Source code for sympy.functions.elementary.exponential

from __future__ import print_function, division

from sympy.core import C, sympify
from sympy.core.add import Add
from sympy.core.function import Lambda, Function, ArgumentIndexError
from sympy.core.cache import cacheit
from sympy.core.singleton import S
from sympy.core.symbol import Wild, Dummy
from sympy.core.mul import Mul

from sympy.functions.elementary.miscellaneous import sqrt
from sympy.ntheory import multiplicity, perfect_power
from sympy.core.compatibility import xrange

# NOTE IMPORTANT
# The series expansion code in this file is an important part of the gruntz
# algorithm for determining limits. _eval_nseries has to return a generalized
# power series with coefficients in C(log(x), log).
# In more detail, the result of _eval_nseries(self, x, n) must be
#   c_0*x**e_0 + ... (finitely many terms)
# where e_i are numbers (not necessarily integers) and c_i involve only
# numbers, the function log, and log(x). [This also means it must not contain
# log(x(1+p)), this *has* to be expanded to log(x)+log(1+p) if x.is_positive and
# p.is_positive.]


class ExpBase(Function):

    nargs = 1
    unbranched = True

    def inverse(self, argindex=1):
        """
        Returns the inverse function of ``exp(x)``.
        """
        return log

    def as_numer_denom(self):
        """
        Returns this with a positive exponent as a 2-tuple (a fraction).

        Examples
        ========

        >>> from sympy.functions import exp
        >>> from sympy.abc import x
        >>> exp(-x).as_numer_denom()
        (1, exp(x))
        >>> exp(x).as_numer_denom()
        (exp(x), 1)
        """
        # this should be the same as Pow.as_numer_denom wrt
        # exponent handling
        exp = self.exp
        neg_exp = exp.is_negative
        if not neg_exp and not (-exp).is_negative:
            neg_exp = _coeff_isneg(exp)
        if neg_exp:
            return S.One, self.func(-exp)
        return self, S.One

    @property
    def exp(self):
        """
        Returns the exponent of the function.
        """
        return self.args[0]

    def as_base_exp(self):
        """
        Returns the 2-tuple (base, exponent).
        """
        return self.func(1), Mul(*self.args)

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate())

    def _eval_is_bounded(self):
        arg = self.args[0]
        if arg.is_unbounded:
            if arg.is_negative:
                return True
            if arg.is_positive:
                return False
        if arg.is_bounded:
            return True

    def _eval_is_rational(self):
        s = self.func(*self.args)
        if s.func == self.func:
            if s.args[0].is_rational:
                return False
        else:
            return s.is_rational

    def _eval_is_zero(self):
        return (self.args[0] is S.NegativeInfinity)

    def _eval_power(b, e):
        """exp(arg)**e -> exp(arg*e) if assumptions allow it.
        """
        f = b.func
        be = b.exp
        rv = f(be*e)
        if e.is_integer:
            return rv
        if be.is_real:
            return rv
        # "is True" needed below; exp.is_polar returns <property object ...>
        if f.is_polar is True:
            return rv
        if e.is_polar:
            return rv
        if be.is_polar:
            return rv
        besmall = abs(be) <= S.Pi
        if besmall is True:
            return rv
        elif besmall is False and e.is_Rational and e.q == 2:
            return -rv

    def _eval_expand_power_exp(self, **hints):
        arg = self.args[0]
        if arg.is_Add and arg.is_commutative:
            expr = 1
            for x in arg.args:
                expr *= self.func(x)
            return expr
        return self.func(arg)


class exp_polar(ExpBase):
    r"""
    Represent a 'polar number' (see g-function Sphinx documentation).

    ``exp_polar`` represents the function
    `Exp: \mathbb{C} \rightarrow \mathcal{S}`, sending the complex number
    `z = a + bi` to the polar number `r = exp(a), \theta = b`. It is one of
    the main functions to construct polar numbers.

