def continuous_domain(f, symbol, domain): """ Returns the intervals in the given domain for which the function is continuous. This method is limited by the ability to determine the various singularities and discontinuities of the given function. Examples ======== >>> from sympy import Symbol, S, tan, log, pi, sqrt >>> from sympy.sets import Interval >>> from sympy.calculus.util import continuous_domain >>> x = Symbol('x') >>> continuous_domain(1/x, x, S.Reals) (-oo, 0) U (0, oo) >>> continuous_domain(tan(x), x, Interval(0, pi)) [0, pi/2) U (pi/2, pi] >>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5)) [2, 5] >>> continuous_domain(log(2*x - 1), x, S.Reals) (1/2, oo) """ from sympy.solvers.inequalities import solve_univariate_inequality from sympy.solvers.solveset import solveset, _has_rational_power if domain.is_subset(S.Reals): constrained_interval = domain for atom in f.atoms(Pow): predicate, denom = _has_rational_power(atom, symbol) constraint = S.EmptySet if predicate and denom == 2: constraint = solve_univariate_inequality( atom.base >= 0, symbol).as_set() constrained_interval = Intersection(constraint, constrained_interval) for atom in f.atoms(log): constraint = solve_univariate_inequality(atom.args[0] > 0, symbol).as_set() constrained_interval = Intersection(constraint, constrained_interval) domain = constrained_interval try: sings = S.EmptySet for atom in f.atoms(Pow): predicate, denom = _has_rational_power(atom, symbol) if predicate and denom == 2: sings = solveset(1 / f, symbol, domain) break else: sings = Intersection(solveset(1 / f, symbol), domain) except: raise NotImplementedError( "Methods for determining the continuous domains" " of this function has not been developed.") return domain - sings
def continuous_domain(f, symbol, domain): """ Returns the intervals in the given domain for which the function is continuous. This method is limited by the ability to determine the various singularities and discontinuities of the given function. Examples ======== >>> from sympy import Symbol, S, tan, log, pi, sqrt >>> from sympy.sets import Interval >>> from sympy.calculus.util import continuous_domain >>> x = Symbol('x') >>> continuous_domain(1/x, x, S.Reals) (-oo, 0) U (0, oo) >>> continuous_domain(tan(x), x, Interval(0, pi)) [0, pi/2) U (pi/2, pi] >>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5)) [2, 5] >>> continuous_domain(log(2*x - 1), x, S.Reals) (1/2, oo) """ from sympy.solvers.inequalities import solve_univariate_inequality from sympy.solvers.solveset import solveset, _has_rational_power if domain.is_subset(S.Reals): constrained_interval = domain for atom in f.atoms(Pow): predicate, denom = _has_rational_power(atom, symbol) constraint = S.EmptySet if predicate and denom == 2: constraint = solve_univariate_inequality(atom.base >= 0, symbol).as_set() constrained_interval = Intersection(constraint, constrained_interval) for atom in f.atoms(log): constraint = solve_univariate_inequality(atom.args[0] > 0, symbol).as_set() constrained_interval = Intersection(constraint, constrained_interval) domain = constrained_interval try: sings = S.EmptySet for atom in f.atoms(Pow): predicate, denom = _has_rational_power(atom, symbol) if predicate and denom == 2: sings = solveset(1/f, symbol, domain) break else: sings = Intersection(solveset(1/f, symbol), domain) except: raise NotImplementedError("Methods for determining the continuous domains" " of this function has not been developed.") return domain - sings
def test__has_rational_power(): from sympy.solvers.solveset import _has_rational_power assert _has_rational_power(sqrt(2), x)[0] is False assert _has_rational_power(x*sqrt(2), x)[0] is False assert _has_rational_power(x**2*sqrt(x), x) == (True, 2) assert _has_rational_power(sqrt(2)*x**(S(1)/3), x) == (True, 3) assert _has_rational_power(sqrt(x)*x**(S(1)/3), x) == (True, 6)
def test__has_rational_power(): from sympy.solvers.solveset import _has_rational_power assert _has_rational_power(sqrt(2), x)[0] is False assert _has_rational_power(x * sqrt(2), x)[0] is False assert _has_rational_power(x**2 * sqrt(x), x) == (True, 2) assert _has_rational_power(sqrt(2) * x**(S(1) / 3), x) == (True, 3) assert _has_rational_power(sqrt(x) * x**(S(1) / 3), x) == (True, 6)
def test_improve_coverage(): from sympy.solvers.solveset import _has_rational_power x = Symbol('x', real=True) y = exp(x+1/x**2) raises(NotImplementedError, lambda: solveset(y**2+y, x)) assert _has_rational_power(sin(x)*exp(x) + 1, x) == (False, S.One) assert _has_rational_power((sin(x)**2)*(exp(x) + 1)**3, x) == (False, S.One)
def test_improve_coverage(): from sympy.solvers.solveset import _has_rational_power x = Symbol('x') y = exp(x+1/x**2) solution = solveset(y**2+y, x, S.Reals) unsolved_object = ConditionSet(x, Eq((exp((x**3 + 1)/x**2) + 1)*exp((x**3 + 1)/x**2), 0), S.Reals) assert solution == unsolved_object assert _has_rational_power(sin(x)*exp(x) + 1, x) == (False, S.One) assert _has_rational_power((sin(x)**2)*(exp(x) + 1)**3, x) == (False, S.One)
