Source code for sympy.functions.elementary.complexes

from sympy.core import S, C
from sympy.core.function import Function, Derivative, ArgumentIndexError
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.core import Add, Mul
from sympy.core.relational import Eq

###############################################################################
######################### REAL and IMAGINARY PARTS ############################
###############################################################################

[docs]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. >>> from sympy import re, im, I, E >>> from sympy.abc import x, y >>> re(2*E) 2*E >>> re(2*I + 17) 17 >>> re(2*I) 0 >>> re(im(x) + x*I + 2) 2 See Also ======== im """ nargs = 1 is_real = True unbranched = True # implicitely works on the projection to C @classmethod def eval(cls, arg): if arg is S.NaN: return S.NaN elif arg.is_real: return arg elif arg.is_imaginary: return S.Zero elif arg.is_Function and arg.func == conjugate: return re(arg.args[0]) else: included, reverted, excluded = [], [], [] args = Add.make_args(arg) for term in args: coeff = term.as_coefficient(S.ImaginaryUnit) if coeff is not None: if not coeff.is_real: reverted.append(coeff) elif not term.has(S.ImaginaryUnit) and term.is_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
[docs] def as_real_imag(self, deep=True, **hints): """ Returns the real number with a zero complex part. """ return (self, S.Zero)
def _eval_derivative(self, x): if x.is_real or self.args[0].is_real: return re(Derivative(self.args[0], x, **{'evaluate': True})) if x.is_imaginary or self.args[0].is_imaginary: return -S.ImaginaryUnit \ * im(Derivative(self.args[0], x, **{'evaluate': 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 ======== >>> from sympy import re, im, E, I >>> from sympy.abc import x, y >>> im(2*E) 0 >>> re(2*I + 17) 17 >>> im(x*I) re(x) >>> im(re(x) + y) im(y) See Also ======== re """ nargs = 1 is_real = True unbranched = True # implicitely works on the projection to C @classmethod def eval(cls, arg): if arg is S.NaN: return S.NaN elif arg.is_real: return S.Zero elif arg.is_imaginary: return -S.ImaginaryUnit * arg elif arg.is_Function and arg.func == conjugate: return -im(arg.args[0]) else: included, reverted, excluded = [], [], [] args = Add.make_args(arg) for term in args: coeff = term.as_coefficient(S.ImaginaryUnit) if coeff is not None: if not coeff.is_real: reverted.append(coeff) else: excluded.append(coeff) elif term.has(S.ImaginaryUnit) or not term.is_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 ======== >>> from sympy.functions import im >>> from sympy import I >>> im(2 + 3*I).as_real_imag() (3, 0) """ return (self, S.Zero) def _eval_derivative(self, x): if x.is_real or self.args[0].is_real: return im(Derivative(self.args[0], x, **{'evaluate': True})) if x.is_imaginary or self.args[0].is_imaginary: return -S.ImaginaryUnit \ * re(Derivative(self.args[0], x, **{'evaluate': True})) ############################################################################### ############### SIGN, ABSOLUTE VALUE, ARGUMENT and CONJUGATION ################ ###############################################################################
[docs]class sign(Function): """ Returns the sign of an expression, that is: * 1 if expression is positive * 0 if expression is equal to zero * -1 if expression is negative Examples ======== >>> from sympy.functions import sign >>> sign(-1) -1 >>> sign(0) 0 See Also ======== Abs, conjugate """ nargs = 1 def doit(self): if self.args[0].is_nonzero: return self.args[0] / Abs(self.args[0]) return self @classmethod def eval(cls, arg): if arg is S.NaN: return S.NaN if arg is S.Zero: return S.Zero if arg.is_positive: return S.One if arg.is_negative: return S.NegativeOne if arg.is_Function: if arg.func is sign: return arg if arg.is_imaginary: arg2 = -S.ImaginaryUnit * arg if arg2.is_positive: return S.ImaginaryUnit if arg2.is_negative: return -S.ImaginaryUnit if arg.is_Mul: c, args = arg.as_coeff_mul() unk = [] is_imag = c.is_imaginary is_neg = c.is_negative for ai in args: ai2 = -S.ImaginaryUnit * ai if ai.is_negative: is_neg = not is_neg elif ai.is_imaginary and ai2.is_positive: is_imag = not is_imag elif ai.is_negative is None or \ (ai.is_imaginary is None or ai2.is_positive is None): unk.append(ai) if c is S.One and len(unk) == len(args): return None return (S.NegativeOne if is_neg else S.One) \ * (S.ImaginaryUnit if is_imag else S.One) \ * cls(arg._new_rawargs(*unk)) is_bounded = True def _eval_Abs(self): if self.args[0].is_nonzero: return S.One def _eval_conjugate(self): return sign(conjugate(self.args[0])) def _eval_derivative(self, x): if self.args[0].is_real: from sympy.functions.special.delta_functions import DiracDelta return 2 * Derivative(self.args[0], x, **{'evaluate': True}) \ * DiracDelta(self.args[0]) elif self.args[0].is_imaginary: from sympy.functions.special.delta_functions import DiracDelta return 2 * Derivative(self.args[0], x, **{'evaluate': True}) \ * DiracDelta(-S.ImaginaryUnit * self.args[0]) def _eval_is_zero(self): return self.args[0].is_zero def _eval_power(self, other): if ( self.args[0].is_real and self.args[0].is_nonzero and other.is_integer and other.is_even ): return S.One def _sage_(self): import sage.all as sage return sage.sgn(self.args[0]._sage_())
[docs]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 SymPy expression to the built-in abs(), it will pass it automatically to Abs(). Examples ======== >>> from sympy import Abs, Symbol, S >>> Abs(-1) 1 >>> x = Symbol('x', real=True) >>> Abs(-x) Abs(x) >>> Abs(x**2) x**2 >>> abs(-x) # The Python built-in Abs(x) Note that the Python built-in will return either an Expr or int depending on the argument:: >>> type(abs(-1)) <... 'int'> >>> type(abs(S.NegativeOne)) <class 'sympy.core.numbers.One'> Abs will always return a sympy object. See Also ======== sign, conjugate """ nargs = 1 is_real = True is_negative = False unbranched = True
[docs] def fdiff(self, argindex=1): """ Get the first derivative of the argument to Abs(). Examples ======== >>> from sympy.abc import x >>> from sympy.functions import Abs >>> Abs(-x).fdiff() sign(x) """ if argindex == 1: return sign(self.args[0]) else: raise ArgumentIndexError(self, argindex)
@classmethod def eval(cls, arg): if hasattr(arg, '_eval_Abs'): obj = arg._eval_Abs() if obj is not None: return obj if arg.is_Mul: known = [] unk = [] for t in arg.args: tnew = cls(t) if tnew.func is cls: unk.append(tnew.args[0]) else: known.append(tnew) known = Mul(*known) unk = cls(Mul(*unk), evaluate=False) if unk else S.One return known*unk if arg is S.NaN: return S.NaN if arg.is_nonnegative: return arg if arg.is_nonpositive: return -arg if arg.is_imaginary: arg2 = -S.ImaginaryUnit * arg if arg2.is_nonnegative: return arg2 if arg.is_real is False and arg.is_imaginary is False: from sympy import expand_mul return sqrt( expand_mul(arg * arg.conjugate()) ) if arg.is_Pow: base, exponent = arg.as_base_exp() if exponent.is_even and base.is_real: return arg def _eval_is_nonzero(self): return self._args[0].is_nonzero def _eval_is_positive(self): return self.is_nonzero def _eval_power(self, other): if self.args[0].is_real and other.is_integer: if other.is_even: return self.args[0]**other elif other is not S.NegativeOne and other.is_Integer: e = other - sign(other) return self.args[0]**e*self return def _eval_nseries(self, x, n, logx): direction = self.args[0].leadterm(x)[0] s = self.args[0]._eval_nseries(x, n=n, logx=logx) when = Eq(direction, 0) return Piecewise( ((s.subs(direction, 0)), when), (sign(direction)*s, True), ) def _sage_(self): import sage.all as sage return sage.abs_symbolic(self.args[0]._sage_()) def _eval_derivative(self, x): if self.args[0].is_real or self.args[0].is_imaginary: return Derivative(self.args[0], x, **{'evaluate': True}) \ * sign(conjugate(self.args[0])) return (re(self.args[0]) * Derivative(re(self.args[0]), x, **{'evaluate': True}) + im(self.args[0]) * Derivative(im(self.args[0]), x, **{'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). if arg.is_real: return arg*(C.Heaviside(arg) - C.Heaviside(-arg)) else: return self
[docs]class arg(Function): """Returns the argument (in radians) of a complex number""" nargs = 1 is_real = True is_bounded = True @classmethod def eval(cls, arg): x, y = re(arg), im(arg) arg = C.atan2(y, x) if arg.is_number: return arg def _eval_derivative(self, t): x, y = re(self.args[0]), im(self.args[0]) return (x * Derivative(y, t, **{'evaluate': True}) - y * Derivative(x, t, **{'evaluate': True})) / (x**2 + y**2)
