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|
--- @class IntegralExpression
--- An expression for the integral of an expression.
--- @field symbol SymbolExpression
--- @field expression Expression
--- @field upper Expression
--- @field lower Expression
--- @field attempts table<number, Expression>
--- @field results table<number, Expression>
--- @field enhancedsubstitution Integer
--- @field recursive boolean
IntegralExpression = {}
__IntegralExpression = {}
--------------------------
-- Static functionality --
--------------------------
--- Recursive part of the indefinite integral operator. Returns nil if the expression could not be integrated.
--- We switch to prodcedural programming here because it is more natural.
--- @param integral IntegralExpression
--- @return Expression|nil
function IntegralExpression.integrate(integral)
integral.expression = integral.expression:autosimplify()
if not integral.recursive and #integral.attempts > 0 then
return Copy(integral:lock(Expression.NIL, true))
end
-- print(integral.expression)
local F = IntegralExpression.table(integral)
if F then return F end
-- If we see the same integrand again, and hasn't been solved already, then the integral can't be solved
local resultindex = Contains(integral.attempts, integral.expression)
if resultindex then
return integral.results[resultindex]
end
local newindex = #integral.attempts+1
integral.attempts[newindex] = integral.expression
-- print("Evalutaing: " .. tostring(integral.expression))
F = IntegralExpression.linearproperties(integral)
if F then
-- print("Linear Properties")
integral.results[newindex] = F
return F
end
-- local exp = integral.expression
-- local sym = integral.symbol
-- local es = integral.enhancedsubstitution
-- integral.enhancedsubstitution = Integer.zero()
F = IntegralExpression.substitutionmethod(integral)
if F then
-- print("u-Substitution")
integral.results[newindex] = F
return F
end
-- integral.expression = exp
-- integral.symbol = sym
-- integral.enhancedsubstitution = es
F = IntegralExpression.rationalfunction(integral)
if F then
-- print("Rational Function")
integral.results[newindex] = F
return F
end
F = IntegralExpression.partsmethod(integral)
if F then
-- print("Parts")
integral.results[newindex] = F
return F
end
F = IntegralExpression.eulersformula(integral)
if F then
-- print("Euler's formula")
integral.results[newindex] = F
return F
end
local expanded = integral.expression:expand()
if integral.expression ~= expanded then
integral.expression = expanded
F = IntegralExpression.integrate(integral)
if F then
-- print("Expanded")
integral.results[newindex] = F
return F
end
end
expanded = (Integer.one()/((Integer.one()/integral.expression):autosimplify():expand())):autosimplify()
if integral.expression ~= expanded then
integral.expression = expanded
F = IntegralExpression.integrate(integral)
if F then
-- print("Inverse Expanded")
integral.results[newindex] = F
return F
end
end
F = IntegralExpression.enhancedsubstitutionmethod(integral)
if F then
-- print("Enhanced u-Substitution")
integral.results[newindex] = F
return F
end
return nil
end
--- A table of basic integrals, returns nil if the integrand isn't in the table.
