log.py

"""
Logarithmic functions

AUTHORS:

- Yoora Yi Tenen (2012-11-16): Add documentation for :meth:`log()` (:trac:`12113`)

"""
from sage.symbolic.function import GinacFunction, BuiltinFunction, is_inexact
from sage.symbolic.constants import e as const_e

from sage.libs.mpmath import utils as mpmath_utils
from sage.structure.coerce import parent as sage_structure_coerce_parent
from sage.symbolic.expression import Expression
from sage.rings.real_double import RDF
from sage.rings.complex_double import CDF
from sage.rings.integer import Integer

class Function_exp(GinacFunction):
    def __init__(self):
        r"""
        The exponential function, `\exp(x) = e^x`.

        EXAMPLES::

            sage: exp(-1)
            e^(-1)
            sage: exp(2)
            e^2
            sage: exp(2).n(100)
            7.3890560989306502272304274606
            sage: exp(x^2 + log(x))
            e^(x^2 + log(x))
            sage: exp(x^2 + log(x)).simplify()
            x*e^(x^2)
            sage: exp(2.5)
            12.1824939607035
            sage: exp(float(2.5))
            12.182493960703473
            sage: exp(RDF('2.5'))
            12.1824939607

        To prevent automatic evaluation, use the ``hold`` parameter::

            sage: exp(I*pi,hold=True)
            e^(I*pi)
            sage: exp(0,hold=True)
            e^0

        To then evaluate again, we currently must use Maxima via
        :meth:`sage.symbolic.expression.Expression.simplify`::

            sage: exp(0,hold=True).simplify()
            1

        ::

            sage: exp(pi*I/2)
            I
            sage: exp(pi*I)
            -1
            sage: exp(8*pi*I)
            1
            sage: exp(7*pi*I/2)
            -I

        TEST::

            sage: latex(exp(x))
            e^{x}
            sage: latex(exp(sqrt(x)))
            e^{\sqrt{x}}
            sage: latex(exp)
            \exp
            sage: latex(exp(sqrt(x))^x)
            \left(e^{\sqrt{x}}\right)^{x}
            sage: latex(exp(sqrt(x)^x))
            e^{\left(\sqrt{x}^{x}\right)}

        Test conjugates::

            sage: conjugate(exp(x))
            e^conjugate(x)

        Test simplifications when taking powers of exp, #7264::

            sage: var('a,b,c,II')
            (a, b, c, II)
            sage: model_exp = exp(II)**a*(b)
            sage: sol1_l={b: 5.0, a: 1.1}
            sage: model_exp.subs(sol1_l)
            5.00000000000000*(e^II)^1.10000000000000

        ::

            sage: exp(3)^II*exp(x)
            (e^3)^II*e^x
            sage: exp(x)*exp(x)
            e^(2*x)
            sage: exp(x)*exp(a)
            e^(a + x)
            sage: exp(x)*exp(a)^2
            e^(2*a + x)

        Another instance of the same problem, #7394::

            sage: 2*sqrt(e)
            2*sqrt(e)
        """
        GinacFunction.__init__(self, "exp", latex_name=r"\exp",
                                   conversions=dict(maxima='exp'))

    def __call__(self, x, coerce=True, hold=False, prec=None,
            dont_call_method_on_arg=False):
        """
        Note that the ``prec`` argument is deprecated. The precision for
        the result is deduced from the precision of the input. Convert
        the input to a higher precision explicitly if a result with higher
        precision is desired.::

            sage: t = exp(RealField(100)(2)); t
            7.3890560989306502272304274606
            sage: t.prec()
            100

        TESTS::

            sage: exp(2,prec=100)
            doctest:...: DeprecationWarning: The prec keyword argument is deprecated. Explicitly set the precision of the input, for example exp(RealField(300)(1)), or use the prec argument to .n() for exact inputs, e.g., exp(1).n(300), instead.
            See http://trac.sagemath.org/7490 for details.
            7.3890560989306502272304274606
        """
        if prec is not None:
            from sage.misc.superseded import deprecation
            deprecation(7490, "The prec keyword argument is deprecated. Explicitly set the precision of the input, for example exp(RealField(300)(1)), or use the prec argument to .n() for exact inputs, e.g., exp(1).n(300), instead.")
            x = GinacFunction.__call__(self, x, coerce=coerce, hold=hold,
                    dont_call_method_on_arg=dont_call_method_on_arg)
            return x.n(prec)
        return GinacFunction.__call__(self, x, coerce=coerce, hold=hold,
                dont_call_method_on_arg=dont_call_method_on_arg)

exp = Function_exp()

class Function_log(GinacFunction):
    def __init__(self):
        r"""
        The natural logarithm of x.  See `log?` for
        more information about its behavior.

