forked from diofant/diofant
/
complexes.py
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/
complexes.py
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from ...core import (Add, Derivative, Dummy, Eq, Expr, Function, I, Integer,
Mul, Rational, Symbol, Tuple, factor_terms, nan, oo, pi,
zoo)
from ...core.function import AppliedUndef, ArgumentIndexError
from ...core.sympify import sympify
from ...logic.boolalg import BooleanAtom
from .exponential import exp, exp_polar, log
from .miscellaneous import sqrt
from .piecewise import ExprCondPair, Piecewise
from .trigonometric import atan2
###############################################################################
# ####################### REAL and IMAGINARY PARTS ########################## #
###############################################################################
class re(Function):
"""Returns real part of expression.
This function performs only elementary analysis and so it will fail to
decompose properly more complicated expressions. If completely simplified
result is needed then use Basic.as_real_imag() or perform complex
expansion on instance of this function.
Examples
========
>>> re(2*E)
2*E
>>> re(2*I + 17)
17
>>> re(2*I)
0
>>> re(im(x) + x*I + 2)
2
See Also
========
diofant.functions.elementary.complexes.im
"""
is_extended_real = True
unbranched = True # implicitely works on the projection to C
@classmethod
def eval(cls, arg):
if arg is zoo:
return nan
if arg.is_extended_real:
return arg
if arg.is_imaginary or (I*arg).is_extended_real:
return Integer(0)
if arg.is_Function and isinstance(arg, conjugate):
return re(arg.args[0])
included, reverted, excluded = [], [], []
args = Add.make_args(arg)
for term in args:
coeff = term.as_coefficient(I)
if coeff is not None:
if not coeff.is_extended_real:
reverted.append(coeff)
elif not term.has(I) and term.is_extended_real:
excluded.append(term)
else:
# Try to do some advanced expansion. If
# impossible, don't try to do re(arg) again
# (because this is what we are trying to do now).
real_imag = term.as_real_imag(ignore=arg)
if real_imag:
excluded.append(real_imag[0])
else:
included.append(term)
if len(args) != len(included):
a, b, c = (Add(*xs) for xs in [included, reverted, excluded])
return cls(a) - im(b) + c
def as_real_imag(self, deep=True, **hints):
"""Returns the real number with a zero imaginary part."""
return self, Integer(0)
def _eval_derivative(self, s):
if s.is_extended_real or self.args[0].is_extended_real:
return re(Derivative(self.args[0], s, evaluate=True))
if s.is_imaginary or self.args[0].is_imaginary:
return -I*im(Derivative(self.args[0], s, evaluate=True))
def _eval_rewrite_as_im(self, arg):
return self.args[0] - I*im(self.args[0])
def _eval_is_algebraic(self):
return self.args[0].is_algebraic
def _eval_is_real(self):
if self.args[0].is_complex:
return True
class im(Function):
"""Returns imaginary part of expression.
This function performs only elementary analysis and so it will fail to
decompose properly more complicated expressions. If completely simplified
result is needed then use Basic.as_real_imag() or perform complex expansion
on instance of this function.
Examples
========
>>> im(2*E)
0
>>> re(2*I + 17)
17
>>> im(x*I)
re(x)
>>> im(re(x) + y)
im(y)
See Also
========
diofant.functions.elementary.complexes.re
"""
is_extended_real = True
unbranched = True # implicitely works on the projection to C
@classmethod
def eval(cls, arg):
if arg is zoo:
return nan
if arg.is_extended_real:
return Integer(0)
if arg.is_imaginary or (I*arg).is_extended_real:
return -I * arg
if arg.is_Function and isinstance(arg, conjugate):
return -im(arg.args[0])
included, reverted, excluded = [], [], []
args = Add.make_args(arg)
for term in args:
coeff = term.as_coefficient(I)
if coeff is not None:
if not coeff.is_extended_real:
reverted.append(coeff)
else:
excluded.append(coeff)
elif term.has(I) or not term.is_extended_real:
# Try to do some advanced expansion. If
# impossible, don't try to do im(arg) again
# (because this is what we are trying to do now).
real_imag = term.as_real_imag(ignore=arg)
if real_imag:
excluded.append(real_imag[1])
else:
included.append(term)
if len(args) != len(included):
a, b, c = (Add(*xs) for xs in [included, reverted, excluded])
return cls(a) + re(b) + c
def as_real_imag(self, deep=True, **hints):
"""
Return the imaginary part with a zero real part.