    >>> from sympy import exp_polar, pi, I, exp

    The main difference is that polar numbers don't "wrap around" at `2 \pi`:

    >>> exp(2*pi*I)
    1
    >>> exp_polar(2*pi*I)
    exp_polar(2*I*pi)

    apart from that they behave mostly like classical complex numbers:

    >>> exp_polar(2)*exp_polar(3)
    exp_polar(5)

    See also
    ========

    sympy.simplify.simplify.powsimp
    sympy.functions.elementary.complexes.polar_lift
    sympy.functions.elementary.complexes.periodic_argument
    sympy.functions.elementary.complexes.principal_branch
    """

    is_polar = True
    is_comparable = False  # cannot be evalf'd

    def _eval_Abs(self):
        from sympy import expand_mul
        return sqrt( expand_mul(self * self.conjugate()) )

    def _eval_evalf(self, prec):
        """ Careful! any evalf of polar numbers is flaky """
        from sympy import im, pi, re
        i = im(self.args[0])
        if i <= -pi or i > pi:
            return self  # cannot evalf for this argument
        res = exp(self.args[0])._eval_evalf(prec)
        if i > 0 and im(res) < 0:
            # i ~ pi, but exp(I*i) evaluated to argument slightly bigger than pi
            return re(res)
        return res

    def _eval_is_real(self):
        if self.args[0].is_real:
            return True

    def as_base_exp(self):
        # XXX exp_polar(0) is special!
        if self.args[0] == 0:
            return self, S(1)
        return ExpBase.as_base_exp(self)