        EXAMPLES::

            sage: ln(e^2)
            2
            sage: ln(2)
            log(2)
            sage: ln(10)
            log(10)

        ::

            sage: ln(RDF(10))
            2.30258509299
            sage: ln(2.718)
            0.999896315728952
            sage: ln(2.0)
            0.693147180559945
            sage: ln(float(-1))
            3.141592653589793j
            sage: ln(complex(-1))
            3.141592653589793j

        We do not currently support a ``hold`` parameter in functional
        notation::

            sage: log(SR(-1),hold=True)
            Traceback (most recent call last):
            ...
            TypeError: log() got an unexpected keyword argument 'hold'

        This is possible with method notation::

            sage: I.log(hold=True)
            log(I)
            sage: I.log(hold=True).simplify()
            1/2*I*pi

        TESTS::

            sage: latex(x.log())
            \log\left(x\right)
            sage: latex(log(1/4))
            \log\left(\frac{1}{4}\right)
            sage: loads(dumps(ln(x)+1))
            log(x) + 1

        ``conjugate(log(x))==log(conjugate(x))`` unless on the branch cut which
        runs along the negative real axis.::

            sage: conjugate(log(x))
            conjugate(log(x))
            sage: var('y', domain='positive')
            y
            sage: conjugate(log(y))
            log(y)
            sage: conjugate(log(y+I))
            conjugate(log(y + I))
            sage: conjugate(log(-1))
            -I*pi
            sage: log(conjugate(-1))
            I*pi

        Check if float arguments are handled properly.::

            sage: from sage.functions.log import function_log as log
            sage: log(float(5))
            1.6094379124341003
            sage: log(float(0))
            -inf
            sage: log(float(-1))
            3.141592653589793j
            sage: log(x).subs(x=float(-1))
            3.141592653589793j
        """
        GinacFunction.__init__(self, 'log', latex_name=r'\log',
                                   conversions=dict(maxima='log'))

ln = function_log = Function_log()

def log(x, base=None):
    """
    Return the logarithm of x to the given base.

    Calls the ``log`` method of the object x when computing
    the logarithm, thus allowing use of logarithm on any object
    containing a ``log`` method. In other words, log works
    on more than just real numbers.

    EXAMPLES::

        sage: log(e^2)
        2

    To change the base of the logarithm, add a second parameter::

        sage: log(1000,10)
        3

    You can use :class:`RDF<sage.rings.real_double.RealDoubleField_class>`,
    :class:`~sage.rings.real_mpfr.RealField` or ``n`` to get a numerical real
    approximation::

        sage: log(1024, 2)
        10
        sage: RDF(log(1024, 2))
        10.0
        sage: log(10, 4)
        log(10)/log(4)
        sage: RDF(log(10, 4))
        1.66096404744
        sage: log(10, 2)
        log(10)/log(2)
        sage: n(log(10, 2))
        3.32192809488736
        sage: log(10, e)
        log(10)
        sage: n(log(10, e))
        2.30258509299405

    The log function works for negative numbers, complex
    numbers, and symbolic numbers too, picking the branch
    with angle between `-pi` and `pi`::

        sage: log(-1+0*I)
        I*pi
        sage: log(CC(-1))
        3.14159265358979*I
        sage: log(-1.0)
        3.14159265358979*I

    For input zero, the following behavior occurs::

        sage: log(0)
        -Infinity
        sage: log(CC(0))
        -infinity
        sage: log(0.0)
        -infinity

    The log function also works in finite fields as long as the argument lies
    in the multiplicative group generated by the base::

        sage: F = GF(13); g = F.multiplicative_generator(); g
        2
        sage: a = F(8)
        sage: log(a,g); g^log(a,g)
        3
        8
        sage: log(a,3)
        Traceback (most recent call last):
        ...
        ValueError: No discrete log of 8 found to base 3
        sage: log(F(9), 3)
        2