Examples
========
>>> im(2 + 3*I).as_real_imag()
(3, 0)
"""
return self, Integer(0)
def _eval_derivative(self, s):
if s.is_extended_real or self.args[0].is_extended_real:
return im(Derivative(self.args[0], s, evaluate=True))
if s.is_imaginary or self.args[0].is_imaginary:
return -I*re(Derivative(self.args[0], s, evaluate=True))
def _eval_rewrite_as_re(self, arg):
return -I*(self.args[0] - re(self.args[0]))
def _eval_is_algebraic(self):
return self.args[0].is_algebraic
def _eval_is_real(self):
if self.args[0].is_complex:
return True
###############################################################################
# ############# SIGN, ABSOLUTE VALUE, ARGUMENT and CONJUGATION ############## #
###############################################################################
class sign(Function):
"""
Returns the complex sign of an expression.
For nonzero complex number z is an equivalent of z/abs(z).
Else returns zero.
Examples
========
>>> sign(-1)
-1
>>> sign(0)
0
>>> sign(-3*I)
-I
>>> sign(1 + I)
sign(1 + I)
>>> _.evalf()
0.707106781186548 + 0.707106781186548*I
See Also
========
Abs
conjugate
"""
is_complex = True
def doit(self, **hints):
if self.args[0].is_nonzero:
return self.args[0] / Abs(self.args[0])
return self
@classmethod
def eval(cls, arg):
# handle what we can
if arg.is_Mul:
c, args = arg.as_coeff_mul()
unk = []
s = sign(c)
for a in args:
if a.is_negative:
s = -s
elif a.is_positive:
pass
elif a.is_imaginary and im(a).is_comparable:
s *= sign(a)
else:
unk.append(a)
if c == 1 and len(unk) == len(args):
return
return s * cls(arg._new_rawargs(*unk))
if arg.is_zero: # it may be an Expr that is zero
return Integer(0)
if arg.is_positive:
return Integer(1)
if arg.is_negative:
return Integer(-1)
if arg.is_Function:
if isinstance(arg, sign):
return arg
if arg.is_imaginary:
if arg.is_Pow and arg.exp == Rational(1, 2):
# we catch this because non-trivial sqrt args are not expanded
# e.g. sqrt(1-sqrt(2)) --x--> to I*sqrt(sqrt(2) - 1)
return I
arg2 = -I * arg
if arg2.is_positive:
return I
def _eval_Abs(self):
if self.args[0].is_nonzero:
return Integer(1)
def _eval_conjugate(self):
return sign(conjugate(self.args[0]))
def _eval_derivative(self, s):
if self.args[0].is_extended_real:
from ..special.delta_functions import DiracDelta
return 2 * Derivative(self.args[0], s, evaluate=True) \
* DiracDelta(self.args[0])
if self.args[0].is_imaginary:
from ..special.delta_functions import DiracDelta
return 2 * Derivative(self.args[0], s, evaluate=True) \
* DiracDelta(-I * self.args[0])
def _eval_is_nonnegative(self):
if self.args[0].is_nonnegative:
return True
def _eval_is_nonpositive(self):
if self.args[0].is_nonpositive:
return True
def _eval_is_imaginary(self):
return self.args[0].is_imaginary
def _eval_is_integer(self):
return self.args[0].is_extended_real
def _eval_is_zero(self):
return self.args[0].is_zero
def _eval_power(self, other):
if (
self.args[0].is_extended_real and
self.args[0].is_nonzero and
other.is_integer and
other.is_even
):
return Integer(1)
def _eval_rewrite_as_Piecewise(self, arg):
if arg.is_extended_real:
return Piecewise((1, arg > 0), (-1, arg < 0), (0, True))
def _eval_rewrite_as_Heaviside(self, arg):
from .. import Heaviside
if arg.is_extended_real:
return Heaviside(arg)*2-1
def _eval_rewrite_as_exp(self, x):
return exp(I*arg(x))
def _eval_simplify(self, ratio, measure):
return self.func(self.args[0].factor())
def _eval_nseries(self, x, n, logx):
direction = self.args[0].as_leading_term(x).as_coeff_exponent(x)[0]
if direction.is_extended_real:
return self.func(direction)
return super()._eval_nseries(x, n, logx)
class Abs(Function):
"""Return the absolute value of the argument.