--- @param integral IntegralExpression
--- @return Expression|nil
function IntegralExpression.table(integral)
local integrand = integral.expression
local symbol = integral.symbol
-- Constant integrand rule - int(c, x) = c*x
if integrand:freeof(symbol) then
return integrand*symbol
end
if integrand:type() == SymbolExpression then
-- int(x, x) = x^2/2
if integrand == symbol then
return integrand ^ Integer(2) / Integer(2)
end
-- Constant integrand rule again
return integrand*symbol
end
if integrand:type() == BinaryOperation then
if integrand.operation == BinaryOperation.POW then
-- int(1/x, x) = ln(x)
if integrand.expressions[1] == symbol and integrand.expressions[2] == Integer(-1) then
return LN(symbol)
end
-- Cavalieri's formula - int(x^n, x) = x^(n+1)/(n+1)
if integrand.expressions[1] == symbol and integrand.expressions[2]:freeof(symbol) then
return symbol ^ (integrand.expressions[2] + Integer.one()) / (integrand.expressions[2] + Integer.one())
end
-- int(n^x, x) = n^x/ln(n)
if integrand.expressions[1]:freeof(symbol) and integrand.expressions[2] == symbol then
return integrand / LN(integrand.expressions[1])
end
-- int(csc(x)^2, x) = -cot(x)
if integrand.expressions[1] == CSC(symbol) and integrand.expressions[2] == Integer(2) then
return -COT(symbol)
end
-- int(sec(x)^2, x) = tan(x)
if integrand.expressions[1] == SEC(symbol) and integrand.expressions[2] == Integer(2) then
return TAN(symbol)
end
end
if integrand.operation == BinaryOperation.MUL and #integrand.expressions == 2 then
-- int(tan(x)sec(x), x) = sec(x)
if integrand.expressions[1] == TAN(symbol) and integrand.expressions[2] == SEC(symbol) then
return SEC(symbol)
end
-- int(csc(x)cot(x), x) = -csc(x)
if integrand.expressions[1] == CSC(symbol) and integrand.expressions[2] == COT(symbol) then
return -CSC(symbol)
end
end
return nil
end
if integrand:type() == Logarithm then
-- int(log_n(x), x) = (x*ln(x)-x)/ln(n)
if integrand.base:freeof(symbol) and integrand.expression == symbol then
return (symbol * LN(symbol) - symbol) / LN(integrand.base)
end
return nil
end
if integrand:type() == TrigExpression then
if integrand == SIN(symbol) then
return -COS(symbol)
end
if integrand == COS(symbol) then
return SIN(symbol)
end
if integrand == TAN(symbol) then
return -LN(COS(symbol))
end
if integrand == CSC(symbol) then
return -LN(CSC(symbol)+COT(symbol))
end
if integrand == SEC(symbol) then
return LN(SEC(symbol) + TAN(symbol))
end
if integrand == COT(symbol) then
return LN(SIN(symbol))
end
if integrand == ARCSIN(symbol) then
return symbol*ARCSIN(symbol) + (Integer.one()-symbol^(Integer(2)))^(Integer.one()/Integer(2))
end
if integrand == ARCCOS(symbol) then
return symbol*ARCCOS(symbol) - (Integer.one()-symbol^(Integer(2)))^(Integer.one()/Integer(2))
end
if integrand == ARCTAN(symbol) then
return symbol*ARCTAN(symbol) - (Integer.one()/Integer(2))*LN(Integer.one()+symbol^Integer(2))
end
if integrand == ARCCSC(symbol) then
return symbol*ARCCSC(symbol) + LN(symbol*(Integer.one()+(Integer.one()-symbol^(Integer(-2)))^(Integer.one()/Integer(2))))
end
if integrand == ARCSEC(symbol) then
return symbol*ARCSEC(symbol) - LN(symbol*(Integer.one()+(Integer.one()-symbol^(Integer(-2)))^(Integer.one()/Integer(2))))
end
if integrand == ARCCOT(symbol) then
return symbol*ARCCOT(symbol) + (Integer.one()/Integer(2))*LN(Integer.one()+symbol^Integer(2))
end
end
return nil
end
--- Uses linearity to break up the integral and integrate each piece.
--- @param integral IntegralExpression
--- @return Expression|nil
function IntegralExpression.linearproperties(integral)
local expression = integral.expression
local symbol = integral.symbol
local es = integral.enhancedsubstitution
if expression:type() == BinaryOperation then
if expression.operation == BinaryOperation.MUL then
local freepart = Integer.one()
local variablepart = Integer.one()
for _, term in ipairs(expression.expressions) do
if term:freeof(symbol) then
freepart = freepart*term
else
variablepart = variablepart*term
end
end
if freepart == Integer.one() then
return nil
end
integral.expression = variablepart
integral.symbol = symbol
integral.enhancedsubstitution = es
local F = IntegralExpression.integrate(integral)
if F then
return freepart*F
end
return nil
end
if expression.operation == BinaryOperation.ADD then
local sum = Integer.zero()
for _, term in ipairs(expression.expressions) do
integral.expression = term
integral.symbol = symbol
integral.enhancedsubstitution = es
local F = IntegralExpression.integrate(integral)
if F then
sum = sum + F
else
return nil
end
end
return sum
end
end
return nil
end
--- Attempts u-substitutions to evaluate the integral.