[docs]class exp(ExpBase): """ The exponential function, :math:`e^x`. See Also ======== log """
[docs] def fdiff(self, argindex=1): """ Returns the first derivative of this function. """ if argindex == 1: return self else: raise ArgumentIndexError(self, argindex)
@classmethod def eval(cls, arg): if arg.is_Number: if arg is S.NaN: return S.NaN elif arg is S.Zero: return S.One elif arg is S.One: return S.Exp1 elif arg is S.Infinity: return S.Infinity elif arg is S.NegativeInfinity: return S.Zero elif arg.func is log: return arg.args[0] elif arg.is_Mul: Ioo = S.ImaginaryUnit*S.Infinity if arg in [Ioo, -Ioo]: return S.NaN coeff = arg.coeff(S.Pi*S.ImaginaryUnit) if coeff: if (2*coeff).is_integer: if coeff.is_even: return S.One elif coeff.is_odd: return S.NegativeOne elif (coeff + S.Half).is_even: return -S.ImaginaryUnit elif (coeff + S.Half).is_odd: return S.ImaginaryUnit # Warning: code in risch.py will be very sensitive to changes # in this (see DifferentialExtension). # look for a single log factor coeff, terms = arg.as_coeff_Mul() # but it can't be multiplied by oo if coeff in [S.NegativeInfinity, S.Infinity]: return None coeffs, log_term = [coeff], None for term in Mul.make_args(terms): if term.func is log: if log_term is None: log_term = term.args[0] else: return None elif term.is_comparable: coeffs.append(term) else: return None return log_term**Mul(*coeffs) if log_term else None elif arg.is_Add: out = [] add = [] for a in arg.args: if a is S.One: add.append(a) continue newa = cls(a) if newa.func is cls: add.append(a) else: out.append(newa) if out: return Mul(*out)*cls(Add(*add), evaluate=False) elif arg.is_Matrix: from sympy import Matrix return arg.exp() @property
[docs] def base(self): """ Returns the base of the exponential function. """ return S.Exp1
@staticmethod @cacheit
[docs] def taylor_term(n, x, *previous_terms): """ Calculates the next term in the Taylor series expansion. """ if n < 0: return S.Zero if n == 0: return S.One x = sympify(x) if previous_terms: p = previous_terms[-1] if p is not None: return p * x / n return x**n/C.factorial()(n)
[docs] def as_real_imag(self, deep=True, **hints): """ Returns this function as a 2-tuple representing a complex number. Examples ======== >>> from sympy import I >>> from sympy.abc import x >>> from sympy.functions import exp >>> exp(x).as_real_imag() (exp(re(x))*cos(im(x)), exp(re(x))*sin(im(x))) >>> exp(1).as_real_imag() (E, 0) >>> exp(I).as_real_imag() (cos(1), sin(1)) >>> exp(1+I).as_real_imag() (E*cos(1), E*sin(1)) See Also ======== sympy.functions.elementary.complexes.re sympy.functions.elementary.complexes.im """ re, im = self.args[0].as_real_imag() if deep: re = re.expand(deep, **hints) im = im.expand(deep, **hints) cos, sin = C.cos(im), C.sin(im) return (exp(re)*cos, exp(re)*sin)
def _eval_subs(self, old, new): arg = self.args[0] o = old if old.is_Pow: # handle (exp(3*log(x))).subs(x**2, z) -> z**(3/2) o = exp(o.exp*log(o.base)) if o.func is exp: # exp(a*expr) .subs( exp(b*expr), y ) -> y ** (a/b) a, expr_terms = self.args[0].as_independent( C.Symbol, as_Add=False) b, expr_terms_ = o.args[0].as_independent( C.Symbol, as_Add=False) if expr_terms == expr_terms_: return new**(a/b) if arg.is_Add: # exp(2*x+a).subs(exp(3*x),y) -> y**(2/3) * exp(a) # exp(exp(x) + exp(x**2)).subs(exp(exp(x)), w) -> w * exp(exp(x**2)) oarg = o.args[0] new_l = [] o_al = [] coeff2, terms2 = oarg.as_coeff_mul() for a in arg.args: a = a._subs(o, new) coeff1, terms1 = a.as_coeff_mul() if terms1 == terms2: new_l.append(new**(coeff1/coeff2)) else: o_al.append(a._subs(o, new)) if new_l: new_l.append(self.func(Add(*o_al))) r = Mul(*new_l) return r if o is S.Exp1: # treat this however Pow is being treated u = C.Dummy('u', positive=True) return (u**self.args[0]).xreplace({u: new}) return Function._eval_subs(self, o, new) def _eval_is_real(self): if self.args[0].is_real: return True elif self.args[0].is_imaginary: arg2 = -S(2) * S.ImaginaryUnit * self.args[0] / S.Pi return arg2.is_even def _eval_is_positive(self): if self.args[0].is_real: return not self.args[0] is S.NegativeInfinity elif self.args[0].is_imaginary: arg2 = -S.ImaginaryUnit * self.args[0] / S.Pi return arg2.is_even def _eval_nseries(self, x, n, logx): # NOTE Please see the comment