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Singularities
=============

This module implements algorithms for finding singularities for a function
and identifying types of functions.

The differential calculus methods in this module include methods to identify
the following function types in the given ``Interval``:
- Increasing
- Strictly Increasing
- Decreasing
- Strictly Decreasing
- Monotonic

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tgt¡ t¡D ]*}|jjrXt‚|jjrH|||j||ƒ7 }qH|  t¡D ]}|||jd ||ƒ7 }q~|W S  ty¼   ttdƒƒ‚Y n0 dS )a¡  
    Find singularities of a given function.

    Parameters
    ==========

    expression : Expr
        The target function in which singularities need to be found.
    symbol : Symbol
        The symbol over the values of which the singularity in
        expression in being searched for.

    Returns
    =======

    Set
        A set of values for ``symbol`` for which ``expression`` has a
        singularity. An ``EmptySet`` is returned if ``expression`` has no
        singularities for any given value of ``Symbol``.

    Raises
    ======

    NotImplementedError
        Methods for determining the singularities of this function have
        not been developed.

    Notes
    =====

    This function does not find non-isolated singularities
    nor does it find branch points of the expression.

    Currently supported functions are:
        - univariate continuous (real or complex) functions

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Mathematical_singularity

    Examples
    ========

    >>> from sympy import singularities, Symbol, log
    >>> x = Symbol('x', real=True)
    >>> y = Symbol('y', real=False)
    >>> singularities(x**2 + x + 1, x)
    EmptySet
    >>> singularities(1/(x + 1), x)
    {-1}
    >>> singularities(1/(y**2 + 1), y)
    {-I, I}
    >>> singularities(1/(y**3 + 1), y)
    {-1, 1/2 - sqrt(3)*I/2, 1/2 + sqrt(3)*I/2}
    >>> singularities(log(x), x)
    {0}

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    Helper function for functions checking function monotonicity.

    Parameters
    ==========

    expression : Expr
        The target function which is being checked
    predicate : function
        The property being tested for. The function takes in an integer
        and returns a boolean. The integer input is the derivative and
        the boolean result should be true if the property is being held,
        and false otherwise.
    interval : Set, optional
        The range of values in which we are testing, defaults to all reals.
    symbol : Symbol, optional
        The symbol present in expression which gets varied over the given range.

    It returns a boolean indicating whether the interval in which
    the function's derivative satisfies given predicate is a superset
    of the given interval.

    Returns
    =======

    Boolean
        True if ``predicate`` is true for all the derivatives when ``symbol``
        is varied in ``range``, False otherwise.

    r   r   Né   zKThe function has not yet been implemented for all multivariate expressions.Úx)r   r   r   Úfree_symbolsÚlenr   Úpopr   Údiffr   r   Z	is_subset)	r   Ú	predicateÚintervalr   r   ÚfreeÚvariableZ
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    Return whether the function is increasing in the given interval.

    Parameters
    ==========

    expression : Expr
        The target function which is being checked.
    interval : Set, optional
        The range of values in which we are testing (defaults to set of
        all real numbers).
    symbol : Symbol, optional
        The symbol present in expression which gets varied over the given range.

    Returns
    =======

    Boolean
        True if ``expression`` is increasing (either strictly increasing or
        constant) in the given ``interval``, False otherwise.

    Examples
    ========

    >>> from sympy import is_increasing
    >>> from sympy.abc import x, y
    >>> 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
    >>> is_increasing(x**2 + y, Interval(1, 2), x)
    True

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    Return whether the function is strictly increasing in the given interval.

    Parameters
    ==========

    expression : Expr
        The target function which is being checked.
    interval : Set, optional
        The range of values in which we are testing (defaults to set of
        all real numbers).
    symbol : Symbol, optional
        The symbol present in expression which gets varied over the given range.

    Returns
    =======

    Boolean
        True if ``expression`` is strictly increasing in the given ``interval``,
        False otherwise.

