l(2) z := l(3) 1/sqrt(4*z^3 - g2*z - g3) derivative(opWeierstrassPInverse, [eWeierstrassPInverseGrad_g2, eWeierstrassPInverseGrad_g3, eWeierstrassPInverseGrad_z]) )endif -- handle WhittakerM opWhittakerM := operator('whittakerM)$CommonOperators whittakerM(k, m, z) == opWhittakerM(k, m, z) eWhittakerM(k : F, m : F, z : F) : F == kernel(opWhittakerM, [k, m, z]) elWhittakerM(l : List F) : F == eWhittakerM(l(1), l(2), l(3)) evaluate(opWhittakerM, elWhittakerM)$BasicOperatorFunctions1(F) eWhittakerMGrad_z(k : F, m : F, z : F) : F == (ahalf - k/z)*whittakerM(k, m, z) + (ahalf + k + m)*whittakerM(k + 1, m, z)/z dWhittakerM(l : List F, t : SE) : F == grad3(l, t, opWhittakerM, eWhittakerMGrad_z) setProperty(opWhittakerM, SPECIALDIFF, dWhittakerM@((List F, SE)->F) pretend None) -- handle WhittakerW opWhittakerW := operator('whittakerW)$CommonOperators whittakerW(k, m, z) == opWhittakerW(k, m, z) eWhittakerW(k : F, m : F, z : F) : F == kernel(opWhittakerW, [k, m, z]) elWhittakerW(l : List F) : F == eWhittakerW(l(1), l(2), l(3)) evaluate(opWhittakerW, elWhittakerW)$BasicOperatorFunctions1(F) eWhittakerWGrad_z(k : F, m : F, z : F) : F == (ahalf - k/z)*whittakerW(k, m, z) - whittakerW(k + 1, m, z)/z dWhittakerW(l : List F, t : SE) : F == grad3(l, t, opWhittakerW, eWhittakerWGrad_z) setProperty(opWhittakerW, SPECIALDIFF, dWhittakerW@((List F, SE)->F) pretend None) -- handle AngerJ opAngerJ := operator('angerJ)$CommonOperators angerJ(v, z) == opAngerJ(v, z) if F has TranscendentalFunctionCategory then eAngerJ(v : F, z : F) : F == z = 0 => sin(v*pi())/(v*pi()) kernel(opAngerJ, [v, z]) elAngerJ(l : List F) : F == eAngerJ(l(1), l(2)) evaluate(opAngerJ, elAngerJ)$BasicOperatorFunctions1(F) eAngerJGrad_z(v : F, z : F) : F == -angerJ(v + 1, z) + v*angerJ(v, z)/z - sin(v*pi())/(pi()*z) dAngerJ(l : List F, t : SE) : F == grad2(l, t, opAngerJ, eAngerJGrad_z) setProperty(opAngerJ, SPECIALDIFF, dAngerJ@((List F, SE)->F) pretend None) else eeAngerJ(l : List F) : F == kernel(opAngerJ, l) evaluate(opAngerJ, eeAngerJ)$BasicOperatorFunctions1(F) -- handle WeberE opWeberE := operator('weberE)$CommonOperators weberE(v, z) == opWeberE(v, z) if F has TranscendentalFunctionCategory then eWeberE(v : F, z : F) : F == z = 0 => 2*sin(ahalf*v*pi())^2/(v*pi()) kernel(opWeberE, [v, z]) elWeberE(l : List F) : F == eWeberE(l(1), l(2)) evaluate(opWeberE, elWeberE)$BasicOperatorFunctions1(F) eWeberEGrad_z(v : F, z : F) : F == -weberE(v + 1, z) + v*weberE(v, z)/z - (1 - cos(v*pi()))/(pi()*z) dWeberE(l : List F, t : SE) : F == grad2(l, t, opWeberE, eWeberEGrad_z) setProperty(opWeberE, SPECIALDIFF, dWeberE@((List F, SE)->F) pretend None) else eeWeberE(l : List F) : F == kernel(opWeberE, l) evaluate(opWeberE, eeWeberE)$BasicOperatorFunctions1(F) -- handle StruveH opStruveH := operator('struveH)$CommonOperators struveH(v, z) == opStruveH(v, z) eStruveH(v : F, z : F) : F == kernel(opStruveH, [v, z]) elStruveH(l : List F) : F == eStruveH(l(1), l(2)) evaluate(opStruveH, elStruveH)$BasicOperatorFunctions1(F) if F has TranscendentalFunctionCategory and F has RadicalCategory then eStruveHGrad_z(v : F, z : F) : F == -struveH(v + 1, z) + v*struveH(v, z)/z + (ahalf*z)^v/(sqrt(pi())*Gamma(v + threehalfs)) dStruveH(l : List F, t : SE) : F == grad2(l, t, opStruveH, eStruveHGrad_z) setProperty(opStruveH, SPECIALDIFF, dStruveH@((List F, SE)->F) pretend None) -- handle StruveL opStruveL := operator('struveL)$CommonOperators struveL(v, z) == opStruveL(v, z) eStruveL(v : F, z : F) : F == kernel(opStruveL, [v, z]) elStruveL(l : List F) : F == eStruveL(l(1), l(2)) evaluate(opStruveL, elStruveL)$BasicOperatorFunctions1(F) if F has TranscendentalFunctionCategory and F has RadicalCategory then eStruveLGrad_z(v : F, z : F) : F == struveL(v + 1, z) + v*struveL(v, z)/z + (ahalf*z)^v/(sqrt(pi())*Gamma(v + threehalfs)) dStruveL(l : List F, t : SE) : F == grad2(l, t, opStruveL, eStruveLGrad_z) setProperty(opStruveL, SPECIALDIFF, dStruveL@((List F, SE)->F) pretend None) -- handle HankelH1 opHankelH1 := operator('hankelH1)$CommonOperators hankelH1(v, z) == opHankelH1(v, z) eHankelH1(v : F, z : F) : F == kernel(opHankelH1, [v, z]) elHankelH1(l : List F) : F == eHankelH1(l(1), l(2)) evaluate(opHankelH1, elHankelH1)$BasicOperatorFunctions1(F) eHankelH1Grad_z(v : F, z : F) : F == -hankelH1(v + 1, z) + v*hankelH1(v, z)/z dHankelH1(l : List F, t : SE) : F == grad2(l, t, opHankelH1, eHankelH1Grad_z) setProperty(opHankelH1, SPECIALDIFF, dHankelH1@((List F, SE)->F) pretend None) -- handle HankelH2 opHankelH2 := operator('hankelH2)$CommonOperators hankelH2(v, z) == opHankelH2(v, z) eHankelH2(v : F, z : F) : F == kernel(opHankelH2, [v, z]) elHankelH2(l : List F) : F == eHankelH2(l(1), l(2)) evaluate(opHankelH2, elHankelH2)$BasicOperatorFunctions1(F) eHankelH2Grad_z(v : F, z : F) : F == -hankelH2(v + 1, z) + v*hankelH2(v, z)/z dHankelH2(l : List F, t : SE) : F == grad2(l, t, opHankelH2, eHankelH2Grad_z) setProperty(opHankelH2, SPECIALDIFF, dHankelH2@((List F, SE)->F) pretend None) -- handle LommelS1 opLommelS1 := operator('lommelS1)$CommonOperators lommelS1(m, v, z) == opLommelS1(m, v, z) eLommelS1(m : F, v : F, z : F) : F == kernel(opLommelS1, [m, v, z]) elLommelS1(l : List F) : F == eLommelS1(l(1), l(2), l(3)) evaluate(opLommelS1, elLommelS1)$BasicOperatorFunctions1(F) eLommelS1Grad_z(m : F, v : F, z : F) : F == -v*lommelS1(m, v, z)/z + (m + v - 1)*lommelS1(m - 1, v - 1, z) dLommelS1(l : List F, t : SE) : F == grad3(l, t, opLommelS1, eLommelS1Grad_z) setProperty(opLommelS1, SPECIALDIFF, dLommelS1@((List F, SE)->F) pretend None) -- handle LommelS2 opLommelS2 := operator('lommelS2)$CommonOperators lommelS2(mu, nu, z) == opLommelS2(mu, nu, z) eLommelS2(mu : F, nu : F, z : F) : F == kernel(opLommelS2, [mu, nu, z]) elLommelS2(l : List F) : F == eLommelS2(l(1), l(2), l(3)) evaluate(opLommelS2, elLommelS2)$BasicOperatorFunctions1(F) eLommelS2Grad_z(m : F, v : F, z : F) : F == -v*lommelS2(m, v, z)/z + (m + v - 1)*lommelS2(m - 1, v - 1, z) dLommelS2(l : List F, t : SE) : F == grad3(l, t, opLommelS2, eLommelS2Grad_z) setProperty(opLommelS2, SPECIALDIFF, dLommelS2@((List F, SE)->F) pretend None) -- handle KummerM opKummerM := operator('kummerM)$CommonOperators kummerM(mu, nu, z) == opKummerM(mu, nu, z) eKummerM(a : F, b : F, z : F) : F == z = 0 => 1 kernel(opKummerM, [a, b, z]) elKummerM(l : List F) : F == eKummerM(l(1), l(2), l(3)) evaluate(opKummerM, elKummerM)$BasicOperatorFunctions1(F) eKummerMGrad_z(a : F, b : F, z : F) : F == ((z + a - b)*kummerM(a, b, z)+(b - a)*kummerM(a - 1, b, z))/z dKummerM(l : List F, t : SE) : F == grad3(l, t, opKummerM, eKummerMGrad_z) setProperty(opKummerM, SPECIALDIFF, dKummerM@((List F, SE)->F) pretend None) -- handle KummerU opKummerU := operator('kummerU)$CommonOperators kummerU(a, b, z) == opKummerU(a, b, z) eKummerU(a : F, b : F, z : F) : F == kernel(opKummerU, [a, b, z]) elKummerU(l : List F) : F == eKummerU(l(1), l(2), l(3)) evaluate(opKummerU, elKummerU)$BasicOperatorFunctions1(F) eKummerUGrad_z(a : F, b : F, z : F) : F == ((z + a - b)*kummerU(a, b, z) - kummerU(a - 1, b, z))/z dKummerU(l : List F, t : SE) : F == grad3(l, t, opKummerU, eKummerUGrad_z) setProperty(opKummerU, SPECIALDIFF, dKummerU@((List F, SE)->F) pretend None) -- handle LegendreP opLegendreP := operator('legendreP)$CommonOperators legendreP(nu, mu, z) == opLegendreP(nu, mu, z) eLegendreP(nu : F, mu : F, z : F) : F == kernel(opLegendreP, [nu, mu, z]) elLegendreP(l : List F) : F == eLegendreP(l(1), l(2), l(3)) evaluate(opLegendreP, elLegendreP)$BasicOperatorFunctions1(F) eLegendrePGrad_z(nu : F, mu : F, z : F) : F == (nu - mu + 1)*legendreP(nu + 1, mu, z) - (nu + 1)*z*legendreP(nu, mu, z) dLegendreP(l : List F, t : SE) : F == grad3(l, t, opLegendreP, eLegendrePGrad_z) setProperty(opLegendreP, SPECIALDIFF, dLegendreP@((List F, SE)->F) pretend None) -- handle LegendreQ opLegendreQ := operator('legendreQ)$CommonOperators legendreQ(nu, mu, z) == opLegendreQ(nu, mu, z) eLegendreQ(nu : F, mu : F, z : F) : F == kernel(opLegendreQ, [nu, mu, z]) elLegendreQ(l : List F) : F == eLegendreQ(l(1), l(2), l(3)) evaluate(opLegendreQ, elLegendreQ)$BasicOperatorFunctions1(F) eLegendreQGrad_z(nu : F, mu : F, z : F) : F == (nu - mu + 1)*legendreQ(nu + 1, mu, z) - (nu + 1)*z*legendreQ(nu, mu, z) dLegendreQ(l : List F, t : SE) : F == grad3(l, t, opLegendreQ, eLegendreQGrad_z) setProperty(opLegendreQ, SPECIALDIFF, dLegendreQ@((List F, SE)->F) pretend None) -- handle KelvinBei opKelvinBei := operator('kelvinBei)$CommonOperators kelvinBei(v, z) == opKelvinBei(v, z) eKelvinBei(v : F, z : F) : F == kernel(opKelvinBei, [v, z]) elKelvinBei(l : List F) : F == eKelvinBei(l(1), l(2)) evaluate(opKelvinBei, elKelvinBei)$BasicOperatorFunctions1(F) if F has RadicalCategory then eKelvinBeiGrad_z(v : F, z : F) : F == ahalf*sqrt(2::F)*(kelvinBei(v + 1, z) - kelvinBer(v + 1, z)) + v*kelvinBei(v, z)/z dKelvinBei(l : List F, t : SE) : F == grad2(l, t, opKelvinBei, eKelvinBeiGrad_z) setProperty(opKelvinBei, SPECIALDIFF, dKelvinBei@((List F, SE)->F) pretend None) -- handle KelvinBer opKelvinBer := operator('kelvinBer)$CommonOperators kelvinBer(v, z) == opKelvinBer(v, z) eKelvinBer(v : F, z : F) : F == kernel(opKelvinBer, [v, z]) elKelvinBer(l : List F) : F == eKelvinBer(l(1), l(2)) evaluate(opKelvinBer, elKelvinBer)$BasicOperatorFunctions1(F) if F has RadicalCategory then eKelvinBerGrad_z(v : F, z : F) : F == ahalf*sqrt(2::F)*(kelvinBer(v + 1, z) + kelvinBei(v + 1, z)) + v*kelvinBer(v, z)/z dKelvinBer(l : List F, t : SE) : F == grad2(l, t, opKelvinBer, eKelvinBerGrad_z) setProperty(opKelvinBer, SPECIALDIFF, dKelvinBer@((List F, SE)->F) pretend None) -- handle KelvinKei opKelvinKei := operator('kelvinKei)$CommonOperators kelvinKei(v, z) == opKelvinKei(v, z) eKelvinKei(v : F, z : F) : F == kernel(opKelvinKei, [v, z]) elKelvinKei(l : List F) : F == eKelvinKei(l(1), l(2)) evaluate(opKelvinKei, elKelvinKei)$BasicOperatorFunctions1(F) if F has RadicalCategory then eKelvinKeiGrad_z(v : F, z : F) : F == ahalf*sqrt(2::F)*(kelvinKei(v + 1, z) - kelvinKer(v + 1, z)) + v*kelvinKei(v, z)/z dKelvinKei(l : List F, t : SE) : F == grad2(l, t, opKelvinKei, eKelvinKeiGrad_z) setProperty(opKelvinKei, SPECIALDIFF, dKelvinKei@((List F, SE)->F) pretend None) -- handle KelvinKer opKelvinKer := operator('kelvinKer)$CommonOperators kelvinKer(v, z) == opKelvinKer(v, z) eKelvinKer(v : F, z : F) : F == kernel(opKelvinKer, [v, z]) elKelvinKer(l : List F) : F == eKelvinKer(l(1), l(2)) evaluate(opKelvinKer, elKelvinKer)$BasicOperatorFunctions1(F) if F has RadicalCategory then eKelvinKerGrad_z(v : F, z : F) : F == ahalf*sqrt(2::F)*(kelvinKer(v + 1, z) + kelvinKei(v + 1, z)) + v*kelvinKer(v, z)/z dKelvinKer(l : List F, t : SE) : F == grad2(l, t, opKelvinKer, eKelvinKerGrad_z) setProperty(opKelvinKer, SPECIALDIFF, dKelvinKer@((List F, SE)->F) pretend None) -- handle EllipticK opEllipticK := operator('ellipticK)$CommonOperators ellipticK(m) == opEllipticK(m) eEllipticK(m : F) : F == kernel(opEllipticK, [m]) elEllipticK(l : List F) : F == eEllipticK(l(1)) evaluate(opEllipticK, elEllipticK)$BasicOperatorFunctions1(F) dEllipticK(m : F) : F == ahalf*(ellipticE(m) - (1 - m)*ellipticK(m))/(m*(1 - m)) derivative(opEllipticK, dEllipticK) -- handle one argument EllipticE opEllipticE := operator('ellipticE)$CommonOperators ellipticE(m) == opEllipticE(m) eEllipticE(m : F) : F == kernel(opEllipticE, [m]) elEllipticE(l : List F) : F == eEllipticE(l(1)) evaluate(opEllipticE, elEllipticE)$BasicOperatorFunctions1(F) dEllipticE(m : F) : F == ahalf*(ellipticE(m) - ellipticK(m))/m derivative(opEllipticE, dEllipticE) -- handle two argument EllipticE opEllipticE2 := operator('ellipticE2)$CommonOperators ellipticE(z, m) == opEllipticE2(z, m) eEllipticE2(z : F, m : F) : F == z = 0 => 0 z = 1 => eEllipticE(m) kernel(opEllipticE2, [z, m]) elEllipticE2(l : List F) : F == eEllipticE2(l(1), l(2)) evaluate(opEllipticE2, elEllipticE2)$BasicOperatorFunctions1(F) if F has RadicalCategory then eEllipticE2Grad_z(l : List F) : F == z := l(1) m := l(2) sqrt(1 - m*z^2)/sqrt(1 - z^2) eEllipticE2Grad_m(l : List F) : F == z := l(1) m := l(2) ahalf*(ellipticE(z, m) - ellipticF(z, m))/m derivative(opEllipticE2, [eEllipticE2Grad_z, eEllipticE2Grad_m]) inEllipticE2(li : List INP) : INP == convert cons(convert('ellipticE), li) input(opEllipticE2, inEllipticE2@((List INP) -> INP)) -- handle EllipticF opEllipticF := operator('ellipticF)$CommonOperators ellipticF(z, m) == opEllipticF(z, m) eEllipticF(z : F, m : F) : F == z = 0 => 0 z = 1 => ellipticK(m) kernel(opEllipticF, [z, m]) elEllipticF(l : List F) : F == eEllipticF(l(1), l(2)) evaluate(opEllipticF, elEllipticF)$BasicOperatorFunctions1(F) if F has RadicalCategory then eEllipticFGrad_z(l : List F) : F == z := l(1) m := l(2) 1/(sqrt(1 - m*z^2)*sqrt(1 - z^2)) eEllipticFGrad_m(l : List F) : F == z := l(1) m := l(2) ahalf*((ellipticE(z, m) - (1 - m)*ellipticF(z, m))/m - z*sqrt(1 - z^2)/sqrt(1 - m*z^2))/(1 - m) derivative(opEllipticF, [eEllipticFGrad_z, eEllipticFGrad_m]) -- handle EllipticPi opEllipticPi := operator('ellipticPi)$CommonOperators ellipticPi(z, n, m) == opEllipticPi(z, n, m) eEllipticPi(z : F, n : F, m : F) : F == z = 0 => 0 kernel(opEllipticPi, [z, n, m]) elEllipticPi(l : List F) : F == eEllipticPi(l(1), l(2), l(3)) evaluate(opEllipticPi, elEllipticPi)$BasicOperatorFunctions1(F) if F has RadicalCategory then eEllipticPiGrad_z(l : List F) : F == z := l(1) n := l(2) m := l(3) 1/((1 - n*z^2)*sqrt(1 - m*z^2)*sqrt(1 - z^2)) eEllipticPiGrad_n(l : List F) : F == z := l(1) n := l(2) m := l(3) t1 := -(n^2 - m)*ellipticPi(z, n, m)/((n - 1)*(n - m)*n) t2 := ellipticF(z, m)/((n - 1)*n) t3 := -ellipticE(z, m)/((n - 1)*(n - m)) t4 := n*z*sqrt(1 - m*z^2)*sqrt(1 - z^2)/ ((1 - n*z^2)*(n - 1)*(n - m)) ahalf*(t1 + t2 + t3 + t4) eEllipticPiGrad_m(l : List F) : F == z := l(1) n := l(2) m := l(3) t1 := m*z*sqrt(1 - z^2)/sqrt(1 - m*z^2) t2 := (-ellipticE(z, m) + t1)/(1 - m) ahalf*(ellipticPi(z, n, m) + t2)/(n - m) derivative(opEllipticPi, [eEllipticPiGrad_z, eEllipticPiGrad_n, eEllipticPiGrad_m]) -- handle JacobiSn opJacobiSn := operator('jacobiSn)$CommonOperators jacobiSn(z, m) == opJacobiSn(z, m) eJacobiSn(z : F, m : F) : F == z = 0 => 0 if is?(z, opEllipticF) then args := argument(retract(z)@K) m = args(2) => return args(1) kernel(opJacobiSn, [z, m]) elJacobiSn : List F -> F elJacobiSn(l : List F) : F == eJacobiSn(l(1), l(2)) evaluate(opJacobiSn, elJacobiSn)$BasicOperatorFunctions1(F) jacobiGradHelper(z : F, m : F) : F == (z - ellipticE(jacobiSn(z, m), m)/(1 - m))/m eJacobiSnGrad_z(l : List F) : F == z := l(1) m := l(2) jacobiCn(z, m)*jacobiDn(z, m) eJacobiSnGrad_m(l : List F) : F == z := l(1) m := l(2) ahalf*(eJacobiSnGrad_z(l)*jacobiGradHelper(z, m) + jacobiSn(z, m)*jacobiCn(z, m)^2/(1 - m)) derivative(opJacobiSn, [eJacobiSnGrad_z, eJacobiSnGrad_m]) -- handle JacobiCn opJacobiCn := operator('jacobiCn)$CommonOperators jacobiCn(z, m) == opJacobiCn(z, m) eJacobiCn(z : F, m : F) : F == z = 0 => 1 kernel(opJacobiCn, [z, m]) elJacobiCn(l : List F) : F == eJacobiCn(l(1), l(2)) evaluate(opJacobiCn, elJacobiCn)$BasicOperatorFunctions1(F) eJacobiCnGrad_z(l : List F) : F == z := l(1) m := l(2) -jacobiSn(z, m)*jacobiDn(z, m) eJacobiCnGrad_m(l : List F) : F == z := l(1) m := l(2) ahalf*(eJacobiCnGrad_z(l)*jacobiGradHelper(z, m) - jacobiSn(z, m)^2*jacobiCn(z, m)/(1 - m)) derivative(opJacobiCn, [eJacobiCnGrad_z, eJacobiCnGrad_m]) -- handle JacobiDn opJacobiDn := operator('jacobiDn)$CommonOperators jacobiDn(z, m) == opJacobiDn(z, m) eJacobiDn(z : F, m : F) : F == z = 0 => 1 kernel(opJacobiDn, [z, m]) elJacobiDn(l : List F) : F == eJacobiDn(l(1), l(2)) evaluate(opJacobiDn, elJacobiDn)$BasicOperatorFunctions1(F) eJacobiDnGrad_z(l : List F) : F == z := l(1) m := l(2) -m*jacobiSn(z, m)*jacobiCn(z, m) eJacobiDnGrad_m(l : List F) : F == z := l(1) m := l(2) ahalf*(eJacobiDnGrad_z(l)*jacobiGradHelper(z, m) - jacobiSn(z, m)^2*jacobiDn(z, m)/(1 - m)) derivative(opJacobiDn, [eJacobiDnGrad_z, eJacobiDnGrad_m]) -- handle JacobiZeta opJacobiZeta := operator('jacobiZeta)$CommonOperators jacobiZeta(z, m) == opJacobiZeta(z, m) eJacobiZeta(z : F, m : F) : F == z = 0 => 0 kernel(opJacobiZeta, [z, m]) elJacobiZeta(l : List F) : F == eJacobiZeta(l(1), l(2)) evaluate(opJacobiZeta, elJacobiZeta)$BasicOperatorFunctions1(F) eJacobiZetaGrad_z(l : List F) : F == z := l(1) m := l(2) dn := jacobiDn(z, m) dn*dn - ellipticE(m)/ellipticK(m) eJacobiZetaGrad_m(l : List F) : F == z := l(1) m := l(2) ek := ellipticK(m) ee := ellipticE(m) er := ee/ek dn := jacobiDn(z, m) res1 := (dn*dn + m - 1)*jacobiZeta(z, m) res2 := res1 + (m - 1)*z*dn*dn res3 := res2 - m*jacobiCn(z, m)*jacobiDn(z, m)*jacobiSn(z, m) res4 := res3 + z*(1 - m + dn*dn)*er ahalf*(res4 - z*er*er)/(m*m - m) derivative(opJacobiZeta, [eJacobiZetaGrad_z, eJacobiZetaGrad_m]) -- handle JacobiTheta opJacobiTheta := operator('jacobiTheta)$CommonOperators jacobiTheta(q, z) == opJacobiTheta(q, z) eJacobiTheta(q : F, z : F) : F == kernel(opJacobiTheta, [q, z]) elJacobiTheta(l : List F) : F == eJacobiTheta(l(1), l(2)) evaluate(opJacobiTheta, elJacobiTheta)$BasicOperatorFunctions1(F) -- handle LerchPhi opLerchPhi := operator('lerchPhi)$CommonOperators lerchPhi(z, s, a) == opLerchPhi(z, s, a) eLerchPhi(z : F, s : F, a : F) : F == -- z = 0 => 1/a^s a = 1 => polylog(s, z)/z kernel(opLerchPhi, [z, s, a]) elLerchPhi(l : List F) : F == eLerchPhi(l(1), l(2), l(3)) evaluate(opLerchPhi, elLerchPhi)$BasicOperatorFunctions1(F) dLerchPhi(l : List F, t : SE) : F == z := l(1) s := l(2) a := l(3) dz := differentiate(z, t)*(lerchPhi(z, s - 1, a) - a*lerchPhi(z, s, a))/z da := -differentiate(a, t)*s*lerchPhi(z, s + 1, a) dm := dummy differentiate(s, t)*kernel(opdiff, [opLerchPhi [z, dm, a], dm, s]) + dz + da setProperty(opLerchPhi, SPECIALDIFF, dLerchPhi@((List F, SE)->F) pretend None) -- handle RiemannZeta opRiemannZeta := operator('riemannZeta)$CommonOperators riemannZeta(z) == opRiemannZeta(z) eRiemannZeta(z : F) : F == kernel(opRiemannZeta, [z]) elRiemannZeta(l : List F) : F == eRiemannZeta(l(1)) evaluate(opRiemannZeta, elRiemannZeta)$BasicOperatorFunctions1(F) -- orthogonal polynomials charlierC(n : F, a : F, z : F) : F == opCharlierC(n, a, z) eCharlierC(n : F, a : F, z : F) : F == n = 0 => 1 n = 1 => (z - a)/a kernel(opCharlierC, [n, a, z]) elCharlierC(l : List F) : F == eCharlierC(l(1), l(2), l(3)) evaluate(opCharlierC, elCharlierC)$BasicOperatorFunctions1(F) hermiteH(n : F, z: F) : F == opHermiteH(n, z) eHermiteH(n : F, z: F) : F == n = -1 => 0 n = 0 => 1 n = 1 => (2::F)*z kernel(opHermiteH, [n, z]) elHermiteH(l : List F) : F == eHermiteH(l(1), l(2)) evaluate(opHermiteH, elHermiteH)$BasicOperatorFunctions1(F) eHermiteHGrad_z(n : F, z : F) : F == (2::F)*n*hermiteH(n - 1, z) dHermiteH(l : List F, t : SE) : F == grad2(l, t, opHermiteH, eHermiteHGrad_z) setProperty(opHermiteH, SPECIALDIFF, dHermiteH@((List F, SE)->F) pretend None) jacobiP(n : F, a : F, b : F, z : F) : F == opJacobiP(n, a, b, z) eJacobiP(n : F, a : F, b : F, z : F) : F == n = -1 => 0 n = 0 => 1 n = 1 => ahalf*(a - b) + (1 + ahalf*(a + b))*z kernel(opJacobiP, [n, a, b, z]) elJacobiP(l : List F) : F == eJacobiP(l(1), l(2), l(3), l(4)) evaluate(opJacobiP, elJacobiP)$BasicOperatorFunctions1(F) eJacobiPGrad_z(n : F, a : F, b : F, z : F) : F == ahalf*(a + b + n + 1)*jacobiP(n - 1, a + 1, b + 1, z) dJacobiP(l : List F, t : SE) : F == grad4(l, t, opJacobiP, eJacobiPGrad_z) setProperty(opJacobiP, SPECIALDIFF, dJacobiP@((List F, SE)->F) pretend None) laguerreL(n : F, a : F, z : F) : F == opLaguerreL(n, a, z) eLaguerreL(n : F, a : F, z : F) : F == n = -1 => 0 n = 0 => 1 n = 1 => (1 + a - z) kernel(opLaguerreL, [n, a, z]) elLaguerreL(l : List F) : F == eLaguerreL(l(1), l(2), l(3)) evaluate(opLaguerreL, elLaguerreL)$BasicOperatorFunctions1(F) eLaguerreLGrad_z(n : F, a : F, z : F) : F == laguerreL(n - 1, a + 1, z) dLaguerreL(l : List F, t : SE) : F == grad3(l, t, opLaguerreL, eLaguerreLGrad_z) setProperty(opLaguerreL, SPECIALDIFF, dLaguerreL@((List F, SE)->F) pretend None) meixnerM(n : F, b : F, c : F, z : F) : F == opMeixnerM(n, b, c, z) eMeixnerM(n : F, b : F, c : F, z : F) : F == n = 0 => 1 n = 1 => (c - 1)*z/(c*b) + 1 kernel(opMeixnerM, [n, b, c, z]) elMeixnerM(l : List F) : F == eMeixnerM(l(1), l(2), l(3), l(4)) evaluate(opMeixnerM, elMeixnerM)$BasicOperatorFunctions1(F) -- belong? op == has?