--- @param integral IntegralExpression
--- @return Expression|nil
function IntegralExpression.substitutionmethod(integral)
local expression = integral.expression
local symbol = integral.symbol
local es = integral.enhancedsubstitution
local P = IntegralExpression.trialsubstitutions(expression)
local F = nil
local i = 1
while not F and i <= #P do
local g = P[i]
if g ~= symbol and not g:freeof(symbol) then
local subsymbol = SymbolExpression("u")
if symbol == SymbolExpression("u") then
subsymbol = SymbolExpression("v")
end
local u = (expression / (DerivativeExpression(g, symbol))):autosimplify()
u = u:substitute({[g]=subsymbol}):autosimplify()
--factor u and cancel like non-constant terms
u = u:factor():autosimplify()
if u:freeof(symbol) then
integral.expression = u
integral.symbol = subsymbol
integral.enhancedsubstitution = es
F = IntegralExpression.integrate(integral)
if F then
if integral.recursive then
F = F:substitute({[subsymbol]=g})
end
return F
end
end
end
i = i + 1
end
return F
end
--- Attempts u-substitutions to evaluate the integral, including solving for the original variable and substituting the result into the expression.
--- @param integral IntegralExpression
--- @return Expression|nil
function IntegralExpression.enhancedsubstitutionmethod(integral)
local expression = integral.expression
local symbol = integral.symbol
local es = integral.enhancedsubstitution
local P = IntegralExpression.trialsubstitutions(expression)
local F = nil
local i = 1
while not F and i <= #P do
local g = P[i]
if g ~= symbol and not g:freeof(symbol) then
local subsymbol = SymbolExpression("u")
if symbol == SymbolExpression("u") then
subsymbol = SymbolExpression("v")
end
local u = (expression / (DerivativeExpression(g, symbol))):autosimplify()
u = u:substitute({[g]=subsymbol}):autosimplify()
--factor u and cancel like non-constant terms
u = u:factor():autosimplify()
if integral.enhancedsubstitution > Integer.zero() then
local f = Equation(subsymbol, g):solvefor(symbol)
if f.lhs == symbol then
u = u:substitute({[symbol]=f.rhs}):autosimplify()
integral.expression = u
integral.symbol = subsymbol
integral.enhancedsubstitution = integral.enhancedsubstitution - Integer.one()
F = IntegralExpression.integrate(integral)
if F then
if integral.recursive then
F = F:substitute({[subsymbol]=g})
end
return F
end
integral.enhancedsubstitution = integral.enhancedsubstitution + Integer.one()
end
end
end
i = i + 1
end
return F
end
--- Generates a list of possible u-substitutions to attempt
--- @param expression Expression
--- @return table<number, Expression>
function IntegralExpression.trialsubstitutions(expression)
local substitutions = {}
-- Recursive part - evaluates each term in a product.
if expression:type() == BinaryOperation and expression.operation == BinaryOperation.MUL then
substitutions[#substitutions+1] = expression
for _, term in ipairs(expression.expressions) do
substitutions = JoinArrays(substitutions, IntegralExpression.trialsubstitutions(term))
end
end
--Recursive part - evaluates each term in a sum.
if expression:type() == BinaryOperation and expression.operation == BinaryOperation.ADD then
substitutions[#substitutions+1] = expression
for _,term in ipairs(expression.expressions) do
substitutions = JoinArrays(substitutions, IntegralExpression.trialsubstitutions(term))
end
end
-- Function forms and arguments of function forms (includes a recursive part)
if expression:type() == TrigExpression or expression:type() == Logarithm then
substitutions[#substitutions+1] = expression
if not expression.expression:isatomic() then
substitutions[#substitutions+1] = expression.expression
end
substitutions = JoinArrays(substitutions, IntegralExpression.trialsubstitutions(expression.expression))
end
-- Bases and exponents of powers
if expression:type() == BinaryOperation and expression.operation == BinaryOperation.POW then
substitutions[#substitutions+1] = expression
-- Atomic expressions are technically valid substitutions, but they won't be useful
if not expression.expressions[1]:isatomic() then
--substitutions[#substitutions+1] = expression.expressions[1]
substitutions = JoinArrays(substitutions, IntegralExpression.trialsubstitutions(expression.expressions[1]))
end
if not expression.expressions[2]:isatomic() then
--substitutions[#substitutions+1] = expression.expressions[2]
substitutions = JoinArrays(substitutions, IntegralExpression.trialsubstitutions(expression.expressions[2]))
end
end
return substitutions
end
--- Uses Lazard, Rioboo, Rothstein, and Trager's method to integrate rational functions.