    The log function also works for p-adics (see documentation for
    p-adics for more information)::

        sage: R = Zp(5); R
        5-adic Ring with capped relative precision 20
        sage: a = R(16); a
        1 + 3*5 + O(5^20)
        sage: log(a)
        3*5 + 3*5^2 + 3*5^4 + 3*5^5 + 3*5^6 + 4*5^7 + 2*5^8 + 5^9 + 5^11 + 2*5^12 + 5^13 + 3*5^15 + 2*5^16 + 4*5^17 + 3*5^18 + 3*5^19 + O(5^20)
    """
    if base is None:
        try:
            return x.log()
        except AttributeError:
            return ln(x)
    else:
        try:
            return x.log(base)
        except (AttributeError, TypeError):
            return log(x) / log(base)


class Function_polylog(GinacFunction):
    def __init__(self):
        r"""
        The polylog function
        `\text{Li}_n(z) = \sum_{k=1}^{\infty} z^k / k^n`.

        INPUT:

        -  ``n`` - object
        -  ``z`` - object

        EXAMPLES::

            sage: polylog(1, x)
            -log(-x + 1)
            sage: polylog(2,1)
            1/6*pi^2
            sage: polylog(2,x^2+1)
            polylog(2, x^2 + 1)
            sage: polylog(4,0.5)
            polylog(4, 0.500000000000000)

            sage: f = polylog(4, 1); f
            1/90*pi^4
            sage: f.n()
            1.08232323371114

            sage: polylog(4, 2).n()
            2.42786280675470 - 0.174371300025453*I
            sage: complex(polylog(4,2))
            (2.4278628067547032-0.17437130002545306j)
            sage: float(polylog(4,0.5))
            0.5174790616738993

            sage: z = var('z')
            sage: polylog(2,z).series(z==0, 5)
            1*z + 1/4*z^2 + 1/9*z^3 + 1/16*z^4 + Order(z^5)

            sage: loads(dumps(polylog))
            polylog

            sage: latex(polylog(5, x))
            {\rm Li}_{5}(x)

        TESTS:

        Check if #8459 is fixed::

            sage: t = maxima(polylog(5,x)).sage(); t
            polylog(5, x)
            sage: t.operator() == polylog
            True
            sage: t.subs(x=.5).n()
            0.508400579242269
        """
        GinacFunction.__init__(self, "polylog", nargs=2)

    def _maxima_init_evaled_(self, *args):
        """
        EXAMPLES:

        These are indirect doctests for this function.::

            sage: polylog(2, x)._maxima_()
            li[2](x)
            sage: polylog(4, x)._maxima_()
            polylog(4,x)
        """
        n, x = args
        if int(n) in [1,2,3]:
            return 'li[%s](%s)'%(n, x)
        else:
            return 'polylog(%s, %s)'%(n, x)


polylog = Function_polylog()

class Function_dilog(GinacFunction):
    def __init__(self):
        r"""
        The dilogarithm function
        `\text{Li}_2(z) = \sum_{k=1}^{\infty} z^k / k^2`.

        This is simply an alias for polylog(2, z).

        EXAMPLES::

            sage: dilog(1)
            1/6*pi^2
            sage: dilog(1/2)
            1/12*pi^2 - 1/2*log(2)^2
            sage: dilog(x^2+1)
            dilog(x^2 + 1)
            sage: dilog(-1)
            -1/12*pi^2
            sage: dilog(-1.1)
            -0.890838090262283
            sage: float(dilog(1))
            1.6449340668482262
            sage: var('z')
            z
            sage: dilog(z).diff(z, 2)
            log(-z + 1)/z^2 - 1/((z - 1)*z)
            sage: dilog(z).series(z==1/2, 3)
            (1/12*pi^2 - 1/2*log(2)^2) + (-2*log(1/2))*(z - 1/2) + (2*log(1/2) + 2)*(z - 1/2)^2 + Order(1/8*(2*z - 1)^3)

            sage: latex(dilog(z))
            {\rm Li}_2\left(z\right)

        TESTS:

        ``conjugate(dilog(x))==dilog(conjugate(x))`` unless on the branch cuts
        which run along the positive real axis beginning at 1.::

            sage: conjugate(dilog(x))
            conjugate(dilog(x))
            sage: var('y',domain='positive')
            y
            sage: conjugate(dilog(y))
            conjugate(dilog(y))
            sage: conjugate(dilog(1/19))
            dilog(1/19)
            sage: conjugate(dilog(1/2*I))
            dilog(-1/2*I)
            sage: dilog(conjugate(1/2*I))
            dilog(-1/2*I)
            sage: conjugate(dilog(2))
            conjugate(dilog(2))
        """
        GinacFunction.__init__(self, 'dilog',
                conversions=dict(maxima='li[2]'))

dilog = Function_dilog()


class Function_lambert_w(BuiltinFunction):
    r"""
    The integral branches of the Lambert W function `W_n(z)`.