at the beginning of this file, labelled # IMPORTANT. from sympy import limit, oo, powsimp arg = self.args[0] arg_series = arg._eval_nseries(x, n=n, logx=logx) if arg_series.is_Order: return 1 + arg_series arg0 = limit(arg_series.removeO(), x, 0) if arg0 in [-oo, oo]: return self t = Dummy("t") exp_series = exp(t)._taylor(t, n) o = exp_series.getO() exp_series = exp_series.removeO() r = exp(arg0)*exp_series.subs(t, arg_series - arg0) r += C.Order(o.expr.subs(t, (arg_series - arg0)), x) r = r.expand() return powsimp(r, deep=True, combine='exp') def _taylor(self, x, n): l = [] g = None for i in xrange(n): g = self.taylor_term(i, self.args[0], g) g = g.nseries(x, n=n) l.append(g) return Add(*l) + C.Order(x**n, x) def _eval_as_leading_term(self, x): arg = self.args[0] if arg.is_Add: return Mul(*[exp(f).as_leading_term(x) for f in arg.args]) arg = self.args[0].as_leading_term(x) if C.Order(1, x).contains(arg): return S.One return exp(arg) def _eval_rewrite_as_sin(self, arg): I = S.ImaginaryUnit return C.sin(I*arg + S.Pi/2) - I*C.sin(I*arg) def _eval_rewrite_as_cos(self, arg): I = S.ImaginaryUnit return C.cos(I*arg) + I*C.cos(I*arg + S.Pi/2) def _sage_(self): import sage.all as sage return sage.exp(self.args[0]._sage_())
[docs]class log(Function): """ The natural logarithm function `\ln(x)` or `\log(x)`. Logarithms are taken with the natural base, `e`. To get a logarithm of a different base ``b``, use ``log(x, b)``, which is essentially short-hand for ``log(x)/log(b)``. See Also ======== exp """ nargs = (1, 2)
[docs] def fdiff(self, argindex=1): """ Returns the first derivative of the function. """ if argindex == 1: return 1/self.args[0] s = C.Dummy('x') return Lambda(s**(-1), s) else: raise ArgumentIndexError(self, argindex)
[docs] def inverse(self, argindex=1): """ Returns `e^x`, the inverse function of `\log(x)`. """ return exp
@classmethod def eval(cls, arg, base=None): from sympy import unpolarify arg = sympify(arg) if base is not None: base = sympify(base) if base == 1: if arg == 1: return S.NaN else: return S.ComplexInfinity try: # handle extraction of powers of the base now # or else expand_log in Mul would have to handle this n = multiplicity(base, arg) if n: den = base**n if den.is_Integer: return n + log(arg // den) / log(base) else: return n + log(arg / den) / log(base) else: return log(arg)/log(base) except ValueError: pass if base is not S.Exp1: return cls(arg)/cls(base) else: return cls(arg) if arg.is_Number: if arg is S.Zero: return S.ComplexInfinity elif arg is S.One: return S.Zero elif arg is S.Infinity: return S.Infinity elif arg is S.NegativeInfinity: return S.Infinity elif arg is S.NaN: return S.NaN elif arg.is_negative: return S.Pi * S.ImaginaryUnit + cls(-arg) elif arg.is_Rational: if arg.q != 1: return cls(arg.p) - cls(arg.q) elif arg is S.ComplexInfinity: return S.ComplexInfinity elif arg is S.Exp1: return S.One elif arg.func is exp and arg.args[0].is_real: return arg.args[0] elif arg.func is exp_polar: return unpolarify(arg.exp) #don't autoexpand Pow or Mul (see the issue 252): elif not arg.is_Add: coeff = arg.as_coefficient(S.ImaginaryUnit) if coeff is not None: if coeff is S.Infinity: return S.Infinity elif coeff is S.NegativeInfinity: return S.Infinity elif coeff.is_Rational: if coeff.is_nonnegative: return S.Pi * S.ImaginaryUnit * S.Half + cls(coeff) else: return -S.Pi * S.ImaginaryUnit * S.Half + cls(-coeff)
[docs] def as_base_exp(self): """ Returns this function in the form (base, exponent). """ return self, S.One
@staticmethod @cacheit
[docs] def taylor_term(n, x, *previous_terms): # of log(1+x) """ Returns the next term in the Taylor series expansion of `\log(1+x)`. """ from sympy import powsimp if n < 0: return S.Zero x = sympify(x) if n == 0: return x if previous_terms: p = previous_terms[-1] if p is not None: return powsimp((-n) * p * x / (n + 1), deep=True, combine='exp') return (1 - 2*(n % 2)) * x**(n + 1)/(n + 1)
def _eval_expand_log(self, deep=True, **hints): from sympy import unpolarify from sympy.concrete import Sum, Product force = hints.get('force', False) arg = self.args[0] if arg.is_Integer: # remove perfect powers p = perfect_power(int(arg)) if p is not False: return p[1]*self.func(p[0]) elif arg.is_Mul: expr = [] nonpos = [] for x in arg.args: if force or x.is_positive or x.is_polar: a = self.func(x) if isinstance(a, log): expr.append(self.func(x)._eval_expand_log(**hints)) else: expr.append(a) elif x.is_negative: a = self.func(-x) expr.append(a) nonpos.append(S.NegativeOne) else: nonpos.append(x) return Add(*expr) + log(Mul(*nonpos)) elif arg.is_Pow: if force or (arg.exp.is_real and arg.base.is_positive) or \ arg.base.is_polar: b = arg.base e = arg.exp a = self.func(b) if isinstance(a, log): return unpolarify(e) * a._eval_expand_log(**hints) else: return unpolarify(e) * a elif isinstance(arg, Product): if arg.function.is_positive: return Sum(log(arg.function), *arg.limits) return self.func(arg) def _eval_simplify(self, ratio, measure): from sympy.simplify.simplify import expand_log, logcombine, simplify expr = self.func(simplify(self.args[0], ratio=ratio, measure=measure)) expr = expand_log(expr, deep=True) return min([expr, self], key=measure)
[docs] def as_real_imag(self, deep=True, **hints): """ Returns this function as a complex coordinate. Examples ======== >>> from sympy import I >>> from sympy.abc import x >>> from sympy.functions import log >>> log(x).as_real_imag() (log(Abs(x)), arg(x)) >>> log(I).as_real_imag() (0, pi/2) >>> log(1+I).as_real_imag() (log(sqrt(2)), pi/4) >>> log(I*x).as_real_imag() (log(Abs(x)), arg(I*x)) """ if deep: abs = C.Abs(self.args[0].expand(deep, **hints)) arg = C.arg(self.args[0].expand(deep, **hints)) else: abs = C.Abs(self.args[0]) arg = C.arg(self.args[0]) if hints.get('log', False): # Expand the log hints['complex'] = False return (log(abs).expand(deep, **hints), arg) else: return (log(abs), arg)
def _eval_is_rational(self): s = self.func(*self.args) if s.func == self.func: if s.args[0].is_rational: return False else: return s.is_rational def _eval_is_real(self): return self.args[0].is_positive def _eval_is_bounded(self): arg = self.args[0] if arg.is_infinitesimal: return False return arg.is_bounded def _eval_is_positive(self): arg = self.args[0] if arg.is_positive: if arg.is_unbounded: return True if arg.is_infinitesimal: return False if arg.is_Number: return arg > 1 def _eval_is_zero(self): # XXX This is not quite useless. Try evaluating log(0.5).is_negative # without it. There's probably a nicer way though. if self.args[0] is S.One: return True elif self.args[0].is_number: return self.args[0].expand() is S.One elif self.args[0].is_negative: return False def _eval_nseries(self, x, n, logx): # NOTE Please see the comment at the beginning of this file, labelled # IMPORTANT. from sympy import cancel if not logx: logx = log(x) if self.args[0] == x: return logx arg = self.args[0] k, l = Wild("k"), Wild("l") r = arg.match(k*x**l) if r is not None: #k = r.get(r, S.One) #l = r.get(l, S.Zero) k, l = r[k], r[l] if l != 0 and not l.has(x) and not k.has(x): r = log(k) + l*logx # XXX true regardless of assumptions? return r # TODO new and probably slow s = self.args[0].nseries(x, n=n, logx=logx) while s.is_Order: n += 1 s = self.args[0].nseries(x, n=n, logx=logx) a, b = s.leadterm(x) p = cancel(s/(a*x**b) - 1) g = None l = [] for i in xrange(n + 2): g = log.taylor_term(i, p, g) g = g.nseries(x, n=n, logx=logx) l.append(g) return log(a) + b*logx + Add(*l) + C.Order(p**n, x) def _eval_as_leading_term(self, x): arg = self.args[0].as_leading_term(x) if arg is S.One: return (self.args[0] - 1).as_leading_term(x) return self.func(arg) def _sage_(self): import sage.all as sage return sage.log(self.args[0]._sage_())
[docs]class LambertW(Function): """Lambert W function, defined as the inverse function of x*exp(x). This function represents the principal branch of this inverse function, which like the natural logarithm is multivalued. For more information, see: http://en.wikipedia.org/wiki/Lambert_W_function """ nargs = 1 @classmethod def eval(cls, x): if x == S.Zero: return S.Zero if x == S.Exp1: return S.One if x == -1/S.Exp1: return S.NegativeOne if x == -log(2)/2: return -log(2) if x == S.Infinity: return S.Infinity
[docs] def fdiff(self, argindex=1): """ Return the first derivative of this function. """ if argindex == 1: x = self.args[0] return LambertW(x)/(x*(1 + LambertW(x))) else: raise ArgumentIndexError(self, argindex)
from sympy.core.function import _coeff_isneg