--- This is mostly to try to avoid factoring and finding the roots of the full denominator whenever possible.
--- @param integral IntegralExpression
--- @return Expression|nil
function IntegralExpression.rationalfunction(integral)
local expression = integral.expression
local symbol = integral.symbol
local es = integral.enhancedsubstitution
-- Type checking and conversion to polynomial type.
local f, g, fstat, gstat
if expression:type() == BinaryOperation and expression.operation == BinaryOperation.POW and expression.expressions[2] == Integer(-1) then
g, gstat = expression.expressions[1]:topolynomial()
if not gstat then
return nil
end
f = PolynomialRing({Integer.one()}, g.symbol)
else
if expression:type() ~= BinaryOperation or expression.operation ~= BinaryOperation.MUL or expression.expressions[3] then
return nil
end
if expression.expressions[2]:type() == BinaryOperation and expression.expressions[2].operation == BinaryOperation.POW and expression.expressions[2].expressions[2] == Integer(-1) then
if expression.expressions[1].topolynomial ~=nil and expression.expressions[2].expressions[1].topolynomial ~=nil then
f, fstat = expression.expressions[1]:topolynomial()
g, gstat = expression.expressions[2].expressions[1]:topolynomial()
end
elseif expression.expressions[1]:type() == BinaryOperation and expression.expressions[1].operation == BinaryOperation.POW and expression.expressions[1].expressions[2] == Integer(-1) then
if expression.expressions[2].topolynomial ~= nil and expression.expressions[1].expressions[1].topolynomial ~= nil then
f, fstat = expression.expressions[2]:topolynomial()
g, gstat = expression.expressions[1].expressions[1]:topolynomial()
end
else
return nil
end
if not fstat or not gstat or f.symbol ~= symbol.symbol or g.symbol ~= symbol.symbol then
return nil
end
end
-- Explicit handling of degree 1 or less over a binomial.
do
local disc = g.coefficients[1]*g.coefficients[1]-Integer(4)*g.coefficients[2]*g.coefficients[0]
if f.degree <= Integer.one() and g.degree == Integer(2) and disc < Integer.zero() then
return (f.coefficients[1] * LN(g.coefficients[0] + g.coefficients[1] * symbol + g.coefficients[2] * symbol ^ Integer(2))/(Integer(2) * g.coefficients[2]) + (Integer(2)*f.coefficients[0]*g.coefficients[2] - f.coefficients[1]*g.coefficients[1]) / (g.coefficients[2] * sqrt(Integer(4)*g.coefficients[0]*g.coefficients[2] - g.coefficients[1] ^ Integer(2))) * ARCTAN((Integer(2)*g.coefficients[2]*symbol+g.coefficients[1]) / sqrt(Integer(4)*g.coefficients[0]*g.coefficients[2]-g.coefficients[1] ^ Integer(2)))):autosimplify()
end
end
-- If the polynomials are not relatively prime, divides out the common factors.
local gcd = PolynomialRing.gcd(f, g)
if gcd ~= Integer.one() then
f, g = f // gcd, g // gcd
end
-- Seperates out the polynomial part and rational part and integrates the polynomial part.
local q, h = f:divremainder(g)
integral.expression = q
integral.symbol = symbol
integral.enhancedsubstitution = es
U = IntegralExpression.integrate(integral)
if h == Integer.zero() then
return U
end
-- Performs partial fraction decomposition into square-free denominators on the rational part.
local gg = g:squarefreefactorization()
local pfd = PolynomialRing.partialfractions(h, g, gg)
-- Hermite reduction.