def continuous_domain(f, symbol, domain): """ Returns the intervals in the given domain for which the function is continuous. This method is limited by the ability to determine the various singularities and discontinuities of the given function. Parameters ========== f : Expr The concerned function. symbol : Symbol The variable for which the intervals are to be determined. domain : Interval The domain over which the continuity of the symbol has to be checked. Examples ======== >>> from sympy import Symbol, S, tan, log, pi, sqrt >>> from sympy.sets import Interval >>> from sympy.calculus.util import continuous_domain >>> x = Symbol('x') >>> continuous_domain(1/x, x, S.Reals) Union(Interval.open(-oo, 0), Interval.open(0, oo)) >>> continuous_domain(tan(x), x, Interval(0, pi)) Union(Interval.Ropen(0, pi/2), Interval.Lopen(pi/2, pi)) >>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5)) Interval(2, 5) >>> continuous_domain(log(2*x - 1), x, S.Reals) Interval.open(1/2, oo) Returns ======= Interval Union of all intervals where the function is continuous. Raises ====== NotImplementedError If the method to determine continuity of such a function has not yet been developed. """ from sympy.solvers.inequalities import solve_univariate_inequality from sympy.solvers.solveset import solveset, _has_rational_power if domain.is_subset(S.Reals): constrained_interval = domain for atom in f.atoms(Pow): predicate, denomin = _has_rational_power(atom, symbol) constraint = S.EmptySet if predicate and denomin == 2: constraint = solve_univariate_inequality(atom.base >= 0, symbol).as_set() constrained_interval = Intersection(constraint, constrained_interval) for atom in f.atoms(log): constraint = solve_univariate_inequality(atom.args[0] > 0, symbol).as_set() constrained_interval = Intersection(constraint, constrained_interval) domain = constrained_interval try: sings = S.EmptySet if f.has(Abs): sings = solveset(1/f, symbol, domain) + \ solveset(denom(together(f)), symbol, domain) else: for atom in f.atoms(Pow): predicate, denomin = _has_rational_power(atom, symbol) if predicate and denomin == 2: sings = solveset(1/f, symbol, domain) +\ solveset(denom(together(f)), symbol, domain) break else: sings = Intersection(solveset(1/f, symbol), domain) + \ solveset(denom(together(f)), symbol, domain) except NotImplementedError: import sys raise (NotImplementedError("Methods for determining the continuous domains" " of this function have not been developed."), None, sys.exc_info()[2]) return domain - sings
def continuous_domain(f, symbol, domain): """ Returns the intervals in the given domain for which the function is continuous. This method is limited by the ability to determine the various singularities and discontinuities of the given function. Parameters ========== f : Expr The concerned function. symbol : Symbol The variable for which the intervals are to be determined. domain : Interval The domain over which the continuity of the symbol has to be checked. Examples ======== >>> from sympy import Symbol, S, tan, log, pi, sqrt >>> from sympy.sets import Interval >>> from sympy.calculus.util import continuous_domain >>> x = Symbol('x') >>> continuous_domain(1/x, x, S.Reals) Union(Interval.open(-oo, 0), Interval.open(0, oo)) >>> continuous_domain(tan(x), x, Interval(0, pi)) Union(Interval.Ropen(0, pi/2), Interval.Lopen(pi/2, pi)) >>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5)) Interval(2, 5) >>> continuous_domain(log(2*x - 1), x, S.Reals) Interval.open(1/2, oo) Returns ======= Interval Union of all intervals where the function is continuous. Raises ====== NotImplementedError If the method to determine continuity of such a function has not yet been developed. """ from sympy.solvers.inequalities import solve_univariate_inequality from sympy.solvers.solveset import solveset, _has_rational_power if domain.is_subset(S.Reals): constrained_interval = domain for atom in f.atoms(Pow): predicate, denomin = _has_rational_power(atom, symbol) constraint = S.EmptySet if predicate and denomin == 2: constraint = solve_univariate_inequality( atom.base >= 0, symbol).as_set() constrained_interval = Intersection(constraint, constrained_interval) for atom in f.atoms(log): constraint = solve_univariate_inequality(atom.args[0] > 0, symbol).as_set() constrained_interval = Intersection(constraint, constrained_interval) domain = constrained_interval try: sings = S.EmptySet if f.has(Abs): sings = solveset(1/f, symbol, domain) + \ solveset(denom(together(f)), symbol, domain) else: for atom in f.atoms(Pow): predicate, denomin = _has_rational_power(atom, symbol) if predicate and denomin == 2: sings = solveset(1/f, symbol, domain) +\ solveset(denom(together(f)), symbol, domain) break else: sings = Intersection(solveset(1/f, symbol), domain) + \ solveset(denom(together(f)), symbol, domain) except NotImplementedError: import sys raise (NotImplementedError( "Methods for determining the continuous domains" " of this function have not been developed."), None, sys.exc_info()[2]) return domain - sings