This is an extension of the built-in function abs() to accept symbolic
values. If you pass a Diofant expression to the built-in abs(), it will
pass it automatically to Abs().
Examples
========
>>> Abs(-1)
1
>>> x = Symbol('x', real=True)
>>> abs(-x) # The Python built-in
Abs(x)
>>> abs(x**2)
x**2
Note that the Python built-in will return either an Expr or int depending on
the argument:
>>> type(abs(-1))
<... 'int'>
>>> type(abs(Integer(-1)))
<class 'diofant.core.numbers.One'>
Abs will always return a diofant object.
See Also
========
diofant.functions.elementary.complexes.sign
diofant.functions.elementary.complexes.conjugate
"""
is_extended_real = True
is_negative = False
unbranched = True
def fdiff(self, argindex=1):
"""
Get the first derivative of the argument to Abs().
Examples
========
>>> abs(-x).fdiff()
sign(x)
"""
if argindex == 1:
return sign(self.args[0])
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
from ...simplify import signsimp
if hasattr(arg, '_eval_Abs'):
obj = arg._eval_Abs()
if obj is not None:
return obj
if not isinstance(arg, Expr):
raise TypeError(f'Bad argument type for Abs(): {type(arg)}')
# handle what we can
arg = signsimp(arg, evaluate=False)
if arg.is_Mul:
known = []
unk = []
for t in arg.args:
tnew = cls(t)
if isinstance(tnew, cls):
unk.append(tnew.args[0])
else:
known.append(tnew)
known = Mul(*known)
unk = cls(Mul(*unk), evaluate=False) if unk else Integer(1)
return known*unk
if arg.is_Pow:
base, exponent = arg.as_base_exp()
if base.is_extended_real:
if exponent.is_integer:
if exponent.is_even:
return arg
if base == -1:
return Integer(1)
if isinstance(base, cls) and exponent == -1:
return arg
return Abs(base)**exponent
if base.is_nonnegative:
return base**re(exponent)
if base.is_negative:
return (-base)**re(exponent)*exp(-pi*im(exponent))
if arg.is_zero: # it may be an Expr that is zero
return Integer(0)
if arg.is_nonnegative:
return arg
if arg.is_nonpositive:
return -arg
if arg.is_imaginary:
arg2 = -I * arg
if arg2.is_nonnegative:
return arg2
if arg.is_Add:
if any(a.is_infinite for a in arg.as_real_imag()):
return oo
if arg.is_extended_real is not True and arg.is_imaginary is None:
if all(a.is_extended_real or a.is_imaginary or (I*a).is_extended_real for a in arg.args):
from ...core import expand_mul
return sqrt(expand_mul(arg*arg.conjugate()))
if arg is zoo:
return oo
if arg.is_extended_real is not True and arg.is_imaginary is False:
from ...core import expand_mul
return sqrt(expand_mul(arg*arg.conjugate()))
def _eval_is_integer(self):
if self.args[0].is_extended_real:
return self.args[0].is_integer
def _eval_is_nonzero(self):
return self.args[0].is_nonzero
def _eval_is_finite(self):
if self.args[0].is_complex:
return True
def _eval_is_positive(self):
return self.is_nonzero
def _eval_is_rational(self):
if self.args[0].is_extended_real:
return self.args[0].is_rational
def _eval_is_even(self):
if self.args[0].is_extended_real:
return self.args[0].is_even
def _eval_is_odd(self):
if self.args[0].is_extended_real:
return self.args[0].is_odd
def _eval_is_algebraic(self):
return self.args[0].is_algebraic
def _eval_power(self, other):
if self.args[0].is_extended_real and other.is_integer:
if other.is_even:
return self.args[0]**other
if other != -1 and other.is_Integer:
return self.args[0]**(other - 1)*self
def _eval_nseries(self, x, n, logx):
direction = self.args[0].as_leading_term(x).as_coeff_exponent(x)[0]
s = self.args[0]._eval_nseries(x, n, logx)
when, lim = Eq(direction, 0), direction.limit(x, 0)
if lim.equals(0) is False:
return s/sign(lim)
return Piecewise((lim, when), (s/sign(direction), True))
def _eval_derivative(self, s):
if self.args[0].is_extended_real or self.args[0].is_imaginary:
return Derivative(self.args[0], s, evaluate=True) \
* sign(conjugate(self.args[0]))
return (re(self.args[0]) * Derivative(re(self.args[0]), s,
evaluate=True) + im(self.args[0]) * Derivative(im(self.args[0]),
s, evaluate=True)) / Abs(self.args[0])
def _eval_rewrite_as_Heaviside(self, arg):
# Note this only holds for real arg (since Heaviside is not defined
# for complex arguments).
from .. import Heaviside
if arg.is_extended_real:
return arg*(Heaviside(arg) - Heaviside(-arg))
def _eval_rewrite_as_Piecewise(self, arg):
if arg.is_extended_real:
return Piecewise((arg, arg >= 0), (-arg, True))
def _eval_rewrite_as_sign(self, arg):
return arg/sign(arg)
def _eval_rewrite_as_tractable(self, arg, wrt=None, **kwargs):
if wrt is not None and (s := sign(arg.limit(wrt, oo))) in (1, -1):
return s*arg
class arg(Function):
"""Returns the argument (in radians) of a complex number.
For a real number, the argument is always 0.
Examples
========
>>> arg(2.0)
0
>>> arg(I)
pi/2
>>> arg(sqrt(2) + I*sqrt(2))
pi/4
"""
is_real = True
@classmethod
def eval(cls, arg):
if not arg.is_Atom:
c, arg_ = factor_terms(arg).as_coeff_Mul()
if arg_.is_Mul:
arg_ = Mul(*[a if (sign(a) not in (-1, 1)) else
sign(a) for a in arg_.args])
arg_ = sign(c)*arg_
else:
arg_ = arg
if arg_.is_zero:
return Integer(0)
x, y = re(arg_), im(arg_)
rv = atan2(y, x)
if rv.is_number and not rv.atoms(AppliedUndef):
return rv
if arg_ != arg:
return cls(arg_, evaluate=False)
def _eval_derivative(self, s):
x, y = re(self.args[0]), im(self.args[0])
return (x * Derivative(y, s, evaluate=True) - y *
Derivative(x, s, evaluate=True)) / (x**2 + y**2)
def _eval_rewrite_as_atan2(self, arg):
x, y = re(self.args[0]), im(self.args[0])
return atan2(y, x)
def _eval_rewrite_as_sign(self, arg):
return -I*log(sign(arg))
class conjugate(Function):
"""Returns the complex conjugate of an argument.
In mathematics, the complex conjugate of a complex number
is given by changing the sign of the imaginary part.