    Examples
    ========

    >>> from sympy import is_strictly_increasing
    >>> from sympy.abc import x, y
    >>> 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
    >>> is_strictly_increasing(-x**2 + y, Interval(-oo, 0), x)
    False

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    Return whether the function is decreasing in the given interval.

    Parameters
    ==========

    expression : Expr
        The target function which is being checked.
    interval : Set, optional
        The range of values in which we are testing (defaults to set of
        all real numbers).
    symbol : Symbol, optional
        The symbol present in expression which gets varied over the given range.

    Returns
    =======

    Boolean
        True if ``expression`` is decreasing (either strictly decreasing or
        constant) in the given ``interval``, False otherwise.

    Examples
    ========

    >>> from sympy import is_decreasing
    >>> from sympy.abc import x, y
    >>> from sympy import S, Interval, oo
    >>> is_decreasing(1/(x**2 - 3*x), Interval.open(S(3)/2, 3))
    True
    >>> 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(1/(x**2 - 3*x), Interval.Ropen(-oo, 1.5))
    False
    >>> is_decreasing(-x**2, Interval(-oo, 0))
    False
    >>> is_decreasing(-x**2 + y, Interval(-oo, 0), x)
    False

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    Return whether the function is strictly decreasing in the given interval.

    Parameters
    ==========

    expression : Expr
        The target function which is being checked.
    interval : Set, optional
        The range of values in which we are testing (defaults to set of
        all real numbers).
    symbol : Symbol, optional
        The symbol present in expression which gets varied over the given range.

    Returns
    =======

    Boolean
        True if ``expression`` is strictly decreasing in the given ``interval``,
        False otherwise.

    Examples
    ========

    >>> from sympy import is_strictly_decreasing
    >>> from sympy.abc import x, y
    >>> from sympy import S, Interval, oo
    >>> is_strictly_decreasing(1/(x**2 - 3*x), Interval.Lopen(3, oo))
    True
    >>> is_strictly_decreasing(1/(x**2 - 3*x), Interval.Ropen(-oo, S(3)/2))
    False
    >>> is_strictly_decreasing(1/(x**2 - 3*x), Interval.Ropen(-oo, 1.5))
    False
    >>> is_strictly_decreasing(-x**2, Interval(-oo, 0))
    False
    >>> is_strictly_decreasing(-x**2 + y, Interval(-oo, 0), x)
    False

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ju S )a³  
    Return whether the function is monotonic in the given interval.

    Parameters
    ==========

    expression : Expr
        The target function which is being checked.
    interval : Set, optional
        The range of values in which we are testing (defaults to set of
        all real numbers).
    symbol : Symbol, optional
        The symbol present in expression which gets varied over the given range.

    Returns
    =======

    Boolean
        True if ``expression`` is monotonic in the given ``interval``,
        False otherwise.

    Raises
    ======

    NotImplementedError
        Monotonicity check has not been implemented for the queried function.

    Examples
    ========

    >>> from sympy import is_monotonic
    >>> from sympy.abc import x, y
    >>> from sympy import S, Interval, oo
    >>> is_monotonic(1/(x**2 - 3*x), Interval.open(S(3)/2, 3))
    True
    >>> 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
    >>> is_monotonic(x**2 + y + 1, Interval(1, 2), x)
    True

    r   r   Nr   zKis_monotonic has not yet been implemented for all multivariate expressions.r   )r   r   r   r    r!   r   r"   r   r#   Úintersectionr   r   )r   r%   r   r   r&   r'   Zturning_pointsr   r   r   Úis_monotonicQ  s    0ÿr4   )N)Ú__doc__Zsympy.core.powerr   Zsympy.core.singletonr   Zsympy.core.symbolr   Zsympy.core.sympifyr   Z&sympy.functions.elementary.exponentialr   Z(sympy.functions.elementary.trigonometricr   r   r	   r
   r   Zsympy.utilities.miscr   r   r   r(   r/   r0   r1   r2   r4   r   r   r   r   Ú<module>   s   
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