local V = Integer.zero()
for _, term in ipairs(pfd.expressions) do
local i = #term.expressions
if i > 1 then
for j = 1, i-1 do
local n = term.expressions[j].expressions[1]
local d = term.expressions[j].expressions[2].expressions[1]
local p = term.expressions[j].expressions[2].expressions[2]
local _, s, t = PolynomialRing.extendedgcd(d, d:derivative())
s = s * n
t = t * n
V = V - t / ((p-Integer.one()) * BinaryOperation.POWEXP({d, p-Integer.one()}))
term.expressions[j+1].expressions[1] = term.expressions[j+1].expressions[1] + s + t:derivative() / (p-Integer.one())
end
end
end
--Lazard-Rioboo-Trager method.
local W = Integer.zero()
for _, term in ipairs(pfd.expressions) do
local a = term.expressions[#term.expressions].expressions[1]
local b = term.expressions[1].expressions[2].expressions[1]
local y = a - b:derivative() * PolynomialRing({Integer.zero(), Integer.one()}, "_")
local r = PolynomialRing.resultant(b, y)
local rr = r:squarefreefactorization()
local remainders = PolynomialRing.monicgcdremainders(b, y)
for pos, factor in ipairs(rr.expressions) do
if pos > 1 then
local re = factor.expressions[1]
local e = factor.expressions[2]
local roots = re:roots()
for _, root in ipairs(roots) do
local w
for _, remainder in ipairs(remainders) do
if remainder.degree == e then
w = remainder
break
end
end
W = W + root*LN(w:substitute({[SymbolExpression("_")] = root}))
end
end
end
end
return U + V + W
end
--- Attempts integration by parts for expressions with a polynomial factor in them. Other product expressions use Euler's formula.
--- @param integral IntegralExpression
--- @return Expression|nil
function IntegralExpression.partsmethod(integral)
local expression = integral.expression
local symbol = integral.symbol
local es = integral.enhancedsubstitution
if expression:type() ~= BinaryOperation or expression.operation ~= BinaryOperation.MUL then
return
end
local u
local vp = Integer.one()
--looking for ILATE
for _, exp in ipairs(expression:subexpressions()) do
if exp:type() == TrigExpression and (exp.name == "arctan" or exp.name == "arccos" or exp.name == "arcsin" or exp.name == "arccot" or exp.name == "arcsec" or exp.name == "arccsc") then
u = exp
else
vp = vp * exp
end
end
if not u or u:freeof(symbol) then
goto skipI
else
vp = vp:autosimplify()
end
--if vp:type() == Logarithm or vp.topolynomial or (vp:type() == TrigExpression and (vp.name == "cos" or vp.name == "sin")) or (vp.operation == BinaryOperation.POW and vp.expressions[1]:freeof(symbol)) then
if select(2,vp:topolynomial()) then
integral.expression = vp
integral.symbol = symbol
integral.enhancedsubstitution = es
local v = IntegralExpression.integrate(integral)
if not v then
goto skipI
end
local up = DerivativeExpression(u, symbol):autosimplify()
integral.expression = v*up
integral.symbol = symbol
integral.enhancedsubstitution = es
local vup = IntegralExpression.integrate(integral)
if not vup then
goto skipI
end
local result = u*v - vup
return result:autosimplify()
end
::skipI::
local u
local vp = Integer.one()
--looking for LATE
for _, exp in ipairs(expression:subexpressions()) do
if exp:type() == Logarithm then
u = exp
else
vp = vp * exp
end
end
if not u or u:freeof(symbol) then
goto skipL
else
vp = vp:autosimplify()
end
--if vp.topolynomial or (vp:type() == TrigExpression and (vp.name == "cos" or vp.name == "sin")) or (vp.operation == BinaryOperation.POW and vp.expressions[1]:freeof(symbol)) then
if select(2,vp:topolynomial()) then
integral.expression = vp
integral.symbol = symbol
integral.enhancedsubstitution = es
local v = IntegralExpression.integrate(integral)
if not v then
goto skipL
end
local up = DerivativeExpression(u, symbol):autosimplify()
integral.expression = v*up
integral.symbol = symbol
integral.enhancedsubstitution = es
local vup = IntegralExpression.integrate(integral)
if not vup then
goto skipL
end
local result = u*v - vup
return result:autosimplify()
end
::skipL::
local u
local vp = Integer.one()
--looking for ATE
for _, exp in ipairs(expression:subexpressions()) do
local _, bool = exp:topolynomial()
if bool then
u = exp
else
vp = vp * exp
end
end
if not u or u:freeof(symbol) then
return
else
vp = vp:autosimplify()
end
if (vp:type() == TrigExpression and (vp.name == "cos" or vp.name == "sin")) or (vp.operation == BinaryOperation.POW and vp.expressions[1]:freeof(symbol)) then
local results = {}
while u ~= Integer.zero() do
integral.expression = vp
integral.symbol = symbol
integral.enhancedsubstitution = es
local v = IntegralExpression.integrate(integral):unlock():autosimplify()
if not v then
return
end
local up = DerivativeExpression(u, symbol):autosimplify()
if not integral.recursive then
return (u*v - IntegralExpression(v*up, symbol):nonrecursive():lock(Expression.NIL, true)):autosimplify()
end
results[#results+1] = u*v
u = up
vp = v
end
local result = results[#results]
for i=#results-1,1,-1 do
result = results[i] - result
end
return result:autosimplify()
end
end
--- Attempts integration using Euler's formula and kind. Alternative for integration by parts for many expressions.
--- @param integral IntegralExpression
--- @return Expression|nil
function IntegralExpression.eulersformula(integral)
local expression = integral.expression
local symbol = integral.symbol
local es = integral.enhancedsubstitution
local new = expression:substitute({[COS(symbol)] = (E^(I*symbol) + E^(-I*symbol))/Integer(2),
[SIN(symbol)] = (E^(I*symbol) - E^(-I*symbol))/(Integer(2)*I)})
if new == expression then
return
end
integral.expression = new:autosimplify():expand()
integral.symbol = symbol
integral.enhancedsubstitution = es
local complexresult = IntegralExpression.integrate(integral)
if not complexresult then
return
end
-- TODO: Proper complex number conversion methods
local function converttorectangular(exp)
exp = exp:expand()
local results = {}
for index, sub in ipairs(exp:subexpressions()) do
results[index] = converttorectangular(sub)
end
local converted = exp:setsubexpressions(results)
if converted.operation == BinaryOperation.POW and converted.expressions[1] == E and converted.expressions[2].operation == BinaryOperation.MUL then
local ipart
local rest = Integer.one()
for _, factor in ipairs(converted.expressions[2]:subexpressions()) do
if factor == I then
ipart = true
else
rest = rest * factor
end
end
if ipart then
return (COS(rest) + I*SIN(rest)):autosimplify()
end
end
return converted
end
return converttorectangular(complexresult:autosimplify()):expand():autosimplify()
end
----------------------------
-- Instance functionality --
----------------------------
--- Creates a new integral operation with the given symbol and expression.
--- @param expression Expression
--- @param symbol SymbolExpression
--- @param lower Expression
--- @param upper Expression
function IntegralExpression:new(expression, symbol, lower, upper)
local o = {}
local __o = Copy(__ExpressionOperations)
if not symbol or not expression then
error("Send wrong number of parameters: integrals must have a variable to integrate with respect to and an expression to integrate.")
end
if lower and not upper then
error("Send wrong number of parameters: definite integrals must have an upper and a lower bound.")
end
o.symbol = symbol
o.expression = Copy(expression)
o.upper = Copy(upper)
o.lower = Copy(lower)
o.recursive = true
o.attempts = {}
o.results = {}
o.enhancedsubstitution = IntegralExpression.ENHANCEDSUBSTITUTIONRECURSIONLIMIT
__o.__index = IntegralExpression
__o.__tostring = function(a)
if a:isdefinite() then
return 'int(' .. tostring(a.expression) .. ", " .. tostring(a.symbol) .. ", ".. tostring(a.lower) .. ', ' .. tostring(a.upper) .. ')'
end
return 'int(' .. tostring(a.expression) .. ", " .. tostring(a.symbol) .. ')'
end
__o.__eq = function(a, b)
-- This shouldn't be needed, since __eq should only fire if both metamethods have the same function, but for some reason Lua always rungs this anyway
if not b:type() == IntegralExpression then
return false
end
return a.symbol == b.symbol and a.expression == b.expression and a.upper == b.upper and a.lower == b.lower
end
o = setmetatable(o, __o)
return o
end
--- Returns true if the integral is definite, i.e., has an upper and lower bound.