Thus, the conjugate of the complex number
`a + i b` (where a and b are real numbers) is `a - i b`
Examples
========
>>> conjugate(2)
2
>>> conjugate(I)
-I
See Also
========
diofant.functions.elementary.complexes.sign
diofant.functions.elementary.complexes.Abs
References
==========
* https://en.wikipedia.org/wiki/Complex_conjugation
"""
@classmethod
def eval(cls, arg):
obj = arg._eval_conjugate()
if obj is not None:
return obj
def _eval_Abs(self):
return Abs(self.args[0], evaluate=True)
def _eval_adjoint(self):
return transpose(self.args[0])
def _eval_conjugate(self):
return self.args[0]
def _eval_derivative(self, s):
if s.is_extended_real:
return conjugate(Derivative(self.args[0], s, evaluate=True))
if s.is_imaginary:
return -conjugate(Derivative(self.args[0], s, evaluate=True))
def _eval_transpose(self):
return adjoint(self.args[0])
def _eval_is_algebraic(self):
return self.args[0].is_algebraic
class transpose(Function):
"""Linear map transposition."""
@classmethod
def eval(cls, arg):
obj = arg._eval_transpose()
if obj is not None:
return obj
def _eval_adjoint(self):
return conjugate(self.args[0])
def _eval_conjugate(self):
return adjoint(self.args[0])
def _eval_transpose(self):
return self.args[0]
class adjoint(Function):
"""Conjugate transpose or Hermite conjugation."""
@classmethod
def eval(cls, arg):
obj = arg._eval_adjoint()
if obj is not None:
return obj
obj = arg._eval_transpose()
if obj is not None:
return conjugate(obj)
def _eval_adjoint(self):
return self.args[0]
def _eval_conjugate(self):
return transpose(self.args[0])
def _eval_transpose(self):
return conjugate(self.args[0])
###############################################################################
# ############# HANDLING OF POLAR NUMBERS ################################### #
###############################################################################
class polar_lift(Function):
"""
Lift argument to the Riemann surface of the logarithm, using the
standard branch.
>>> p = Symbol('p', polar=True)
>>> polar_lift(4)
4*exp_polar(0)
>>> polar_lift(-4)
4*exp_polar(I*pi)
>>> polar_lift(-I)
exp_polar(-I*pi/2)
>>> polar_lift(I + 2)
polar_lift(2 + I)
>>> polar_lift(4*x)
4*polar_lift(x)
>>> polar_lift(4*p)
4*p
See Also
========
diofant.functions.elementary.exponential.exp_polar
diofant.functions.elementary.complexes.periodic_argument
"""
is_polar = True
is_comparable = False # Cannot be evalf'd.
@classmethod
def eval(cls, arg):
from .. import arg as argument
from .exponential import exp_polar
if arg.is_number and (arg.is_finite or arg.is_extended_real):
ar = argument(arg)
# In general we want to affirm that something is known,
# e.g. `not ar.has(argument) and not ar.has(atan)`
# but for now we will just be more restrictive and
# see that it has evaluated to one of the known values.
if ar in (0, pi/2, -pi/2, pi):
return exp_polar(I*ar)*abs(arg)
if arg.is_Mul:
args = arg.args
else:
args = [arg]
included = []
excluded = []
positive = []
for arg in args:
if arg.is_polar:
included += [arg]
elif arg.is_positive:
positive += [arg]
else:
excluded += [arg]
if len(excluded) < len(args):
if excluded:
return Mul(*(included + positive))*polar_lift(Mul(*excluded))
if included:
return Mul(*(included + positive))
return Mul(*positive)*exp_polar(0)
def _eval_evalf(self, prec):
"""Careful! any evalf of polar numbers is flaky."""
return self.args[0]._eval_evalf(prec)
def _eval_Abs(self):
return Abs(self.args[0], evaluate=True)
class periodic_argument(Function):
"""
Represent the argument on a quotient of the Riemann surface of the
logarithm. That is, given a period P, always return a value in
(-P/2, P/2], by using exp(P*I) == 1.