    This function satisfies the equation

    .. math::

        z = W_n(z) e^{W_n(z)}

    INPUT:

    - ``n`` - an integer. `n=0` corresponds to the principal branch.

    - ``z`` - a complex number

    If called with a single argument, that argument is ``z`` and the branch ``n`` is
    assumed to be 0 (the principal branch).

    ALGORITHM:

    Numerical evaluation is handled using the mpmath and SciPy libraries.

    REFERENCES:

    - :wikipedia:`Lambert_W_function`

    EXAMPLES:

    Evaluation of the principal branch::

        sage: lambert_w(1.0)
        0.567143290409784
        sage: lambert_w(-1).n()
        -0.318131505204764 + 1.33723570143069*I
        sage: lambert_w(-1.5 + 5*I)
        1.17418016254171 + 1.10651494102011*I

    Evaluation of other branches::

        sage: lambert_w(2, 1.0)
        -2.40158510486800 + 10.7762995161151*I

    Solutions to certain exponential equations are returned in terms of lambert_w::

        sage: S = solve(e^(5*x)+x==0, x, to_poly_solve=True)
        sage: z = S[0].rhs(); z
        -1/5*lambert_w(5)
        sage: N(z)
        -0.265344933048440

    Check the defining equation numerically at `z=5`::

        sage: N(lambert_w(5)*exp(lambert_w(5)) - 5)
        0.000000000000000

    There are several special values of the principal branch which
    are automatically simplified::

        sage: lambert_w(0)
        0
        sage: lambert_w(e)
        1
        sage: lambert_w(-1/e)
        -1

    Integration (of the principal branch) is evaluated using Maxima::

        sage: integrate(lambert_w(x), x)
        (lambert_w(x)^2 - lambert_w(x) + 1)*x/lambert_w(x)
        sage: integrate(lambert_w(x), x, 0, 1)
        (lambert_w(1)^2 - lambert_w(1) + 1)/lambert_w(1) - 1
        sage: integrate(lambert_w(x), x, 0, 1.0)
        0.330366124762

    Warning: The integral of a non-principal branch is not implemented,
    neither is numerical integration using GSL. The :meth:`numerical_integral`
    function does work if you pass a lambda function::

        sage: numerical_integral(lambda x: lambert_w(x), 0, 1)
        (0.33036612476168054, 3.667800782666048e-15)
    """

    def __init__(self):
        r"""
        See the docstring for :meth:`Function_lambert_w`.

        EXAMPLES::

            sage: lambert_w(0, 1.0)
            0.567143290409784
        """
        BuiltinFunction.__init__(self, "lambert_w", nargs=2,
                                 conversions={'mathematica':'ProductLog',
                                              'maple':'LambertW'})

    def __call__(self, *args, **kwds):
        r"""
        Custom call method allows the user to pass one argument or two. If
        one argument is passed, we assume it is ``z`` and that ``n=0``.

        EXAMPLES::

            sage: lambert_w(1)
            lambert_w(1)
            sage: lambert_w(1, 2)
            lambert_w(1, 2)
        """
        if len(args) == 2:
            return BuiltinFunction.__call__(self, *args, **kwds)
        elif len(args) == 1:
            return BuiltinFunction.__call__(self, 0, args[0], **kwds)
        else:
            raise TypeError("lambert_w takes either one or two arguments.")

    def _eval_(self, n, z):
        """
        EXAMPLES::

            sage: lambert_w(6.0)
            1.43240477589830
            sage: lambert_w(1)
            lambert_w(1)
            sage: lambert_w(x+1)
            lambert_w(x + 1)

        There are three special values which are automatically simplified::

            sage: lambert_w(0)
            0
            sage: lambert_w(e)
            1
            sage: lambert_w(-1/e)
            -1
            sage: lambert_w(SR(0))
            0