--- @return boolean
function IntegralExpression:isdefinite()
return self.upper ~= nil
end
--- Sets the integral to not autosimplify other integral expressions that are produced by the integration process.
--- THIS METHOD MUTATES THE OBJECT IT IS CALLED ON.
function IntegralExpression:nonrecursive()
self.recursive = false
return self
end
--- @return Expression
function IntegralExpression:autosimplify()
local arg = IntegralExpression(self.expression, self.symbol)
local integrated = IntegralExpression.integrate(arg)
-- Our expression could not be integrated.
if not integrated then
return self
end
if not self.recursive then
return integrated:autosimplify():unlock(true)
end
if self:isdefinite() then
return (integrated:substitute({[self.symbol]=self.upper}) - integrated:substitute({[self.symbol]=self.lower})):autosimplify()
end
return integrated:autosimplify()
end
--- @return table<number, Expression>
function IntegralExpression:subexpressions()
if self:isdefinite() then
return {self.expression, self.symbol, self.lower, self.upper}
end
return {self.expression, self.symbol}
end
--- @param subexpressions table<number, Expression>
--- @return IntegralExpression
function IntegralExpression:setsubexpressions(subexpressions)
local out = IntegralExpression(subexpressions[1], subexpressions[2], subexpressions[3], subexpressions[4])
return out;
end
-- function IntegralExpression:freeof(symbol)
-- if self:isdefinite() then
-- return self.expression:freeof(symbol) and self.upper:freeof(symbol) and self.lower:freeof(symbol)
-- end
-- return self.expression:freeof(symbol)
-- end
-- -- Substitutes each expression for a new one.
-- function IntegralExpression:substitute(map)
-- for expression, replacement in pairs(map) do
-- if self == expression then
-- return replacement
-- end
-- end
-- -- Typically, we only perform substitution on autosimplified expressions, so this won't get called. May give strange results, i.e.,
-- -- substituting and then evaluating the integral may not return the same thing as evaluating the integral and then substituting.
-- if self:isdefinite() then
-- return IntegralExpression(self.symbol, self.expression:substitute(map), self.upper:substitute(map), self.lower:substitute(map))
-- end
-- return IntegralExpression(self.symbol, self.expression:substitute(map))
-- end
--- @param other Expression
--- @return boolean
function IntegralExpression:order(other)
if other:type() ~= IntegralExpression then
return false
end
if self.symbol ~= other.symbol then
return self.symbol:order(other.symbol)
end
return self.expression:order(other.expression)
end
--- @return string
function IntegralExpression:tolatex()
if self:isdefinite() then
return '\\int_{' .. self.lower:tolatex() .. '}^{' .. self.upper:tolatex() .. '}{' .. self.expression:tolatex() .. '\\mathop{d' .. self.symbol:tolatex() .. '}}'
end
return '\\int{' .. self.expression:tolatex() .. '\\mathop{d' .. self.symbol:tolatex() .. '}}'
end
-----------------
-- Inheritance --
-----------------
__IntegralExpression.__index = CompoundExpression
__IntegralExpression.__call = IntegralExpression.new
IntegralExpression = setmetatable(IntegralExpression, __IntegralExpression)
----------------------
-- Static constants --
----------------------
INT = function(symbol, expression, lower, upper)
return IntegralExpression(symbol, expression, lower, upper)
end
----------------------
-- Static constants --
----------------------
-- Limit for the maximum number of full u-subs to attempts for any integral.
-- This should be low, since integrals are highly unlikely to need more than 1 or 2 u-subs, and gives exponentially worse performance the higher the number is.
IntegralExpression.ENHANCEDSUBSTITUTIONRECURSIONLIMIT = Integer(2)
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