>>> unbranched_argument(exp(5*I*pi))
pi
>>> unbranched_argument(exp_polar(5*I*pi))
5*pi
>>> periodic_argument(exp_polar(5*I*pi), 2*pi)
pi
>>> periodic_argument(exp_polar(5*I*pi), 3*pi)
-pi
>>> periodic_argument(exp_polar(5*I*pi), pi)
0
See Also
========
diofant.functions.elementary.exponential.exp_polar
diofant.functions.elementary.complexes.polar_lift : Lift argument to the Riemann surface of the logarithm
diofant.functions.elementary.complexes.principal_branch
"""
@classmethod
def _getunbranched(cls, ar):
if ar.is_Mul:
args = ar.args
else:
args = [ar]
unbranched = 0
for a in args:
if not a.is_polar:
unbranched += arg(a)
elif isinstance(a, exp_polar):
unbranched += a.exp.as_real_imag()[1]
elif a.is_Pow:
re, im = a.exp.as_real_imag()
unbranched += re*unbranched_argument(
a.base) + im*log(abs(a.base))
elif isinstance(a, polar_lift):
unbranched += arg(a.args[0])
else:
return
return unbranched
@classmethod
def eval(cls, ar, period):
# Our strategy is to evaluate the argument on the Riemann surface of the
# logarithm, and then reduce.
# NOTE evidently this means it is a rather bad idea to use this with
# period != 2*pi and non-polar numbers.
from .integers import ceiling
from .trigonometric import atan, atan2
if not period.is_positive:
return
if period == oo and isinstance(ar, principal_branch):
return periodic_argument(*ar.args)
if isinstance(ar, polar_lift) and period >= 2*pi:
return periodic_argument(ar.args[0], period)
if ar.is_Mul:
newargs = [x for x in ar.args if not x.is_positive]
if len(newargs) != len(ar.args):
return periodic_argument(Mul(*newargs), period)
unbranched = cls._getunbranched(ar)
if unbranched is None:
return
if unbranched.has(periodic_argument, atan2, arg, atan):
return
if period == oo:
return unbranched
n = ceiling(unbranched/period - Rational(1, 2))*period
assert not n.has(ceiling)
return unbranched - n
def _eval_is_real(self):
if self.args[1].is_real and self.args[1].is_positive:
return True
def unbranched_argument(arg):
return periodic_argument(arg, oo)
class principal_branch(Function):
"""
Represent a polar number reduced to its principal branch on a quotient
of the Riemann surface of the logarithm.
This is a function of two arguments. The first argument is a polar
number `z`, and the second one a positive real number of infinity, `p`.
The result is "z mod exp_polar(I*p)".
