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'))
            12.182493960703473

        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

        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
            sage: exp(2).n(100)
            7.3890560989306502272304274606

        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'))

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))
            2.302585092994046
            sage: ln(2.718)
            0.999896315728952
            sage: ln(2.0)
            0.693147180559945
            sage: ln(float(-1))
            3.141592653589793j
            sage: ln(complex(-1))
            3.141592653589793j

        The ``hold`` parameter can be used to prevent automatic evaluation::

            sage: log(-1,hold=True)
            log(-1)
            sage: log(-1)
            I*pi
            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'))

    def __call__(self, *args, **kwds):
        """
        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))
            1.6609640474436813
            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)


        TESTS:

        Check if :trac:`10136` is fixed::

            sage: log(x).operator() is log
            True
            sage: log(x).operator() is ln
            True

            sage: log(1000, 10, base=5)
            Traceback (most recent call last):
            ...
            TypeError: Symbolic function log must be called as log(x),
            log(x, base=b) or log(x, b)
        """
        base = kwds.pop('base', None)
        if base is None:
            if len(args) == 1:
                return GinacFunction.__call__(self, *args, **kwds)
            # second argument is base
            base = args[1]
            args = args[:1]

        if len(args) != 1:
            raise TypeError("Symbolic function log must be called as "
                    "log(x), log(x, base=b) or log(x, b)")

        try:
            return args[0].log(base)
        except (AttributeError, TypeError):
            return GinacFunction.__call__(self, *args, **kwds) / \
                GinacFunction.__call__(self, base, **kwds)

ln = log = function_log = Function_log()


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](_SAGE_VAR_x)
            sage: polylog(4, x)._maxima_()
            polylog(4,_SAGE_VAR_x)
        """
        args_maxima = []
        for a in args:
            if isinstance(a, str):
                args_maxima.append(a)
            elif hasattr(a, '_maxima_init_'):
                args_maxima.append(a._maxima_init_())
            else:
                args_maxima.append(str(a))

        n, x = args_maxima
        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)
        0.3303661247616807

    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, algorithm=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))
            0.5671432904097838
        """
        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(_SAGE_VAR_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:
            if isinstance(z,str):
                maxima_z=z
            elif hasattr(z,'_maxima_init_'):
                maxima_z=z._maxima_init_()
            else:
                maxima_z=str(z)
            return "lambert_w(%s)" % maxima_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()