(op, SPECIAL) operator op == is?(op, 'abs) => opabs is?(op, 'conjugate)=> opconjugate is?(op, 'Gamma) => opGamma is?(op, 'Gamma2) => opGamma2 is?(op, 'Beta) => opBeta is?(op, 'digamma) => opdigamma is?(op, 'polygamma)=> oppolygamma is?(op, 'besselJ) => opBesselJ is?(op, 'besselY) => opBesselY is?(op, 'besselI) => opBesselI is?(op, 'besselK) => opBesselK is?(op, 'airyAi) => opAiryAi is?(op, 'airyAiPrime) => opAiryAiPrime is?(op, 'airyBi) => opAiryBi is?(op, 'airyBiPrime) => opAiryBiPrime is?(op, 'lambertW) => opLambertW is?(op, 'polylog) => opPolylog is?(op, 'weierstrassP) => opWeierstrassP is?(op, 'weierstrassPPrime) => opWeierstrassPPrime is?(op, 'weierstrassSigma) => opWeierstrassSigma is?(op, 'weierstrassZeta) => opWeierstrassZeta is?(op, 'hypergeometricF) => opHypergeometricF is?(op, 'meijerG) => opMeijerG -- is?(op, 'weierstrassPInverse) => opWeierstrassPInverse is?(op, 'whittakerM) => opWhittakerM is?(op, 'whittakerW) => opWhittakerW is?(op, 'angerJ) => opAngerJ is?(op, 'weberE) => opWeberE is?(op, 'struveH) => opStruveH is?(op, 'struveL) => opStruveL is?(op, 'hankelH1) => opHankelH1 is?(op, 'hankelH2) => opHankelH2 is?(op, 'lommelS1) => opLommelS1 is?(op, 'lommelS2) => opLommelS2 is?(op, 'kummerM) => opKummerM is?(op, 'kummerU) => opKummerU is?(op, 'legendreP) => opLegendreP is?(op, 'legendreQ) => opLegendreQ is?(op, 'kelvinBei) => opKelvinBei is?(op, 'kelvinBer) => opKelvinBer is?(op, 'kelvinKei) => opKelvinKei is?(op, 'kelvinKer) => opKelvinKer is?(op, 'ellipticK) => opEllipticK is?(op, 'ellipticE) => opEllipticE is?(op, 'ellipticE2) => opEllipticE2 is?(op, 'ellipticF) => opEllipticF is?(op, 'ellipticPi) => opEllipticPi is?(op, 'jacobiSn) => opJacobiSn is?(op, 'jacobiCn) => opJacobiCn is?(op, 'jacobiDn) => opJacobiDn is?(op, 'jacobiZeta) => opJacobiZeta is?(op, 'jacobiTheta) => opJacobiTheta is?(op, 'lerchPhi) => opLerchPhi is?(op, 'riemannZeta) => opRiemannZeta is?(op, 'charlierC) => opCharlierC is?(op, 'hermiteH) => opHermiteH is?(op, 'jacobiP) => opJacobiP is?(op, 'laguerreL) => opLaguerreL is?(op, 'meixnerM) => opMeixnerM is?(op, '%logGamma) => op_log_gamma is?(op, '%eis) => op_eis is?(op, '%erfs) => op_erfs is?(op, '%erfis) => op_erfis error "Not a special operator" -- Could put more unconditional special rules for other functions here iGamma x == -- one? x => x (x = 1) => x kernel(opGamma, x) iabs x == zero? x => 0 one? x => 1 is?(x, opabs) => x is?(x, opconjugate) => kernel(opabs, argument(retract(x)@K)(1)) smaller?(x, 0) => kernel(opabs, -x) kernel(opabs, x) if R has abs : R -> R then import from Polynomial R iiabs x == (r := retractIfCan(x)@Union(Fraction Polynomial R, "failed")) case "failed" => iabs x f := r::Fraction Polynomial R (a := retractIfCan(numer f)@Union(R, "failed")) case "failed" or (b := retractIfCan(denom f)@Union(R,"failed")) case "failed" => iabs x abs(a::R)::F / abs(b::R)::F else if F has RadicalCategory and R has conjugate : R -> R then iiabs x == (r := retractIfCan(x)@Union(R, "failed")) case "failed" => iabs x sqrt( (r::R*conjugate(r::R))::F) else iiabs x == iabs x iconjugate(k:K):F == --output("in: ",k::OutputForm)$OutputPackage if symbolIfCan(k) case Symbol then x:=kernel(opconjugate, k::F) else op:=operator(operator(name(k))$CommonOperators)$Expression(R) --output("op: ",properties(op)::OutputForm)$OutputPackage --output("conj op: ",properties(conjugate op)::OutputForm)$OutputPackage x := conjugate(op) argument(k) --output("out: ",x::OutputForm)$OutputPackage return x iiconjugate(x:F):F == --output("iin: ",x::OutputForm)$OutputPackage if (s:=isPlus(x)) case List F then --output("isPlus: ",s::OutputForm)$OutputPackage x := reduce(_+$F,map(iiconjugate,s)) else if (s:=isTimes(x)) case List F then --output("isTimes: ",s::OutputForm)$OutputPackage x:= reduce(_*$F,map(iiconjugate,s)) else if #(ks:List K:=kernels(x))>0 then --output("kernels: ",ks::OutputForm)$OutputPackage x := eval(x,ks, _ map((k:Kernel F):F +-> (height(k)=0 =>k::F;iconjugate k), _ ks)$ListFunctions2(Kernel F,F)) else if R has conjugate:R->R and F has RetractableTo(R) then if (r:=retractIfCan(x)@Union(R,"failed")) case R then x:=conjugate(r::R)::F --output("iout: ",x::OutputForm)$OutputPackage return x iBeta(x, y) == kernel(opBeta, [x, y]) idigamma x == kernel(opdigamma, x) iiipolygamma(n, x) == kernel(oppolygamma, [n, x]) iiiBesselJ(x, y) == kernel(opBesselJ, [x, y]) iiiBesselY(x, y) == kernel(opBesselY, [x, y]) iiiBesselI(x, y) == kernel(opBesselI, [x, y]) iiiBesselK(x, y) == kernel(opBesselK, [x, y]) import from Fraction(Integer) if F has ElementaryFunctionCategory then iAiryAi x == zero?(x) => 1::F/((3::F)^twothirds*Gamma(twothirds)) kernel(opAiryAi, x) iAiryAiPrime x == zero?(x) => -1::F/((3::F)^athird*Gamma(athird)) kernel(opAiryAiPrime, x) iAiryBi x == zero?(x) => 1::F/((3::F)^asixth*Gamma(twothirds)) kernel(opAiryBi, x) iAiryBiPrime x == zero?(x) => (3::F)^asixth/Gamma(athird) kernel(opAiryBiPrime, x) else iAiryAi x == kernel(opAiryAi, x) iAiryAiPrime x == kernel(opAiryAiPrime, x) iAiryBi x == kernel(opAiryBi, x) iAiryBiPrime x == kernel(opAiryBiPrime, x) if F has ElementaryFunctionCategory then iLambertW(x) == zero?(x) => 0 x = exp(1$F) => 1$F x = -exp(-1$F) => -1$F kernel(opLambertW, x) else iLambertW(x) == zero?(x) => 0 kernel(opLambertW, x) if F has ElementaryFunctionCategory then if F has LiouvillianFunctionCategory then iiPolylog(s, x) == s = 1 => -log(1 - x) s = 2::F => dilog(1 - x) kernel(opPolylog, [s, x]) else iiPolylog(s, x) == s = 1 => -log(1 - x) kernel(opPolylog, [s, x]) else iiPolylog(s, x) == kernel(opPolylog, [s, x]) iPolylog(l) == iiPolylog(first l, second l) iWeierstrassP(g2, g3, x) == kernel(opWeierstrassP, [g2, g3, x]) iWeierstrassPPrime(g2, g3, x) == kernel(opWeierstrassPPrime, [g2, g3, x]) iWeierstrassSigma(g2, g3, x) == x = 0 => 0 kernel(opWeierstrassSigma, [g2, g3, x]) iWeierstrassZeta(g2, g3, x) == kernel(opWeierstrassZeta, [g2, g3, x]) -- Could put more conditional special rules for other functions here if R has SpecialFunctionCategory then iiGamma x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iGamma x Gamma(r::R)::F iiBeta l == (r := retractIfCan(first l)@Union(R,"failed")) case "failed" or _ (s := retractIfCan(second l)@Union(R,"failed")) case "failed" _ => iBeta(first l, second l) Beta(r::R, s::R)::F iidigamma x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => idigamma x digamma(r::R)::F iipolygamma l == (s := retractIfCan(first l)@Union(R,"failed")) case "failed" or _ (r := retractIfCan(second l)@Union(R,"failed")) case "failed" _ => iiipolygamma(first l, second l) polygamma(s::R, r::R)::F iiBesselJ l == (r := retractIfCan(first l)@Union(R,"failed")) case "failed" or _ (s := retractIfCan(second l)@Union(R,"failed")) case "failed" _ => iiiBesselJ(first l, second l) besselJ(r::R, s::R)::F iiBesselY l == (r := retractIfCan(first l)@Union(R,"failed")) case "failed" or _ (s := retractIfCan(second l)@Union(R,"failed")) case "failed" _ => iiiBesselY(first l, second l) besselY(r::R, s::R)::F iiBesselI l == (r := retractIfCan(first l)@Union(R,"failed")) case "failed" or _ (s := retractIfCan(second l)@Union(R,"failed")) case "failed" _ => iiiBesselI(first l, second l) besselI(r::R, s::R)::F iiBesselK l == (r := retractIfCan(first l)@Union(R,"failed")) case "failed" or _ (s := retractIfCan(second l)@Union(R,"failed")) case "failed" _ => iiiBesselK(first l, second l) besselK(r::R, s::R)::F iiAiryAi x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iAiryAi x airyAi(r::R)::F iiAiryAiPrime x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iAiryAiPrime x airyAiPrime(r::R)::F iiAiryBi x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iAiryBi x airyBi(r::R)::F iiAiryBi x == (r := retractIfCan(x)@Union(R,"failed")) case "failed" => iAiryBiPrime x airyBiPrime(r::R)::F else if R has RetractableTo Integer then iiGamma x == (r := retractIfCan(x)@Union(Integer, "failed")) case Integer and (r::Integer >= 1) => factorial(r::Integer - 1)::F iGamma x else iiGamma x == iGamma x iiBeta l == iBeta(first l, second l) iidigamma x == idigamma x iipolygamma l == iiipolygamma(first l, second l) iiBesselJ l == iiiBesselJ(first l, second l) iiBesselY l == iiiBesselY(first l, second l) iiBesselI l == iiiBesselI(first l, second l) iiBesselK l == iiiBesselK(first l, second l) iiAiryAi x == iAiryAi x iiAiryAiPrime x == iAiryAiPrime x iiAiryBi x == iAiryBi x iiAiryBiPrime x == iAiryBiPrime x iiWeierstrassP l == iWeierstrassP(first l, second l, third l) iiWeierstrassPPrime l == iWeierstrassPPrime(first l, second l, third l) iiWeierstrassSigma l == iWeierstrassSigma(first l, second l, third l) iiWeierstrassZeta l == iWeierstrassZeta(first l, second l, third l) -- Default behaviour is to build a kernel evaluate(opGamma, iiGamma)$BasicOperatorFunctions1(F) evaluate(opabs, iiabs)$BasicOperatorFunctions1(F) evaluate(conjugate(opabs), (x:F):F +-> iiabs(x))$BasicOperatorFunctions1(F) evaluate(opconjugate, iiconjugate)$BasicOperatorFunctions1(F) evaluate(conjugate(opconjugate), (x:F):F +-> x)$BasicOperatorFunctions1(F) -- evaluate(opGamma2 , iiGamma2 )$BasicOperatorFunctions1(F) evaluate(opBeta , iiBeta )$BasicOperatorFunctions1(F) evaluate(opdigamma , iidigamma )$BasicOperatorFunctions1(F) evaluate(oppolygamma , iipolygamma)$BasicOperatorFunctions1(F) evaluate(opBesselJ , iiBesselJ )$BasicOperatorFunctions1(F) evaluate(opBesselY , iiBesselY )$BasicOperatorFunctions1(F) evaluate(opBesselI , iiBesselI )$BasicOperatorFunctions1(F) evaluate(opBesselK , iiBesselK )$BasicOperatorFunctions1(F) evaluate(opAiryAi , iiAiryAi )$BasicOperatorFunctions1(F) evaluate(opAiryAiPrime, iiAiryAiPrime)$BasicOperatorFunctions1(F) evaluate(opAiryBi , iiAiryBi )$BasicOperatorFunctions1(F) evaluate(opAiryBiPrime, iiAiryBiPrime)$BasicOperatorFunctions1(F) evaluate(opLambertW, iLambertW)$BasicOperatorFunctions1(F) evaluate(opPolylog, iPolylog)$BasicOperatorFunctions1(F) evaluate(opWeierstrassP, iiWeierstrassP)$BasicOperatorFunctions1(F) evaluate(opWeierstrassPPrime, iiWeierstrassPPrime)$BasicOperatorFunctions1(F) evaluate(opWeierstrassSigma, iiWeierstrassSigma)$BasicOperatorFunctions1(F) evaluate(opWeierstrassZeta, iiWeierstrassZeta)$BasicOperatorFunctions1(F) evaluate(opHypergeometricF, iiHypergeometricF)$BasicOperatorFunctions1(F) evaluate(opMeijerG, iiMeijerG)$BasicOperatorFunctions1(F) diff1(op : OP, n : F, x : F) : F == dm := dummy kernel(opdiff, [op [dm, x], dm, n]) iBesselJ(l : List F, t : SE) : F == n := first l; x := second l differentiate(n, t)*diff1(opBesselJ, n, x) + differentiate(x, t) * ahalf * (besselJ (n-1, x) - besselJ (n+1, x)) iBesselY(l : List F, t : SE) : F == n := first l; x := second l differentiate(n, t)*diff1(opBesselY, n, x) + differentiate(x, t) * ahalf * (besselY (n-1, x) - besselY (n+1, x)) iBesselI(l : List F, t : SE) : F == n := first l; x := second l differentiate(n, t)*diff1(opBesselI, n, x) + differentiate(x, t)* ahalf * (besselI (n-1, x) + besselI (n+1, x)) iBesselK(l : List F, t : SE) : F == n := first l; x := second l differentiate(n, t)*diff1(opBesselK, n, x) - differentiate(x, t)* ahalf * (besselK (n-1, x) + besselK (n+1, x)) dPolylog(l : List F, t : SE) : F == s := first l; x := second l differentiate(s, t)*diff1(opPolylog, s, x) + differentiate(x, t)*polylog(s-1, x)/x ipolygamma(l : List F, x : SE) : F == import from List(Symbol) member?(x, variables first l) => error "cannot differentiate polygamma with respect to the first argument" n := first l; y := second l differentiate(y, x)*polygamma(n+1, y) iBetaGrad1(l : List F) : F == x := first l; y := second l Beta(x, y)*(digamma x - digamma(x+y)) iBetaGrad2(l : List F) : F == x := first l; y := second l Beta(x, y)*(digamma y - digamma(x+y)) if F has ElementaryFunctionCategory then iGamma2(l : List F, t : SE) : F == a := first l; x := second l differentiate(a, t)*diff1(opGamma2, a, x) - differentiate(x, t)* x ^ (a - 1) * exp(-x) setProperty(opGamma2, SPECIALDIFF, iGamma2@((List F, SE)->F) pretend None) inGamma2(li : List INP) : INP == convert cons(convert('Gamma), li) input(opGamma2, inGamma2@((List INP) -> INP)) dLambertW(x : F) : F == lw := lambertW(x) lw/(x*(1+lw)) iWeierstrassPGrad1(l : List F) : F == g2 := first l g3 := second l x := third l delta := g2^3 - 27*g3^2 wp := weierstrassP(g2, g3, x) (weierstrassPPrime(g2, g3, x)*(-9*ahalf*g3 *weierstrassZeta(g2, g3, x) + afourth*g2^2*x) - 9*g3*wp^2 + ahalf*g2^2*wp + 3*ahalf*g2*g3)/delta iWeierstrassPGrad2(l : List F) : F == g2 := first l g3 := second l x := third l delta := g2^3 - 27*g3^2 wp := weierstrassP(g2, g3, x) (weierstrassPPrime(g2, g3, x)*(3*g2*weierstrassZeta(g2, g3, x) - 9*ahalf*g3*x) + 6*g2*wp^2 - 9*g3*wp-g2^2)/delta iWeierstrassPGrad3(l : List F) : F == weierstrassPPrime(first l, second l, third l) iWeierstrassPPrimeGrad1(l : List F) : F == g2 := first l g3 := second l x := third l delta := g2^3 - 27*g3^2 wp := weierstrassP(g2, g3, x) wpp := weierstrassPPrime(g2, g3, x) wpp2 := 6*wp^2 - ahalf*g2 (wpp2*(-9*ahalf*g3*weierstrassZeta(g2, g3, x) + afourth*g2^2*x) + wpp*(9*ahalf*g3*wp + afourth*g2^2) - 18*g3*wp*wpp + ahalf*g2^2*wpp)/delta iWeierstrassPPrimeGrad2(l : List F) : F == g2 := first l g3 := second l x := third l delta := g2^3 - 27*g3^2 wp := weierstrassP(g2, g3, x) wpp := weierstrassPPrime(g2, g3, x) wpp2 := 6*wp^2 - ahalf*g2 (wpp2*(3*g2*weierstrassZeta(g2, g3, x) - 9*ahalf*g3*x) + wpp*(-3*g2*wp - 9*ahalf*g3) + 12*g2*wp*wpp - 9*g3*wpp)/delta iWeierstrassPPrimeGrad3(l : List F) : F == g2 := first l 6*weierstrassP(g2, second l, third l)^2 - ahalf*g2 iWeierstrassSigmaGrad1(l : List F) : F == g2 := first l g3 := second l x := third l delta := g2^3 - 27*g3^2 ws := weierstrassSigma(g2, g3, x) wz := weierstrassZeta(g2, g3, x) wsp := wz*ws wsp2 := - weierstrassP(g2, g3, x)*ws + wz^2*ws afourth*(-9*g3*wsp2 - g2^2*ws - 3*afourth*g2*g3*x^2*ws + g2^2*x*wsp)/delta iWeierstrassSigmaGrad2(l : List F) : F == g2 := first l g3 := second l x := third l delta := g2^3 - 27*g3^2 ws := weierstrassSigma(g2, g3, x) wz := weierstrassZeta(g2, g3, x) wsp := wz*ws wsp2 := - weierstrassP(g2, g3, x)*ws + wz^2*ws ahalf*(3*g2*wsp2 + 9*g3*ws + afourth*g2^2*x^2*ws - 9*g3*x*wsp)/delta iWeierstrassSigmaGrad3(l : List F) : F == g2 := first l g3 := second l x := third l weierstrassZeta(g2, g3, x)*weierstrassSigma(g2, g3, x) iWeierstrassZetaGrad1(l : List F) : F == g2 := first l g3 := second l x := third l delta := g2^3 - 27*g3^2 wp := weierstrassP(g2, g3, x) (ahalf*weierstrassZeta(g2, g3, x)*(9*g3*wp + ahalf*g2^2) - ahalf*g2*x*(ahalf*g2*wp+3*afourth*g3) + 9*afourth*g3*weierstrassPPrime(g2, g3, x))/delta iWeierstrassZetaGrad2(l : List F) : F == g2 := first l g3 := second l x := third l delta := g2^3 - 27*g3^2 wp := weierstrassP(g2, g3, x) (-3*weierstrassZeta(g2, g3, x)*(g2*wp + 3*ahalf*g3) + ahalf*x*(9*g3*wp + ahalf*g2^2) - 3*ahalf*g2*weierstrassPPrime(g2, g3, x))/delta iWeierstrassZetaGrad3(l : List F) : F == -weierstrassP(first l, second l, third l) OF ==> OutputForm SEX ==> SExpression NNI ==> NonNegativeInteger dconjugate(lo : List OF) : OF == overbar lo.1 display(opconjugate,dconjugate) if F has RetractableTo(Integer) then get_int_listf : List F -> List Integer get_int_listo : (Integer, List OF) -> List Integer get_int_listi : (Integer, List INP) -> List Integer get_int_listf(lf : List F) : List Integer == map(z +-> retract(z)@Integer, lf)$ListFunctions2(F, Integer) replace_i(lp : List F, v : F, i : NNI) : List F == concat(first(lp, (i - 1)::NNI), cons(v, rest(lp, i))) iiHypergeometricF(l) == n := #l z := l(n-2) if z = 0 then nn := (n - 2)::NNI pq := rest(l, nn) pqi := get_int_listf(pq) p := first(pqi) q := first(rest(pqi)) p <= q + 1 => return 1 kernel(opHypergeometricF, l) idvsum(op : BasicOperator, n : Integer, l : List F, x : Symbol) : F == res : F := 0 for i in 1..n for a in l repeat dm := dummy nl := replace_i(l, dm, i) res := res + differentiate(a, x)*kernel(opdiff, [op nl, dm, a]) res dvhypergeom(l : List F, x : Symbol) : F == n := #l nn := (n - 2)::NNI pq := rest(l, nn) pqi := get_int_listf(pq) ol := l l := first(l, nn) l1 := reverse(l) z := first(l1) p := first(pqi) q := first(rest(pqi)) aprod := 1@F nl := []@(List F) for i in 1..p repeat a := first(l) nl := cons(a + 1, nl) aprod := aprod * a l := rest(l) bprod := 1@F for i in 1..q repeat b := first(l) nl := cons(b + 1, nl) bprod := bprod * b l := rest(l) nl0 := reverse!(nl) nl1 := cons(z, pq) nl := concat(nl0, nl1) aprod := aprod/bprod idvsum(opHypergeometricF, nn - 1, ol, x) + differentiate(z, x)*aprod*opHypergeometricF(nl) add_pairs_to_list(lp : List List F, l : List F) : List F == for p in lp repeat #p ~= 2 => error "not a list of pairs" l := cons(p(2), cons(p(1), l)) l dvmeijer(l : List F, x : Symbol) : F == n := #l nn := (n - 4)::NNI l0 := l nl := rest(l, nn) nli := get_int_listf(nl) l := first(l, nn) l1 := reverse(l) z := first(l1) n1 := first(nli) n2 := nli(2) a := first l sign : F := 1 if n1 > 0 or n2 > 0 then na := a - 1 if n1 = 0 then sign := -1 l2 := cons(na, rest l) else na := a if nli(3) > 0 then sign := -1 l2 := cons(a + 1, rest l) nm : F := opMeijerG(concat(l2, nl)) om : F := opMeijerG(l0) idvsum(opMeijerG, nn - 1, l0, x) + differentiate(z, x)*(sign*nm + na*om)/z get_if_list(n : Integer, lf : List INP) : List List INP == a := []@(List INP) for i in 1..n repeat a := cons(first(lf), a) lf := rest(lf) a := cons(convert('construct), reverse!(a)) [a, lf] get_if_lists(ln : List Integer, lf : List INP) : List List INP == rl := []@(List List INP) for n in ln repeat al := get_if_list(n, lf) rl := cons(first(al), rl) lf := first(rest(al)) rl := reverse!(rl) cons(lf, rl) get_int_listi(n : Integer, lo : List INP) : List Integer == n0 := (#lo - n)::NNI lo := rest(lo, n0) rl := []@(List Integer) for i in 1..n repeat p := integer(first(lo) pretend SEX)$SEX rl := cons(p, rl) lo := rest(lo) rl := reverse!(rl) rl get_of_list(n : Integer, lo : List OF) : List List OF == a := []@(List OF) for i in 1..n repeat a := cons(first(lo), a) lo := rest(lo) a := reverse!(a) [a, lo] get_of_lists(ln : List Integer, lo : List OF) : List List OF == rl := []@(List List OF) for n in ln repeat al := get_of_list(n, lo) rl := cons(first(al), rl) lo := first(rest(al)) rl := reverse!(rl) cons(lo, rl) get_int_listo(n : Integer, lo : List OF) : List Integer == n0 := (#lo - n)::NNI lo := rest(lo, n0) rl := []@(List Integer) for i in 1..n repeat p := integer(first(lo) pretend SEX)$SEX rl := cons(p, rl) lo := rest(lo) rl := reverse!(rl) rl dhyper0(op : OF, lo : List OF) : OF == n0 := (#lo - 2)::NNI pql := get_int_listo(2, lo) lo := first(lo, n0) al := get_of_lists(pql, lo) lo := first(al) al := rest(al) a := first al b := first(rest(al)) z := first(lo) prefix(op, [bracket a, bracket b, z]) dhyper(lo : List OF) : OF == dhyper0("hypergeometricF"::Symbol::OF, lo) ddhyper(lo : List OF) : OF == dhyper0(first lo, rest lo) dmeijer0(op : OF, lo : List OF) : OF == n0 := (#lo - 4)::NNI nl := get_int_listo(4, lo) lo := first(lo, n0) al := get_of_lists(nl, lo) lo := first(al) al := rest(al) z := first(lo) prefix(op, concat( map(bracket, al)$ListFunctions2(List OF, OF), [z])) dmeijer(lo : List OF) : OF == dmeijer0('meijerG::OF, lo) ddmeijer(lo : List OF) : OF == dmeijer0(first lo, rest lo) setProperty(opHypergeometricF, '%diffDisp, ddhyper@(List OF -> OF) pretend None) setProperty(opMeijerG, '%diffDisp, ddmeijer@(List OF -> OF) pretend None) display(opHypergeometricF, dhyper) display(opMeijerG, dmeijer) setProperty(opHypergeometricF, SPECIALDIFF, dvhypergeom@((List F, Symbol)->F) pretend None) setProperty(opMeijerG, SPECIALDIFF, dvmeijer@((List F, Symbol)->F) pretend None) inhyper(lf : List INP) : INP == pqi := get_int_listi(2, lf) al := get_if_lists(pqi, lf) lf := first(al) al := rest(al) a := first al ai : INP := convert(a) b := first(rest(al)) bi : INP := convert(b) zi := first(lf) li : List INP := [convert('hypergeometricF), ai, bi, zi] convert(li) input(opHypergeometricF, inhyper@((List INP) -> INP)) inmeijer(lf : List INP) : INP == pqi := get_int_listi(4, lf) al := get_if_lists(pqi, lf) lf := first(al) al := rest(al) a := first al ai : INP := convert(a) al := rest(al) b := first(al) bi : INP := convert(b) al := rest(al) c := first(al) ci : INP := convert(c) al := rest(al) d := first(al) di : INP := convert(d) zi := first(lf) li : List INP := [convert('meijerG), ai, bi, ci, di, zi] convert(li) input(opMeijerG, inmeijer@((List INP) -> INP)) else iiHypergeometricF(l) == kernel(opHypergeometricF, l) iiMeijerG(l) == kernel(opMeijerG, l) d_eis(x : F) : F == -kernel(op_eis, x) + 1/x if F has TranscendentalFunctionCategory and F has RadicalCategory then d_erfs(x : F) : F == 2*x*kernel(op_erfs, x) - 2::F/sqrt(pi()) d_erfis(x : F) : F == -2*x*kernel(op_erfis, x) + 2::F/sqrt(pi()) derivative(op_erfs, d_erfs) derivative(op_erfis, d_erfis) -- differentiate(abs(g(z)),z) --dvabs : (List F, SE) -> F --dvabs(arg, z) == -- g := first arg -- conjugate(g)*inv(2*abs(g))*differentiate(g,z)+g*inv(2*abs(g))*differentiate(conjugate(g),z) --setProperty(opabs, SPECIALDIFF, dvabs@((List F, SE)->F) pretend None) derivative(opabs, (x : F) : F +-> conjugate(x)*inv(2*abs(x))) -- Wirtinger derivate of non-holomorphic functions derivative(opconjugate, (x : F) : F +-> 0) derivative(conjugate opconjugate, (x : F) : F +-> 1) derivative(opGamma, (x : F) : F +-> digamma(x)*Gamma(x)) derivative(op_log_gamma, (x : F) : F +-> digamma(x)) derivative(opBeta, [iBetaGrad1, iBetaGrad2]) derivative(opdigamma, (x : F) : F +-> polygamma(1, x)) derivative(op_eis, d_eis) derivative(opAiryAi, (x : F) : F +-> airyAiPrime(x)) derivative(opAiryAiPrime, (x : F) : F +-> x*airyAi(x)) derivative(opAiryBi, (x : F) : F +-> airyBiPrime(x)) derivative(opAiryBiPrime, (x : F) : F +-> x*airyBi(x)) derivative(opLambertW, dLambertW) derivative(opWeierstrassP, [iWeierstrassPGrad1, iWeierstrassPGrad2, iWeierstrassPGrad3]) derivative(opWeierstrassPPrime, [iWeierstrassPPrimeGrad1, iWeierstrassPPrimeGrad2, iWeierstrassPPrimeGrad3]) derivative(opWeierstrassSigma, [iWeierstrassSigmaGrad1, iWeierstrassSigmaGrad2, iWeierstrassSigmaGrad3]) derivative(opWeierstrassZeta, [iWeierstrassZetaGrad1, iWeierstrassZetaGrad2, iWeierstrassZetaGrad3]) setProperty(oppolygamma, SPECIALDIFF, ipolygamma@((List F, SE)->F) pretend None) setProperty(opBesselJ, SPECIALDIFF, iBesselJ@((List F, SE)->F) pretend None) setProperty(opBesselY, SPECIALDIFF, iBesselY@((List F, SE)->F) pretend None) setProperty(opBesselI, SPECIALDIFF, iBesselI@((List F, SE)->F) pretend None) setProperty(opBesselK, SPECIALDIFF, iBesselK@((List F, SE)->F) pretend None) setProperty(opPolylog, SPECIALDIFF, dPolylog@((List F, SE)->F) pretend None) )abbrev package SUMFS FunctionSpaceSum ++ Top-level sum function ++ Author: Manuel Bronstein ++ Date Created: ??? ++ Date Last Updated: 19 April 1991 ++ Description: computes sums of top-level expressions; FunctionSpaceSum(R, F) : Exports == Implementation where R : Join(IntegralDomain, Comparable, RetractableTo Integer, LinearlyExplicitRingOver Integer) F : Join(FunctionSpace R, CombinatorialOpsCategory, AlgebraicallyClosedField, TranscendentalFunctionCategory) SE ==> Symbol K ==> Kernel F Exports ==> with sum : (F, SE) -> F ++ sum(a(n), n) returns A(n) such that A(n+1) - A(n) = a(n); sum : (F, SegmentBinding F) -> F ++ sum(f(n), n = a..b) returns f(a) + f(a+1) + ... + f(b); Implementation ==> add import from ElementaryFunctionStructurePackage(R, F) import from GosperSummationMethod(IndexedExponents K, K, R, SparseMultivariatePolynomial(R, K), F) import from Segment(F) innersum : (F, K) -> Union(F, "failed") notRF? : (F, K) -> Boolean newk : () -> K newk() == kernel(new()$SE) sum(x : F, s : SegmentBinding F) == k := kernel(variable s)@K (u := innersum(x, k)) case "failed" => summation(x, s) eval(u::F, k, 1 + hi segment s) - eval(u::F, k, lo segment s) sum(x : F, v : SE) == (u := innersum(x, kernel(v)@K)) case "failed" => summation(x,v) u::F notRF?(f, k) == for kk in tower f repeat member?(k, tower(kk::F)) and (symbolIfCan(kk) case "failed") => return true false innersum(x, k) == zero? x => 0 notRF?(f := normalize(x / (x1 := eval(x, k, k::F - 1))), k) => "failed" (u := GospersMethod(f, k, newk)) case "failed" => "failed" x1 * eval(u::F, k, k::F - 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 function funcpkgs manip algfunc -- elemntry constant funceval COMBFUNC fe spad Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/554697800083835704-25px002.spad using old system compiler. COMBOPC abbreviates category CombinatorialOpsCategory ------------------------------------------------------------------------ initializing NRLIB COMBOPC for CombinatorialOpsCategory compiling into NRLIB COMBOPC ;;; *** |CombinatorialOpsCategory| REDEFINED Time: 0 SEC. finalizing NRLIB COMBOPC Processing CombinatorialOpsCategory for Browser database: --------constructor--------- --------(factorials (% %))--------- --------(factorials (% % (Symbol)))--------- --------(summation (% % (Symbol)))--------- --------(summation (% % (SegmentBinding %)))--------- --------(product (% % (Symbol)))--------- --------(product (% % (SegmentBinding %)))--------- ; compiling file "/var/aw/var/LatexWiki/COMBOPC.NRLIB/COMBOPC.lsp" (written 08 APR 2017 01:10:39 PM): ; /var/aw/var/LatexWiki/COMBOPC.NRLIB/COMBOPC.fasl written ; compilation finished in 0:00:00.005 ------------------------------------------------------------------------ CombinatorialOpsCategory is now explicitly exposed in frame initial CombinatorialOpsCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/COMBOPC.NRLIB/COMBOPC COMBF abbreviates package CombinatorialFunction ------------------------------------------------------------------------ initializing NRLIB COMBF for CombinatorialFunction compiling into NRLIB COMBF ****** Domain: R already in scope processing macro definition dummy ==> ::((Sel (Symbol) new),F) compiling exported factorial : F -> F Time: 0.04 SEC. compiling exported binomial : (F,F) -> F Time: 0 SEC. compiling exported permutation : (F,F) -> F Time: 0 SEC. importing F importing Kernel F compiling local number? : F -> Boolean ****** Domain: R already in scope augmenting R: (RetractableTo (Integer)) Time: 0 SEC. compiling exported ^ : (F,F) -> F Time: 0.05 SEC. compiling exported belong? : BasicOperator -> Boolean Time: 0 SEC. compiling local fourth : List F -> F Time: 0 SEC. compiling local dvpow1 : List F -> F Time: 0.01 SEC. compiling exported factorials : F -> F Time: 0 SEC. compiling exported factorials : (F,Symbol) -> F Time: 0 SEC. compiling local facts : (F,List Symbol) -> F Time: 0.01 SEC. compiling local summand : List F -> F Time: 0 SEC. compiling exported product : (F,Symbol) -> F Time: 0 SEC. compiling exported summation : (F,Symbol) -> F Time: 0 SEC. compiling local dvsum : (List F,Symbol) -> F Time: 0.01 SEC. compiling local dvdsum : (List F,Symbol) -> F Time: 0 SEC. compiling local dvprod : (List F,Symbol) -> F Time: 0.08 SEC. compiling local dvdprod : (List F,Symbol) -> F Time: 0.08 SEC. compiling local dprod : List F -> OutputForm Time: 0 SEC. compiling local ddprod : List F -> OutputForm Time: 0 SEC. compiling local dsum : List F -> OutputForm Time: 0.01 SEC. compiling local ddsum : List F -> OutputForm Time: 0 SEC. compiling local equalsumprod : (Kernel F,Kernel F) -> Boolean Time: 0 SEC. compiling local equaldsumprod : (Kernel F,Kernel F) -> Boolean Time: 0.01 SEC. compiling exported product : (F,SegmentBinding F) -> F Time: 0 SEC. compiling exported summation : (F,SegmentBinding F) -> F Time: 0 SEC. compiling local smpfact : (SparseMultivariatePolynomial(R,Kernel F),List Symbol) -> F Time: 0.01 SEC. compiling local K2fact : (Kernel F,List Symbol) -> F Time: 0.01 SEC. compiling exported operator : BasicOperator -> BasicOperator Time: 0 SEC. compiling local iprod : List F -> F Time: 0.01 SEC. compiling local isum : List F -> F Time: 0 SEC. compiling local idprod : List F -> F Time: 0.01 SEC. compiling local idsum : List F -> F Time: 0.02 SEC. compiling local ifact : F -> F Time: 0 SEC. compiling local ibinom : List F -> F Time: 0 SEC. compiling local iperm : List F -> F Time: 0 SEC. ****** Domain: R already in scope augmenting R: (RetractableTo (Integer)) compiling exported iidsum : List F -> F Time: 0.01 SEC. compiling exported iidprod : List F -> F Time: 0.01 SEC. compiling local iiipow : List F -> F Time: 0.02 SEC. ****** Domain: F already in scope augmenting F: (RadicalCategory) compiling exported ipow : List F -> F Time: 0.01 SEC. compiling exported ipow : List F -> F Time: 0 SEC. compiling exported ipow : List F -> F Time: 0.04 SEC. ****** Domain: R already in scope augmenting R: (CombinatorialFunctionCategory) compiling exported iifact : F -> F Time: 0.01 SEC. compiling exported iiperm : List F -> F Time: 0 SEC. ****** Domain: R already in scope augmenting R: (RetractableTo (Integer)) compiling exported iibinom : List F -> F Time: 0.03 SEC. compiling exported iibinom : List F -> F Time: 0.01 SEC. compiling exported iibinom : List F -> F Time: 0 SEC. compiling exported iifact : F -> F Time: 0 SEC. compiling exported iibinom : List F -> F Time: 0 SEC. compiling exported iiperm : List F -> F Time: 0 SEC. ****** Domain: R already in scope augmenting R: (ElementaryFunctionCategory) compiling exported iipow : List F -> F Time: 0.01 SEC. compiling exported iipow : List F -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (ElementaryFunctionCategory) compiling local dvpow2 : List F -> F Time: 0.01 SEC. (time taken in buildFunctor: 0) ;;; *** |CombinatorialFunction| REDEFINED ;;; *** |CombinatorialFunction| REDEFINED Time: 0.04 SEC. Warnings: [1] ^: val has no value [2] ^: exponent has no value [3] iiipow: var has no value [4] iiipow: exponent has no value [5] ipow: n has no value [6] ipow: var has no value [7] ipow: exponent has no value Cumulative Statistics for Constructor CombinatorialFunction Time: 0.56 seconds finalizing NRLIB COMBF Processing CombinatorialFunction for Browser database: --------constructor--------- --------(belong? ((Boolean) (BasicOperator)))--------- --------(operator ((BasicOperator) (BasicOperator)))--------- --------(^ (F F F))--------- --------(binomial (F F F))--------- --------(permutation (F F F))--------- --------(factorial (F F))--------- --------(factorials (F F))--------- --------(factorials (F F (Symbol)))--------- --------(summation (F F (Symbol)))--------- --------(summation (F F (SegmentBinding F)))--------- --------(product (F F (Symbol)))--------- --------(product (F F (SegmentBinding F)))--------- --------(iifact (F F))--------- --------(iibinom (F (List F)))--------- --------(iiperm (F (List F)))--------- --------(iipow (F (List F)))--------- --------(iidsum (F (List F)))--------- --------(iidprod (F (List F)))--------- --------(ipow (F (List F)))--------- ; compiling file "/var/aw/var/LatexWiki/COMBF.NRLIB/COMBF.lsp" (written 08 APR 2017 01:10:40 PM): ; /var/aw/var/LatexWiki/COMBF.NRLIB/COMBF.fasl written ; compilation finished in 0:00:00.317 ------------------------------------------------------------------------ CombinatorialFunction is now explicitly exposed in frame initial CombinatorialFunction will be automatically loaded when needed from /var/aw/var/LatexWiki/COMBF.NRLIB/COMBF FSPECF abbreviates package FunctionalSpecialFunction (EVAL-WHEN (EVAL LOAD) (SETQ |$tryRecompileArguments| NIL)) Value = NIL ------------------------------------------------------------------------ initializing NRLIB FSPECF for FunctionalSpecialFunction compiling into NRLIB FSPECF ****** Domain: R already in scope processing macro definition INP ==> InputForm processing macro definition SPECIALINPUT ==> QUOTE %specialInput compiling exported abs : F -> F Time: 0.01 SEC. compiling exported conjugate : F -> F Time: 0 SEC. compiling exported Gamma : F -> F Time: 0 SEC. compiling exported Gamma : (F,F) -> F Time: 0 SEC. compiling exported Beta : (F,F) -> F Time: 0 SEC. compiling exported digamma : F -> F Time: 0 SEC. compiling exported polygamma : (F,F) -> F Time: 0 SEC. compiling exported besselJ : (F,F) -> F Time: 0 SEC. compiling exported besselY : (F,F) -> F Time: 0 SEC. compiling exported besselI : (F,F) -> F Time: 0 SEC. compiling exported besselK : (F,F) -> F Time: 0 SEC. compiling exported airyAi : F -> F Time: 0 SEC. compiling exported airyAiPrime : F -> F Time: 0 SEC. compiling exported airyBi : F -> F Time: 0 SEC. compiling exported airyBiPrime : F -> F Time: 0 SEC. compiling exported lambertW : F -> F Time: 0 SEC. compiling exported polylog : (F,F) -> F Time: 0 SEC. compiling exported weierstrassP : (F,F,F) -> F Time: 0 SEC. compiling exported weierstrassPPrime : (F,F,F) -> F Time: 0 SEC. compiling exported weierstrassSigma : (F,F,F) -> F Time: 0 SEC. compiling exported weierstrassZeta : (F,F,F) -> F Time: 0.01 SEC. ****** Domain: F already in scope augmenting F: (RetractableTo (Integer)) augmenting $: (SIGNATURE$ hypergeometricF (F (List F) (List F) F)) augmenting $: (SIGNATURE$ meijerG (F (List F) (List F) (List F) (List F) F)) compiling exported hypergeometricF : (List F,List F,F) -> F Time: 0 SEC. compiling exported meijerG : (List F,List F,List F,List F,F) -> F Time: 0.01 SEC. importing List Kernel F processing macro definition dummy ==> ::((Sel (Symbol) new),F) compiling local grad2 : (List F,Symbol,BasicOperator,(F,F) -> F) -> F Time: 0.01 SEC. compiling local grad3 : (List F,Symbol,BasicOperator,(F,F,F) -> F) -> F Time: 0.03 SEC. compiling local grad4 : (List F,Symbol,BasicOperator,(F,F,F,F) -> F) -> F Time: 0.03 SEC. compiling exported whittakerM : (F,F,F) -> F Time: 0 SEC. compiling local eWhittakerM : (F,F,F) -> F Time: 0 SEC. compiling local elWhittakerM : List F -> F Time: 0 SEC. compiling local eWhittakerMGrad_z : (F,F,F) -> F Time: 0.02 SEC. compiling local dWhittakerM : (List F,Symbol) -> F Time: 0 SEC. compiling exported whittakerW : (F,F,F) -> F Time: 0.01 SEC. compiling local eWhittakerW : (F,F,F) -> F Time: 0 SEC. compiling local elWhittakerW : List F -> F Time: 0 SEC. compiling local eWhittakerWGrad_z : (F,F,F) -> F Time: 0.01 SEC. compiling local dWhittakerW : (List F,Symbol) -> F Time: 0 SEC. compiling exported angerJ : (F,F) -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (TranscendentalFunctionCategory) compiling local eAngerJ : (F,F) -> F Time: 0.01 SEC. compiling local elAngerJ : List F -> F Time: 0 SEC. compiling local eAngerJGrad_z : (F,F) -> F Time: 0.01 SEC. compiling local dAngerJ : (List F,Symbol) -> F Time: 0.01 SEC. compiling local eeAngerJ : List F -> F Time: 0 SEC. compiling exported weberE : (F,F) -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (TranscendentalFunctionCategory) compiling local eWeberE : (F,F) -> F Time: 0.01 SEC. compiling local elWeberE : List F -> F Time: 0 SEC. compiling local eWeberEGrad_z : (F,F) -> F Time: 0.02 SEC. compiling local dWeberE : (List F,Symbol) -> F Time: 0 SEC. compiling local eeWeberE : List F -> F Time: 0 SEC. compiling exported struveH : (F,F) -> F Time: 0.01 SEC. compiling local eStruveH : (F,F) -> F Time: 0.01 SEC. compiling local elStruveH : List F -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (TranscendentalFunctionCategory) ****** Domain: F already in scope augmenting F: (RadicalCategory) compiling local eStruveHGrad_z : (F,F) -> F Time: 0.02 SEC. compiling local dStruveH : (List F,Symbol) -> F Time: 0 SEC. compiling exported struveL : (F,F) -> F Time: 0 SEC. compiling local eStruveL : (F,F) -> F Time: 0 SEC. compiling local elStruveL : List F -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (TranscendentalFunctionCategory) ****** Domain: F already in scope augmenting F: (RadicalCategory) compiling local eStruveLGrad_z : (F,F) -> F Time: 0.02 SEC. compiling local dStruveL : (List F,Symbol) -> F Time: 0 SEC. compiling exported hankelH1 : (F,F) -> F Time: 0.01 SEC. compiling local eHankelH1 : (F,F) -> F Time: 0 SEC. compiling local elHankelH1 : List F -> F Time: 0 SEC. compiling local eHankelH1Grad_z : (F,F) -> F Time: 0.01 SEC. compiling local dHankelH1 : (List F,Symbol) -> F Time: 0 SEC. compiling exported hankelH2 : (F,F) -> F Time: 0 SEC. compiling local eHankelH2 : (F,F) -> F Time: 0 SEC. compiling local elHankelH2 : List F -> F Time: 0.01 SEC. compiling local eHankelH2Grad_z : (F,F) -> F Time: 0 SEC. compiling local dHankelH2 : (List F,Symbol) -> F Time: 0 SEC. compiling exported lommelS1 : (F,F,F) -> F Time: 0 SEC. compiling local eLommelS1 : (F,F,F) -> F Time: 0 SEC. compiling local elLommelS1 : List F -> F Time: 0 SEC. compiling local eLommelS1Grad_z : (F,F,F) -> F Time: 0.02 SEC. compiling local dLommelS1 : (List F,Symbol) -> F Time: 0 SEC. compiling exported lommelS2 : (F,F,F) -> F Time: 0.01 SEC. compiling local eLommelS2 : (F,F,F) -> F Time: 0 SEC. compiling local elLommelS2 : List F -> F Time: 0 SEC. compiling local eLommelS2Grad_z : (F,F,F) -> F Time: 0.05 SEC. compiling local dLommelS2 : (List F,Symbol) -> F Time: 0 SEC. compiling exported kummerM : (F,F,F) -> F Time: 0 SEC. compiling local eKummerM : (F,F,F) -> F Time: 0 SEC. compiling local elKummerM : List F -> F Time: 0 SEC. compiling local eKummerMGrad_z : (F,F,F) -> F Time: 0.01 SEC. compiling local dKummerM : (List F,Symbol) -> F Time: 0.01 SEC. compiling exported kummerU : (F,F,F) -> F Time: 0 SEC. compiling local eKummerU : (F,F,F) -> F Time: 0 SEC. compiling local elKummerU : List F -> F Time: 0.01 SEC. compiling local eKummerUGrad_z : (F,F,F) -> F Time: 0.01 SEC. compiling local dKummerU : (List F,Symbol) -> F Time: 0 SEC. compiling exported legendreP : (F,F,F) -> F Time: 0 SEC. compiling local eLegendreP : (F,F,F) -> F Time: 0 SEC. compiling local elLegendreP : List F -> F Time: 0.01 SEC. compiling local eLegendrePGrad_z : (F,F,F) -> F Time: 0.04 SEC. compiling local dLegendreP : (List F,Symbol) -> F Time: 0 SEC. compiling exported legendreQ : (F,F,F) -> F Time: 0 SEC. compiling local eLegendreQ : (F,F,F) -> F Time: 0 SEC. compiling local elLegendreQ : List F -> F Time: 0 SEC. compiling local eLegendreQGrad_z : (F,F,F) -> F Time: 0.04 SEC. compiling local dLegendreQ : (List F,Symbol) -> F Time: 0.01 SEC. compiling exported kelvinBei : (F,F) -> F Time: 0 SEC. compiling local eKelvinBei : (F,F) -> F Time: 0 SEC. compiling local elKelvinBei : List F -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (RadicalCategory) compiling local eKelvinBeiGrad_z : (F,F) -> F Time: 0.01 SEC. compiling local dKelvinBei : (List F,Symbol) -> F Time: 0.01 SEC. compiling exported kelvinBer : (F,F) -> F Time: 0 SEC. compiling local eKelvinBer : (F,F) -> F Time: 0 SEC. compiling local elKelvinBer : List F -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (RadicalCategory) compiling local eKelvinBerGrad_z : (F,F) -> F Time: 0.02 SEC. compiling local dKelvinBer : (List F,Symbol) -> F Time: 0 SEC. compiling exported kelvinKei : (F,F) -> F Time: 0 SEC. compiling local eKelvinKei : (F,F) -> F Time: 0 SEC. compiling local elKelvinKei : List F -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (RadicalCategory) compiling local eKelvinKeiGrad_z : (F,F) -> F Time: 0.02 SEC. compiling local dKelvinKei : (List F,Symbol) -> F Time: 0 SEC. compiling exported kelvinKer : (F,F) -> F Time: 0 SEC. compiling local eKelvinKer : (F,F) -> F Time: 0 SEC. compiling local elKelvinKer : List F -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (RadicalCategory) compiling local eKelvinKerGrad_z : (F,F) -> F Time: 0.02 SEC. compiling local dKelvinKer : (List F,Symbol) -> F Time: 0 SEC. compiling exported ellipticK : F -> F Time: 0 SEC. compiling local eEllipticK : F -> F Time: 0 SEC. compiling local elEllipticK : List F -> F Time: 0 SEC. compiling local dEllipticK : F -> F Time: 0.02 SEC. compiling exported ellipticE : F -> F Time: 0.01 SEC. compiling local eEllipticE : F -> F Time: 0 SEC. compiling local elEllipticE : List F -> F Time: 0 SEC. compiling local dEllipticE : F -> F Time: 0.01 SEC. compiling exported ellipticE : (F,F) -> F Time: 0 SEC. compiling local eEllipticE2 : (F,F) -> F Time: 0 SEC. compiling local elEllipticE2 : List F -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (RadicalCategory) compiling local eEllipticE2Grad_z : List F -> F Time: 0.01 SEC. compiling local eEllipticE2Grad_m : List F -> F Time: 0.01 SEC. compiling local inEllipticE2 : List InputForm -> InputForm Time: 0.01 SEC. compiling exported ellipticF : (F,F) -> F Time: 0 SEC. compiling local eEllipticF : (F,F) -> F Time: 0.01 SEC. compiling local elEllipticF : List F -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (RadicalCategory) compiling local eEllipticFGrad_z : List F -> F Time: 0.01 SEC. compiling local eEllipticFGrad_m : List F -> F Time: 0.03 SEC. compiling exported ellipticPi : (F,F,F) -> F Time: 0 SEC. compiling local eEllipticPi : (F,F,F) -> F Time: 0.01 SEC. compiling local elEllipticPi : List F -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (RadicalCategory) compiling local eEllipticPiGrad_z : List F -> F Time: 0.23 SEC. compiling local eEllipticPiGrad_n : List F -> F Time: 0.31 SEC. compiling local eEllipticPiGrad_m : List F -> F Time: 0.02 SEC. compiling exported jacobiSn : (F,F) -> F Time: 0.01 SEC. compiling local eJacobiSn : (F,F) -> F Time: 0 SEC. compiling local elJacobiSn : List F -> F Time: 0 SEC. compiling local jacobiGradHelper : (F,F) -> F Time: 0.01 SEC. compiling local eJacobiSnGrad_z : List F -> F Time: 0 SEC. compiling local eJacobiSnGrad_m : List F -> F Time: 0.02 SEC. compiling exported jacobiCn : (F,F) -> F Time: 0 SEC. compiling local eJacobiCn : (F,F) -> F Time: 0.01 SEC. compiling local elJacobiCn : List F -> F Time: 0 SEC. compiling local eJacobiCnGrad_z : List F -> F Time: 0.01 SEC. compiling local eJacobiCnGrad_m : List F -> F Time: 0.03 SEC. compiling exported jacobiDn : (F,F) -> F Time: 0 SEC. compiling local eJacobiDn : (F,F) -> F Time: 0 SEC. compiling local elJacobiDn : List F -> F Time: 0 SEC. compiling local eJacobiDnGrad_z : List F -> F Time: 0.02 SEC. compiling local eJacobiDnGrad_m : List F -> F Time: 0.02 SEC. compiling exported jacobiZeta : (F,F) -> F Time: 0 SEC. compiling local eJacobiZeta : (F,F) -> F Time: 0 SEC. compiling local elJacobiZeta : List F -> F Time: 0.01 SEC. compiling local eJacobiZetaGrad_z : List F -> F Time: 0 SEC. compiling local eJacobiZetaGrad_m : List F -> F Time: 0.30 SEC. compiling exported jacobiTheta : (F,F) -> F Time: 0 SEC. compiling local eJacobiTheta : (F,F) -> F Time: 0.01 SEC. compiling local elJacobiTheta : List F -> F Time: 0 SEC. compiling exported lerchPhi : (F,F,F) -> F Time: 0 SEC. compiling local eLerchPhi : (F,F,F) -> F Time: 0.01 SEC. compiling local elLerchPhi : List F -> F Time: 0 SEC. compiling local dLerchPhi : (List F,Symbol) -> F Time: 0.13 SEC. compiling exported riemannZeta : F -> F Time: 0.01 SEC. compiling local eRiemannZeta : F -> F Time: 0 SEC. compiling local elRiemannZeta : List F -> F Time: 0 SEC. compiling exported charlierC : (F,F,F) -> F Time: 0.01 SEC. compiling local eCharlierC : (F,F,F) -> F Time: 0 SEC. compiling local elCharlierC : List F -> F Time: 0.01 SEC. compiling exported hermiteH : (F,F) -> F Time: 0 SEC. compiling local eHermiteH : (F,F) -> F Time: 0.01 SEC. compiling local elHermiteH : List F -> F Time: 0 SEC. compiling local eHermiteHGrad_z : (F,F) -> F Time: 0.01 SEC. compiling local dHermiteH : (List F,Symbol) -> F Time: 0 SEC. compiling exported jacobiP : (F,F,F,F) -> F Time: 0 SEC. compiling local eJacobiP : (F,F,F,F) -> F Time: 0.02 SEC. compiling local elJacobiP : List F -> F Time: 0 SEC. compiling local eJacobiPGrad_z : (F,F,F,F) -> F Time: 0.03 SEC. compiling local dJacobiP : (List F,Symbol) -> F Time: 0 SEC. compiling exported laguerreL : (F,F,F) -> F Time: 0.01 SEC. compiling local eLaguerreL : (F,F,F) -> F Time: 0.01 SEC. compiling local elLaguerreL : List F -> F Time: 0 SEC. compiling local eLaguerreLGrad_z : (F,F,F) -> F Time: 0 SEC. compiling local dLaguerreL : (List F,Symbol) -> F Time: 0 SEC. compiling exported meixnerM : (F,F,F,F) -> F Time: 0.01 SEC. compiling local eMeixnerM : (F,F,F,F) -> F Time: 0.03 SEC. compiling local elMeixnerM : List F -> F Time: 0 SEC. compiling exported belong? : BasicOperator -> Boolean Time: 0 SEC. compiling exported operator : BasicOperator -> BasicOperator Time: 0.03 SEC. compiling local iGamma : F -> F Time: 0 SEC. compiling local iabs : F -> F Time: 0 SEC. augmenting R: (SIGNATURE R abs (R R)) importing Polynomial R compiling exported iiabs : F -> F Time: 0.01 SEC. ****** Domain: F already in scope augmenting F: (RadicalCategory) augmenting R: (SIGNATURE R conjugate (R R)) compiling exported iiabs : F -> F Time: 0.01 SEC. compiling exported iiabs : F -> F Time: 0 SEC. compiling exported iiabs : F -> F Time: 0 SEC. compiling local iconjugate : Kernel F -> F Time: 0 SEC. compiling exported iiconjugate : F -> F augmenting R: (SIGNATURE R conjugate (R R)) Time: 0.01 SEC. compiling local iBeta : (F,F) -> F Time: 0.01 SEC. compiling local idigamma : F -> F Time: 0 SEC. compiling local iiipolygamma : (F,F) -> F Time: 0 SEC. compiling local iiiBesselJ : (F,F) -> F Time: 0 SEC. compiling local iiiBesselY : (F,F) -> F Time: 0 SEC. compiling local iiiBesselI : (F,F) -> F Time: 0.01 SEC. compiling local iiiBesselK : (F,F) -> F Time: 0.01 SEC. importing Fraction Integer ****** Domain: F already in scope augmenting F: (ElementaryFunctionCategory) compiling exported iAiryAi : F -> F Time: 0.01 SEC. compiling exported iAiryAiPrime : F -> F Time: 0.01 SEC. compiling exported iAiryBi : F -> F Time: 0 SEC. compiling exported iAiryBiPrime : F -> F Time: 0.01 SEC. compiling exported iAiryAi : F -> F Time: 0 SEC. compiling exported iAiryAiPrime : F -> F Time: 0 SEC. compiling exported iAiryBi : F -> F Time: 0 SEC. compiling exported iAiryBiPrime : F -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (ElementaryFunctionCategory) compiling exported iLambertW : F -> F Time: 0 SEC. compiling exported iLambertW : F -> F Time: 0.01 SEC. ****** Domain: F already in scope augmenting F: (ElementaryFunctionCategory) ****** Domain: F already in scope augmenting F: (LiouvillianFunctionCategory) compiling exported iiPolylog : (F,F) -> F Time: 0 SEC. compiling exported iiPolylog : (F,F) -> F Time: 0.01 SEC. compiling exported iiPolylog : (F,F) -> F Time: 0.01 SEC. compiling local iPolylog : List F -> F Time: 0 SEC. compiling local iWeierstrassP : (F,F,F) -> F Time: 0.01 SEC. compiling local iWeierstrassPPrime : (F,F,F) -> F Time: 0 SEC. compiling local iWeierstrassSigma : (F,F,F) -> F Time: 0.01 SEC. compiling local iWeierstrassZeta : (F,F,F) -> F Time: 0.01 SEC. ****** Domain: R already in scope augmenting R: (SpecialFunctionCategory) compiling exported iiGamma : F -> F Time: 0 SEC. compiling exported iiBeta : List F -> F Time: 0 SEC. compiling exported iidigamma : F -> F Time: 0.01 SEC. compiling exported iipolygamma : List F -> F Time: 0 SEC. compiling exported iiBesselJ : List F -> F Time: 0 SEC. compiling exported iiBesselY : List F -> F Time: 0 SEC. compiling exported iiBesselI : List F -> F Time: 0 SEC. compiling exported iiBesselK : List F -> F Time: 0.01 SEC. compiling exported iiAiryAi : F -> F Time: 0.01 SEC. compiling exported iiAiryAiPrime : F -> F Time: 0 SEC. compiling exported iiAiryBi : F -> F Time: 0 SEC. compiling exported iiAiryBi : F -> F Time: 0 SEC. ****** Domain: R already in scope augmenting R: (RetractableTo (Integer)) compiling exported iiGamma : F -> F Time: 0.01 SEC. compiling exported iiGamma : F -> F Time: 0 SEC. compiling exported iiBeta : List F -> F Time: 0 SEC. compiling exported iidigamma : F -> F Time: 0 SEC. compiling exported iipolygamma : List F -> F Time: 0 SEC. compiling exported iiBesselJ : List F -> F Time: 0 SEC. compiling exported iiBesselY : List F -> F Time: 0 SEC. compiling exported iiBesselI : List F -> F Time: 0 SEC. compiling exported iiBesselK : List F -> F Time: 0 SEC. compiling exported iiAiryAi : F -> F Time: 0 SEC. compiling exported iiAiryAiPrime : F -> F Time: 0.01 SEC. compiling exported iiAiryBi : F -> F Time: 0 SEC. compiling exported iiAiryBiPrime : F -> F Time: 0 SEC. compiling local iiWeierstrassP : List F -> F Time: 0 SEC. compiling local iiWeierstrassPPrime : List F -> F Time: 0 SEC. compiling local iiWeierstrassSigma : List F -> F Time: 0 SEC. compiling local iiWeierstrassZeta : List F -> F Time: 0 SEC. compiling local diff1 : (BasicOperator,F,F) -> F Time: 0.09 SEC. compiling local iBesselJ : (List F,Symbol) -> F Time: 0.02 SEC. compiling local iBesselY : (List F,Symbol) -> F Time: 0.03 SEC. compiling local iBesselI : (List F,Symbol) -> F Time: 0.02 SEC. compiling local iBesselK : (List F,Symbol) -> F Time: 0.03 SEC. compiling local dPolylog : (List F,Symbol) -> F Time: 0.01 SEC. compiling local ipolygamma : (List F,Symbol) -> F Time: 0.01 SEC. compiling local iBetaGrad1 : List F -> F Time: 0.01 SEC. compiling local iBetaGrad2 : List F -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (ElementaryFunctionCategory) compiling local iGamma2 : (List F,Symbol) -> F Time: 0.03 SEC. compiling local inGamma2 : List InputForm -> InputForm Time: 0.01 SEC. compiling local dLambertW : F -> F Time: 0.01 SEC. compiling local iWeierstrassPGrad1 : List F -> F Time: 0.47 SEC. compiling local iWeierstrassPGrad2 : List F -> F Time: 0.13 SEC. compiling local iWeierstrassPGrad3 : List F -> F Time: 0 SEC. compiling local iWeierstrassPPrimeGrad1 : List F -> F Time: 0.72 SEC. compiling local iWeierstrassPPrimeGrad2 : List F -> F Time: 0.24 SEC. compiling local iWeierstrassPPrimeGrad3 : List F -> F Time: 0 SEC. compiling local iWeierstrassSigmaGrad1 : List F -> F Time: 1.81 SEC. compiling local iWeierstrassSigmaGrad2 : List F -> F Time: 0.28 SEC. compiling local iWeierstrassSigmaGrad3 : List F -> F Time: 0.01 SEC. compiling local iWeierstrassZetaGrad1 : List F -> F Time: 0.20 SEC. compiling local iWeierstrassZetaGrad2 : List F -> F Time: 0.15 SEC. compiling local iWeierstrassZetaGrad3 : List F -> F Time: 0 SEC. processing macro definition OF ==> OutputForm processing macro definition SEX ==> SExpression processing macro definition NNI ==> NonNegativeInteger compiling local dconjugate : List OutputForm -> OutputForm Time: 0 SEC. ****** Domain: F already in scope augmenting F: (RetractableTo (Integer)) augmenting $: (SIGNATURE$ hypergeometricF (F (List F) (List F) F)) augmenting $: (SIGNATURE$ meijerG (F (List F) (List F) (List F) (List F) F)) compiling local get_int_listf : List F -> List Integer Time: 0.01 SEC. compiling local replace_i : (List F,F,NonNegativeInteger) -> List F Time: 0.01 SEC. compiling exported iiHypergeometricF : List F -> F Time: 0 SEC. compiling local idvsum : (BasicOperator,Integer,List F,Symbol) -> F Time: 0.05 SEC. compiling local dvhypergeom : (List F,Symbol) -> F Time: 0.03 SEC. compiling local add_pairs_to_list : (List List F,List F) -> List F Time: 0.01 SEC. compiling local dvmeijer : (List F,Symbol) -> F Time: 0.05 SEC. compiling local get_if_list : (Integer,List InputForm) -> List List InputForm Time: 0.01 SEC. compiling local get_if_lists : (List Integer,List InputForm) -> List List InputForm Time: 0 SEC. compiling local get_int_listi : (Integer,List InputForm) -> List Integer Time: 0.01 SEC. compiling local get_of_list : (Integer,List OutputForm) -> List List OutputForm Time: 0.01 SEC. compiling local get_of_lists : (List Integer,List OutputForm) -> List List OutputForm Time: 0 SEC. compiling local get_int_listo : (Integer,List OutputForm) -> List Integer Time: 0.01 SEC. compiling local dhyper0 : (OutputForm,List OutputForm) -> OutputForm Time: 0.01 SEC. compiling local dhyper : List OutputForm -> OutputForm Time: 0 SEC. compiling local ddhyper : List OutputForm -> OutputForm Time: 0 SEC. compiling local dmeijer0 : (OutputForm,List OutputForm) -> OutputForm Time: 0.02 SEC. compiling local dmeijer : List OutputForm -> OutputForm Time: 0 SEC. compiling local ddmeijer : List OutputForm -> OutputForm Time: 0 SEC. compiling local inhyper : List InputForm -> InputForm Time: 0 SEC. compiling local inmeijer : List InputForm -> InputForm Time: 0 SEC. compiling exported iiHypergeometricF : List F -> F Time: 0 SEC. compiling local iiMeijerG : List F -> F Time: 0.01 SEC. compiling local d_eis : F -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (TranscendentalFunctionCategory) ****** Domain: F already in scope augmenting F: (RadicalCategory) compiling local d_erfs : F -> F Time: 0.01 SEC. compiling local d_erfis : F -> F Time: 0 SEC. ****** Domain: F already in scope augmenting F: (RetractableTo (Integer)) augmenting $: (SIGNATURE$ hypergeometricF (F (List F) (List F) F)) augmenting $: (SIGNATURE$ meijerG (F (List F) (List F) (List F) (List F) F)) (time taken in buildFunctor: 10) ;;; *** |FunctionalSpecialFunction| REDEFINED ;;; *** |FunctionalSpecialFunction| REDEFINED Time: 0.08 SEC. Warnings: [1] iiabs: not known that (Ring) is of mode (CATEGORY R (SIGNATURE abs (R R))) Cumulative Statistics for Constructor FunctionalSpecialFunction Time: 6.87 seconds finalizing NRLIB FSPECF Processing FunctionalSpecialFunction for Browser database: --------constructor--------- --------(belong? ((Boolean) (BasicOperator)))--------- --------(operator ((BasicOperator) (BasicOperator)))--------- --------(abs (F F))--------- --------(conjugate (F F))--------- --------(Gamma (F F))--------- --------(Gamma (F F F))--------- --------(Beta (F F F))--------- --------(digamma (F F))--------- --------(polygamma (F F F))--------- --------(besselJ (F F F))--------- --------(besselY (F F F))--------- --------(besselI (F F F))--------- --------(besselK (F F F))--------- --------(airyAi (F F))--------- --------(airyAiPrime (F F))--------- --------(airyBi (F F))--------- --------(airyBiPrime (F F))--------- --------(lambertW (F F))--------- --------(polylog (F F F))--------- --------(weierstrassP (F F F F))--------- --------(weierstrassPPrime (F F F F))--------- --------(weierstrassSigma (F F F F))--------- --------(weierstrassZeta (F F F F))--------- --------(whittakerM (F F F F))--------- --------(whittakerW (F F F F))--------- --------(angerJ (F F F))--------- --------(weberE (F F F))--------- --------(struveH (F F F))--------- --------(struveL (F F F))--------- --------(hankelH1 (F F F))--------- --------(hankelH2 (F F F))--------- --------(lommelS1 (F F F F))--------- --------(lommelS2 (F F F F))--------- --------(kummerM (F F F F))--------- --------(kummerU (F F F F))--------- --------(legendreP (F F F F))--------- --------(legendreQ (F F F F))--------- --------(kelvinBei (F F F))--------- --------(kelvinBer (F F F))--------- --------(kelvinKei (F F F))--------- --------(kelvinKer (F F F))--------- --------(ellipticK (F F))--------- --------(ellipticE (F F))--------- --------(ellipticE (F F F))--------- --------(ellipticF (F F F))--------- --------(ellipticPi (F F F F))--------- --------(jacobiSn (F F F))--------- --------(jacobiCn (F F F))--------- --------(jacobiDn (F F F))--------- --------(jacobiZeta (F F F))--------- --------(jacobiTheta (F F F))--------- --------(lerchPhi (F F F F))--------- --------(riemannZeta (F F))--------- --------(charlierC (F F F F))--------- --------(hermiteH (F F F))--------- --------(jacobiP (F F F F F))--------- --------(laguerreL (F F F F))--------- --------(meixnerM (F F F F F))--------- --------(hypergeometricF (F (List F) (List F) F))--------- --------(meijerG (F (List F) (List F) (List F) (List F) F))--------- --------(iiGamma (F F))--------- --------(iiabs (F F))--------- --------(iiconjugate (F F))--------- --------(iiBeta (F (List F)))--------- --------(iidigamma (F F))--------- --------(iipolygamma (F (List F)))--------- --------(iiBesselJ (F (List F)))--------- --------(iiBesselY (F (List F)))--------- --------(iiBesselI (F (List F)))--------- --------(iiBesselK (F (List F)))--------- --------(iiAiryAi (F F))--------- --------(iiAiryAiPrime (F F))--------- --------(iiAiryBi (F F))--------- --------(iiAiryBiPrime (F F))--------- --------(iAiryAi (F F))--------- --------(iAiryAiPrime (F F))--------- --------(iAiryBi (F F))--------- --------(iAiryBiPrime (F F))--------- --------(iiHypergeometricF (F (List F)))--------- --------(iiPolylog (F F F))--------- --------(iLambertW (F F))--------- ; compiling file "/var/aw/var/LatexWiki/FSPECF.NRLIB/FSPECF.lsp" (written 08 APR 2017 01:10:47 PM): ; /var/aw/var/LatexWiki/FSPECF.NRLIB/FSPECF.fasl written ; compilation finished in 0:00:06.863 ------------------------------------------------------------------------ FunctionalSpecialFunction is now explicitly exposed in frame initial FunctionalSpecialFunction will be automatically loaded when needed from /var/aw/var/LatexWiki/FSPECF.NRLIB/FSPECF SUMFS abbreviates package FunctionSpaceSum ------------------------------------------------------------------------ initializing NRLIB SUMFS for FunctionSpaceSum compiling into NRLIB SUMFS ****** Domain: R already in scope ****** Domain: R already in scope ****** Domain: R already in scope ****** Domain: F already in scope ****** Domain: F already in scope ****** Domain: F already in scope importing ElementaryFunctionStructurePackage(R,F) importing GosperSummationMethod(IndexedExponents Kernel F,Kernel F,R,SparseMultivariatePolynomial(R,Kernel F),F) importing Segment F compiling local newk : () -> Kernel F Time: 0 SEC. compiling exported sum : (F,SegmentBinding F) -> F Time: 0 SEC. compiling exported sum : (F,Symbol) -> F Time: 0 SEC. compiling local notRF? : (F,Kernel F) -> Boolean Time: 0.01 SEC. compiling local innersum : (F,Kernel F) -> Union(F,failed) ****** comp fails at level 1 with expression: ****** error in function innersum (R) ****** level 1 ****** $x:= R$m:= (Join (IntegralDomain) (Comparable) (RetractableTo (Integer)) (LinearlyExplicitOver (Integer))) \$f:= ((((|x1| #) (#:G2128 #) (#:G2127 # #) (|k| # #) ...))) >> Apparent user error: Cannot coerce R of mode (Join (IntegralDomain) (Comparable) (RetractableTo (Integer)) (LinearlyExplicitRingOver (Integer))) to mode (Join (IntegralDomain) (Comparable) (RetractableTo (Integer)) (LinearlyExplicitOver (Integer))) Tests from GouldBk?.pdf --rrogers, Fri, 07 Apr 2017 16:22:25 +0000 replyMoved to: SandBoxCombfuncDiscussion

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