>>> principal_branch(z, oo)
z
>>> principal_branch(exp_polar(2*pi*I)*3, 2*pi)
3*exp_polar(0)
>>> principal_branch(exp_polar(2*pi*I)*3*z, 2*pi)
3*principal_branch(z, 2*pi)
See Also
========
diofant.functions.elementary.exponential.exp_polar
diofant.functions.elementary.complexes.polar_lift : Lift argument to the Riemann surface of the logarithm
diofant.functions.elementary.complexes.periodic_argument
"""
is_polar = True
is_comparable = False # cannot always be evalf'd
@classmethod
def eval(cls, x, period):
if isinstance(x, polar_lift):
return principal_branch(x.args[0], period)
if period == oo:
return x
ub = periodic_argument(x, oo)
barg = periodic_argument(x, period)
if ub != barg and not ub.has(periodic_argument) \
and not barg.has(periodic_argument):
pl = polar_lift(x)
def mr(expr):
if not isinstance(expr, Symbol):
return polar_lift(expr)
return expr
pl = pl.replace(polar_lift, mr)
if not pl.has(polar_lift):
res = exp_polar(I*(barg - ub))*pl
if not res.is_polar and not res.has(exp_polar):
res *= exp_polar(0)
return res
if not x.free_symbols:
c, m = x, ()
else:
c, m = x.as_coeff_mul(*x.free_symbols)
others = []
for y in m:
if y.is_positive:
c *= y
else:
others += [y]
m = tuple(others)
arg = periodic_argument(c, period)
if arg.has(periodic_argument):
return
if arg.is_number and (unbranched_argument(c) != arg or
(arg == 0 and m and c != 1)):
if arg == 0:
return abs(c)*principal_branch(Mul(*m), period)
return principal_branch(exp_polar(I*arg)*Mul(*m), period)*abs(c)
if arg.is_number and ((abs(arg) - period/2).is_negative or arg == period/2) \
and not m:
return exp_polar(arg*I)*abs(c)
def _eval_evalf(self, prec):
from .exponential import exp
z, period = self.args
p = periodic_argument(z, period)._eval_evalf(prec)
if p is None or abs(p) > pi or p == -pi:
return self # Cannot evalf for this argument.
return (abs(z)*exp(I*p))._eval_evalf(prec)
def _polarify(eq, lift, pause=False):
from ...integrals import Integral
if isinstance(eq, Tuple):
return eq.func(*[_polarify(arg, lift, pause=False) for arg in eq.args])
if eq.is_polar:
return eq
if isinstance(eq, BooleanAtom):
return eq
if eq.is_number and not pause:
return polar_lift(eq)
if isinstance(eq, (Dummy, Symbol)) and not pause and lift:
return polar_lift(eq)
if eq.is_Atom:
return eq
if eq.is_Add:
r = eq.func(*[_polarify(arg, lift, pause=True) for arg in eq.args])
if lift:
return polar_lift(r)
return r
if eq.is_Function:
return eq.func(*[_polarify(arg, lift, pause=False) for arg in eq.args])
if eq.is_Exp:
return eq.func(eq.base, _polarify(eq.exp, lift, pause=False))
if isinstance(eq, Integral):
# Don't lift the integration variable
func = _polarify(eq.function, lift, pause=pause)
limits = []
for limit in eq.args[1:]:
var = _polarify(limit[0], lift=False, pause=pause)
rest = tuple(_polarify(x, lift=lift, pause=pause) for x in limit[1:])
limits.append((var,) + rest)
return Integral(*((func,) + tuple(limits)))
return eq.func(*[_polarify(arg, lift, pause=pause)
if isinstance(arg, Expr) else arg for arg in eq.args])
def polarify(eq, subs=True, lift=False):
"""
Turn all numbers in eq into their polar equivalents (under the standard
choice of argument).
Note that no attempt is made to guess a formal convention of adding
polar numbers, expressions like 1 + x will generally not be altered.
Note also that this function does not promote exp(x) to exp_polar(x).
If ``subs`` is True, all symbols which are not already polar will be
substituted for polar dummies; in this case the function behaves much
like posify.
If ``lift`` is True, both addition statements and non-polar symbols are
changed to their polar_lift()ed versions.
Note that lift=True implies subs=False.
>>> expr = (-x)**y
>>> expr.expand()
(-x)**y
>>> polarify(expr)[0]
(_x*exp_polar(I*pi))**_y
>>> sorted(polarify(expr)[1].items(), key=default_sort_key)
[(_x, x), (_y, y)]
>>> polarify(expr)[0].expand()
_x**_y*exp_polar(I*pi*_y)
>>> polarify(x, lift=True)
polar_lift(x)
>>> polarify(x*(1+y), lift=True)
polar_lift(x)*polar_lift(y + 1)
Adds are treated carefully:
>>> polarify(1 + sin((1 + I)*x))
(sin(_x*polar_lift(1 + I)) + 1, {_x: x})
"""
if lift: