Source code for sympy.calculus.singularities

from sympy.solvers.solveset import solveset
from sympy.simplify import simplify
from sympy import S


[docs]def singularities(expr, sym): """ Finds singularities for a function. Currently supported functions are: - univariate real rational functions Examples ======== >>> from sympy.calculus.singularities import singularities >>> from sympy import Symbol >>> x = Symbol('x', real=True) >>> singularities(x**2 + x + 1, x) () >>> singularities(1/(x + 1), x) (-1,) References ========== .. [1] http://en.wikipedia.org/wiki/Mathematical_singularity """ if not expr.is_rational_function(sym): raise NotImplementedError("Algorithms finding singularities for" " non rational functions are not yet" " implemented") else: return tuple(sorted(solveset(simplify(1/expr), sym)))
########################################################################### ###################### DIFFERENTIAL CALCULUS METHODS ###################### ###########################################################################
[docs]def is_increasing(f, interval=S.Reals): """ Returns if a function is increasing or not, in the given ``Interval``. Examples ======== >>> from sympy import is_increasing >>> from sympy.abc import x >>> from sympy import S, Interval, oo >>> is_increasing(x**3 - 3*x**2 + 4*x, S.Reals) True >>> is_increasing(-x**2, Interval(-oo, 0)) True >>> is_increasing(-x**2, Interval(0, oo)) False >>> is_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval(-2, 3)) False """ if len(f.free_symbols) > 1: raise NotImplementedError('is_increasing has not yet been implemented ' 'for multivariate expressions') symbol = f.free_symbols.pop() df = f.diff(symbol) df_nonneg_interval = solveset(df >= 0, symbol, domain=S.Reals) return interval.is_subset(df_nonneg_interval)
[docs]def is_strictly_increasing(f, interval=S.Reals): """ Returns if a function is strictly increasing or not, in the given ``Interval``. Examples ======== >>> from sympy import is_strictly_increasing >>> from sympy.abc import x >>> from sympy import Interval, oo >>> is_strictly_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval.Ropen(-oo, -2)) True >>> is_strictly_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval.Lopen(3, oo)) True >>> is_strictly_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval.open(-2, 3)) False >>> is_strictly_increasing(-x**2, Interval(0, oo)) False """ if len(f.free_symbols) > 1: raise NotImplementedError('is_strictly_increasing has not yet been ' 'implemented for multivariate expressions') symbol = f.free_symbols.pop() df = f.diff(symbol) df_pos_interval = solveset(df > 0, symbol, domain=S.Reals) return interval.is_subset(df_pos_interval)
[docs]def is_decreasing(f, interval=S.Reals): """ Returns if a function is decreasing or not, in the given ``Interval``. Examples ======== >>> from sympy import is_decreasing >>> from sympy.abc import x >>> from sympy import S, Interval, oo >>> is_decreasing(1/(x**2 - 3*x), Interval.open(1.5, 3)) True >>> is_decreasing(1/(x**2 - 3*x), Interval.Lopen(3, oo)) True >>> is_decreasing(1/(x**2 - 3*x), Interval.Ropen(-oo, S(3)/2)) False >>> is_decreasing(-x**2, Interval(-oo, 0)) False """ if len(f.free_symbols) > 1: raise NotImplementedError('is_decreasing has not yet been implemented ' 'for multivariate expressions') symbol = f.free_symbols.pop() df = f.diff(symbol) df_nonpos_interval = solveset(df <= 0, symbol, domain=S.Reals) return interval.is_subset(df_nonpos_interval)
[docs]def is_strictly_decreasing(f, interval=S.Reals): """ Returns if a function is decreasing or not, in the given ``Interval``. Examples ======== >>> from sympy import is_decreasing >>> from sympy.abc import x >>> from sympy import S, Interval, oo >>> is_decreasing(1/(x**2 - 3*x), Interval.open(1.5, 3)) True >>> is_decreasing(1/(x**2 - 3*x), Interval.Lopen(3, oo)) True >>> is_decreasing(1/(x**2 - 3*x), Interval.Ropen(-oo, S(3)/2)) False >>> is_decreasing(-x**2, Interval(-oo, 0)) False """ if len(f.free_symbols) > 1: raise NotImplementedError('is_strictly_decreasing has not yet been ' 'implemented for multivariate expressions') symbol = f.free_symbols.pop() df = f.diff(symbol) df_neg_interval = solveset(df < 0, symbol, domain=S.Reals) return interval.is_subset(df_neg_interval)
[docs]def is_monotonic(f, interval=S.Reals): """ Returns if a function is monotonic or not, in the given ``Interval``. Examples ======== >>> from sympy import is_monotonic >>> from sympy.abc import x >>> from sympy import S, Interval, oo >>> is_monotonic(1/(x**2 - 3*x), Interval.open(1.5, 3)) True >>> is_monotonic(1/(x**2 - 3*x), Interval.Lopen(3, oo)) True >>> is_monotonic(x**3 - 3*x**2 + 4*x, S.Reals) True >>> is_monotonic(-x**2, S.Reals) False """ from sympy.core.logic import fuzzy_or if len(f.free_symbols) > 1: raise NotImplementedError('is_monotonic has not yet been ' 'implemented for multivariate expressions') inc = is_increasing(f, interval) dec = is_decreasing(f, interval) return fuzzy_or([inc, dec])