        The special values only hold on the principal branch::

            sage: lambert_w(1,e)
            lambert_w(1, e)
            sage: lambert_w(1, e.n())
            -0.532092121986380 + 4.59715801330257*I

        TESTS:

        When automatic simplication occurs, the parent of the output value should be
        either the same as the parent of the input, or a Sage type::

            sage: parent(lambert_w(int(0)))
            <type 'int'>
            sage: parent(lambert_w(Integer(0)))
            Integer Ring
            sage: parent(lambert_w(e))
            Integer Ring
        """
        if not isinstance(z, Expression):
            if is_inexact(z):
                return self._evalf_(n, z, parent=sage_structure_coerce_parent(z))
            elif n == 0 and z == 0:
                return sage_structure_coerce_parent(z)(Integer(0))
        elif n == 0:
            if z.is_trivial_zero():
                return sage_structure_coerce_parent(z)(Integer(0))
            elif (z-const_e).is_trivial_zero():
                return sage_structure_coerce_parent(z)(Integer(1))
            elif (z+1/const_e).is_trivial_zero():
                return sage_structure_coerce_parent(z)(Integer(-1))
        return None

    def _evalf_(self, n, z, parent=None):
        """
        EXAMPLES::

            sage: N(lambert_w(1))
            0.567143290409784
            sage: lambert_w(RealField(100)(1))
            0.56714329040978387299996866221

        SciPy is used to evaluate for float, RDF, and CDF inputs::

            sage: lambert_w(RDF(1))
            0.56714329041
        """
        R = parent or sage_structure_coerce_parent(z)
        if R is float or R is complex or R is RDF or R is CDF:
            import scipy.special
            return scipy.special.lambertw(z, n)
        else:
            import mpmath
            return mpmath_utils.call(mpmath.lambertw, z, n, parent=R)

    def _derivative_(self, n, z, diff_param=None):
        """
        The derivative of `W_n(x)` is `W_n(x)/(x \cdot W_n(x) + x)`.

        EXAMPLES::

            sage: x = var('x')
            sage: derivative(lambert_w(x), x)
            lambert_w(x)/(x*lambert_w(x) + x)

            sage: derivative(lambert_w(2, exp(x)), x)
            e^x*lambert_w(2, e^x)/(e^x*lambert_w(2, e^x) + e^x)

        TESTS:

        Differentiation in the first parameter raises an error :trac:`14788`::

            sage: n = var('n')
            sage: lambert_w(n, x).diff(n)
            Traceback (most recent call last):
            ...
            ValueError: cannot differentiate lambert_w in the first parameter
        """
        if diff_param == 0:
            raise ValueError("cannot differentiate lambert_w in the first parameter")

        return lambert_w(n, z)/(z*lambert_w(n, z)+z)

    def _maxima_init_evaled_(self, n, z):
        """
        EXAMPLES:

        These are indirect doctests for this function.::

            sage: lambert_w(0, x)._maxima_()
            lambert_w(x)
            sage: lambert_w(1, x)._maxima_()
            Traceback (most recent call last):
            ...
            NotImplementedError: Non-principal branch lambert_w[1](x) is not implemented in Maxima
        """
        if n == 0:
            return "lambert_w(%s)" % z
        else:
            raise NotImplementedError("Non-principal branch lambert_w[%s](%s) is not implemented in Maxima" % (n, z))


    def _print_(self, n, z):
        """
        Custom _print_ method to avoid printing the branch number if
        it is zero.

        EXAMPLES::

            sage: lambert_w(1)
            lambert_w(1)
            sage: lambert_w(0,x)
            lambert_w(x)
        """
        if n == 0:
            return "lambert_w(%s)" % z
        else:
            return "lambert_w(%s, %s)" % (n, z)

    def _print_latex_(self, n, z):
        """
        Custom _print_latex_ method to avoid printing the branch
        number if it is zero.

        EXAMPLES::

            sage: latex(lambert_w(1))
            \operatorname{W_0}(1)
            sage: latex(lambert_w(0,x))
            \operatorname{W_0}(x)
            sage: latex(lambert_w(1,x))
            \operatorname{W_{1}}(x)
        """
        if n == 0:
            return r"\operatorname{W_0}(%s)" % z
        else:
            return r"\operatorname{W_{%s}}(%s)" % (n, z)

lambert_w = Function_lambert_w()