from __future__ import print_function, division
from sympy.core import sympify
from sympy.core.add import Add
from sympy.core.cache import cacheit
from sympy.core.compatibility import range
from sympy.core.function import Function, ArgumentIndexError, _coeff_isneg
from sympy.core.logic import fuzzy_not
from sympy.core.mul import Mul
from sympy.core.numbers import Integer
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.core.symbol import Wild, Dummy
from sympy.functions.combinatorial.factorials import factorial
from sympy.ntheory import multiplicity, perfect_power
# 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):
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_finite(self):
arg = self.args[0]
if arg.is_infinite:
if arg.is_negative:
return True
if arg.is_positive:
return False
if arg.is_finite:
return True
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.exp is S.Zero:
return True
elif s.exp.is_rational and fuzzy_not(s.exp.is_zero):
return False
else:
return s.is_rational
def _eval_is_zero(self):
return (self.args[0] is S.NegativeInfinity)
def _eval_power(self, other):
"""exp(arg)**e -> exp(arg*e) if assumptions allow it.
"""
b, e = self.as_base_exp()
return Pow._eval_power(Pow(b, e, evaluate=False), other)
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): # Abs is never a polar number
from sympy.functions.elementary.complexes import re
return exp(re(self.args[0]))
def _eval_evalf(self, prec):
""" Careful! any evalf of polar numbers is flaky """
from sympy import im, pi, re
i = im(self.args[0])
try:
bad = (i <= -pi or i > pi)
except TypeError:
bad = True
if bad:
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_power(self, other):
return self.func(self.args[0]*other)
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
"""
def fdiff(self, argindex=1):
"""
Returns the first derivative of this function.
"""
if argindex == 1:
return self
else:
raise ArgumentIndexError(self, argindex)
def _eval_refine(self, assumptions):
from sympy.assumptions import ask, Q
arg = self.args[0]
if arg.is_Mul:
Ioo = S.ImaginaryUnit*S.Infinity
if arg in [Ioo, -Ioo]:
return S.NaN
coeff = arg.as_coefficient(S.Pi*S.ImaginaryUnit)
if coeff:
if ask(Q.integer(2*coeff)):
if ask(Q.even(coeff)):
return S.One
elif ask(Q.odd(coeff)):
return S.NegativeOne
elif ask(Q.even(coeff + S.Half)):
return -S.ImaginaryUnit
elif ask(Q.odd(coeff + S.Half)):
return S.ImaginaryUnit
@classmethod
def eval(cls, arg):
from sympy.assumptions import ask, Q
from sympy.calculus import AccumBounds
from sympy.sets.setexpr import SetExpr
from sympy.matrices.matrices import MatrixBase
from sympy import logcombine
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 is S.ComplexInfinity:
return S.NaN
elif isinstance(arg, log):
return arg.args[0]
elif isinstance(arg, AccumBounds):
return AccumBounds(exp(arg.min), exp(arg.max))
elif isinstance(arg, SetExpr):
return arg._eval_func(cls)
elif arg.is_Mul:
if arg.is_number or arg.is_Symbol:
coeff = arg.coeff(S.Pi*S.ImaginaryUnit)
if coeff:
if ask(Q.integer(2*coeff)):
if ask(Q.even(coeff)):
return S.One
elif ask(Q.odd(coeff)):
return S.NegativeOne
elif ask(Q.even(coeff + S.Half)):
return -S.ImaginaryUnit
elif ask(Q.odd(coeff + S.Half)):
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):
term_ = logcombine(term)
if isinstance(term_, 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 isinstance(newa, cls):
add.append(a)
else:
out.append(newa)
if out:
return Mul(*out)*cls(Add(*add), evaluate=False)
elif isinstance(arg, MatrixBase):
return arg.exp()
@property
def base(self):
"""
Returns the base of the exponential function.
"""
return S.Exp1
@staticmethod
@cacheit
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/factorial(n)
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
"""
import sympy
re, im = self.args[0].as_real_imag()
if deep:
re = re.expand(deep, **hints)
im = im.expand(deep, **hints)
cos, sin = sympy.cos(im), sympy.sin(im)
return (exp(re)*cos, exp(re)*sin)
def _eval_subs(self, old, new):
# keep processing of power-like args centralized in Pow
if old.is_Pow: # handle (exp(3*log(x))).subs(x**2, z) -> z**(3/2)
old = exp(old.exp*log(old.base))
elif old is S.Exp1 and new.is_Function:
old = exp
if isinstance(old, exp) or old is S.Exp1:
f = lambda a: Pow(*a.as_base_exp(), evaluate=False) if (
a.is_Pow or isinstance(a, exp)) else a
return Pow._eval_subs(f(self), f(old), new)
if old is exp and not new.is_Function:
return new**self.exp._subs(old, new)
return Function._eval_subs(self, old, 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_algebraic(self):
s = self.func(*self.args)
if s.func == self.func:
if fuzzy_not(self.exp.is_zero):
if self.exp.is_algebraic:
return False
elif (self.exp/S.Pi).is_rational:
return False
else:
return s.is_algebraic
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, Order, 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 += Order(o.expr.subs(t, (arg_series - arg0)), x)
r = r.expand()
return powsimp(r, deep=True, combine='exp')
def _taylor(self, x, n):
from sympy import Order
l = []
g = None
for i in range(n):
g = self.taylor_term(i, self.args[0], g)
g = g.nseries(x, n=n)
l.append(g)
return Add(*l) + Order(x**n, x)
def _eval_as_leading_term(self, x):
from sympy import Order
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 Order(1, x).contains(arg):
return S.One
return exp(arg)
def _eval_rewrite_as_sin(self, arg, **kwargs):
from sympy import sin
I = S.ImaginaryUnit
return sin(I*arg + S.Pi/2) - I*sin(I*arg)
def _eval_rewrite_as_cos(self, arg, **kwargs):
from sympy import cos
I = S.ImaginaryUnit
return cos(I*arg) + I*cos(I*arg + S.Pi/2)
def _eval_rewrite_as_tanh(self, arg, **kwargs):
from sympy import tanh
return (1 + tanh(arg/2))/(1 - tanh(arg/2))
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
from sympy.functions.elementary.trigonometric import sin, cos
if arg.is_Mul:
coeff = arg.coeff(S.Pi*S.ImaginaryUnit)
if coeff and coeff.is_number:
cosine, sine = cos(S.Pi*coeff), sin(S.Pi*coeff)
if not isinstance(cosine, cos) and not isinstance (sine, sin):
return cosine + S.ImaginaryUnit*sine
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if arg.is_Mul:
logs = [a for a in arg.args if isinstance(a, log) and len(a.args) == 1]
if logs:
return Pow(logs[0].args[0], arg.coeff(logs[0]))
[docs]class log(Function):
r"""
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
"""
def fdiff(self, argindex=1):
"""
Returns the first derivative of the function.
"""
if argindex == 1:
return 1/self.args[0]
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
r"""
Returns `e^x`, the inverse function of `\log(x)`.
"""
return exp
@classmethod
def eval(cls, arg, base=None):
from sympy import unpolarify
from sympy.calculus import AccumBounds
from sympy.sets.setexpr import SetExpr
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_Rational and arg.p == 1:
return -cls(arg.q)
if arg is S.ComplexInfinity:
return S.ComplexInfinity
if isinstance(arg, exp) and arg.args[0].is_real:
return arg.args[0]
elif isinstance(arg, exp_polar):
return unpolarify(arg.exp)
elif isinstance(arg, AccumBounds):
if arg.min.is_positive:
return AccumBounds(log(arg.min), log(arg.max))
else:
return
elif isinstance(arg, SetExpr):
return arg._eval_func(cls)
if arg.is_number:
if arg.is_negative:
return S.Pi * S.ImaginaryUnit + cls(-arg)
elif arg is S.ComplexInfinity:
return S.ComplexInfinity
elif arg is S.Exp1:
return S.One
# don't autoexpand Pow or Mul (see the issue 3351):
if 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)
def as_base_exp(self):
"""
Returns this function in the form (base, exponent).
"""
return self, S.One
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms): # of log(1+x)
r"""
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, expand_log
from sympy.concrete import Sum, Product
force = hints.get('force', False)
if (len(self.args) == 2):
return expand_log(self.func(*self.args), deep=deep, force=force)
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_Rational:
return log(arg.p) - log(arg.q)
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 or isinstance(arg, exp):
if force or (arg.exp.is_real and (arg.base.is_positive or ((arg.exp+1)
.is_positive and (arg.exp-1).is_nonpositive))) 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, rational, inverse):
from sympy.simplify.simplify import expand_log, simplify, inversecombine
if (len(self.args) == 2):
return simplify(self.func(*self.args), ratio=ratio, measure=measure,
rational=rational, inverse=inverse)
expr = self.func(simplify(self.args[0], ratio=ratio, measure=measure,
rational=rational, inverse=inverse))
if inverse:
expr = inversecombine(expr)
expr = expand_log(expr, deep=True)
return min([expr, self], key=measure)
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))
"""
from sympy import Abs, arg
if deep:
abs = Abs(self.args[0].expand(deep, **hints))
arg = arg(self.args[0].expand(deep, **hints))
else:
abs = Abs(self.args[0])
arg = 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 (self.args[0] - 1).is_zero:
return True
if s.args[0].is_rational and fuzzy_not((self.args[0] - 1).is_zero):
return False
else:
return s.is_rational
def _eval_is_algebraic(self):
s = self.func(*self.args)
if s.func == self.func:
if (self.args[0] - 1).is_zero:
return True
elif fuzzy_not((self.args[0] - 1).is_zero):
if self.args[0].is_algebraic:
return False
else:
return s.is_algebraic
def _eval_is_real(self):
return self.args[0].is_positive
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_zero:
return False
return arg.is_finite
def _eval_is_positive(self):
return (self.args[0] - 1).is_positive
def _eval_is_zero(self):
return (self.args[0] - 1).is_zero
def _eval_is_nonnegative(self):
return (self.args[0] - 1).is_nonnegative
def _eval_nseries(self, x, n, logx):
# NOTE Please see the comment at the beginning of this file, labelled
# IMPORTANT.
from sympy import cancel, Order
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, 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 range(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) + 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)
[docs]class LambertW(Function):
r"""
The Lambert W function `W(z)` is defined as the inverse
function of `w \exp(w)` [1]_.
In other words, the value of `W(z)` is such that `z = W(z) \exp(W(z))`
for any complex number `z`. The Lambert W function is a multivalued
function with infinitely many branches `W_k(z)`, indexed by
`k \in \mathbb{Z}`. Each branch gives a different solution `w`
of the equation `z = w \exp(w)`.
The Lambert W function has two partially real branches: the
principal branch (`k = 0`) is real for real `z > -1/e`, and the
`k = -1` branch is real for `-1/e < z < 0`. All branches except
`k = 0` have a logarithmic singularity at `z = 0`.
Examples
========
>>> from sympy import LambertW
>>> LambertW(1.2)
0.635564016364870
>>> LambertW(1.2, -1).n()
-1.34747534407696 - 4.41624341514535*I
>>> LambertW(-1).is_real
False
References
==========
.. [1] https://en.wikipedia.org/wiki/Lambert_W_function
"""
@classmethod
def eval(cls, x, k=None):
if k is S.Zero:
return cls(x)
elif k is None:
k = S.Zero
if k is S.Zero:
if x is S.Zero:
return S.Zero
if x is S.Exp1:
return S.One
if x == -1/S.Exp1:
return S.NegativeOne
if x == -log(2)/2:
return -log(2)
if x is S.Infinity:
return S.Infinity
if fuzzy_not(k.is_zero):
if x is S.Zero:
return S.NegativeInfinity
if k is S.NegativeOne:
if x == -S.Pi/2:
return -S.ImaginaryUnit*S.Pi/2
elif x == -1/S.Exp1:
return S.NegativeOne
elif x == -2*exp(-2):
return -Integer(2)
def fdiff(self, argindex=1):
"""
Return the first derivative of this function.
"""
x = self.args[0]
if len(self.args) == 1:
if argindex == 1:
return LambertW(x)/(x*(1 + LambertW(x)))
else:
k = self.args[1]
if argindex == 1:
return LambertW(x, k)/(x*(1 + LambertW(x, k)))
raise ArgumentIndexError(self, argindex)
def _eval_is_real(self):
x = self.args[0]
if len(self.args) == 1:
k = S.Zero
else:
k = self.args[1]
if k.is_zero:
if (x + 1/S.Exp1).is_positive:
return True
elif (x + 1/S.Exp1).is_nonpositive:
return False
elif (k + 1).is_zero:
if x.is_negative and (x + 1/S.Exp1).is_positive:
return True
elif x.is_nonpositive or (x + 1/S.Exp1).is_nonnegative:
return False
elif fuzzy_not(k.is_zero) and fuzzy_not((k + 1).is_zero):
if x.is_real:
return False
def _eval_is_algebraic(self):
s = self.func(*self.args)
if s.func == self.func:
if fuzzy_not(self.args[0].is_zero) and self.args[0].is_algebraic:
return False
else:
return s.is_algebraic