[docs]class conjugate(Function): """ Changes the sign of the imaginary part of a complex number. Examples ======== >>> from sympy import conjugate, I >>> conjugate(1 + I) 1 - I See Also ======== sign, Abs """ nargs = 1 @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, x): if x.is_real: return conjugate(Derivative(self.args[0], x, **{'evaluate': True})) elif x.is_imaginary: return -conjugate(Derivative(self.args[0], x, **{'evaluate': True})) def _eval_transpose(self): return conjugate(transpose(self.args[0]))
class transpose(Function): """ Linear map transposition. """ nargs = 1 @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_derivative(self, x): return transpose(Derivative(self.args[0], x, **{'evaluate': True})) def _eval_transpose(self): return self.args[0] class adjoint(Function): """ Conjugate transpose or Hermite conjugation. """ nargs = 1 @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_derivative(self, x): return adjoint(Derivative(self.args[0], x, **{'evaluate': True})) def _eval_transpose(self): return conjugate(self.args[0]) def _latex(self, printer, exp=None, *args): arg = printer._print(self.args[0]) tex = r'%s^{\dag}' % arg if exp: tex = r'\left(%s\right)^{%s}' % (tex, printer._print(exp)) return tex def _pretty(self, printer, *args): from sympy.printing.pretty.stringpict import prettyForm pform = printer._print(self.args[0], *args) if printer._use_unicode: pform = pform**prettyForm('\u2020') else: pform = pform**prettyForm('+') return pform ############################################################################### ############### HANDLING OF POLAR NUMBERS ##################################### ############################################################################### class polar_lift(Function): """ Lift argument to the riemann surface of the logarithm, using the standard branch. >>> from sympy import Symbol, polar_lift, I >>> p = Symbol('p', polar=True) >>> x = Symbol('x') >>> 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 ======== sympy.functions.elementary.exponential.exp_polar periodic_argument """ nargs = 1 is_polar = True is_comparable = False # Cannot be evalf'd. @classmethod def eval(cls, arg): from sympy import exp_polar, pi, I, arg as argument if arg.is_number: ar = argument(arg) #if not ar.has(argument) and not ar.has(atan): 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)) elif included: return Mul(*(included + positive)) else: 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) 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. >>> from sympy import exp, exp_polar, periodic_argument, unbranched_argument >>> from sympy import I, pi >>> 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 ======== sympy.functions.elementary.exponential.exp_polar polar_lift : Lift argument to the riemann surface of the logarithm principal_branch """ nargs = 2 @classmethod def _getunbranched(cls, ar): from sympy import exp_polar, log, polar_lift 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 a.func is 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 a.func is polar_lift: unbranched += arg(a.args[0]) else: return None 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 sympy import ceiling, oo, atan2, atan, polar_lift, pi, Mul if not period.is_positive: return None if period == oo and isinstance(ar, principal_branch): return periodic_argument(*ar.args) if ar.func is 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 None if unbranched.has(periodic_argument, atan2, arg, atan): return None if period == oo: return unbranched if period != oo: n = ceiling(unbranched/period - S(1)/2)*period if not n.has(ceiling): return unbranched - n def _eval_evalf(self, prec): from sympy import ceiling, oo z, period = self.args if period == oo: unbranched = periodic_argument._getunbranched(z) if unbranched is None: return self return unbranched._eval_evalf(prec) ub = periodic_argument(z, oo)._eval_evalf(prec) return (ub - ceiling(ub/period - S(1)/2)*period)._eval_evalf(prec) def unbranched_argument(arg): from sympy import oo 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)". >>> from sympy import exp_polar, principal_branch, oo, I, pi >>> from sympy.abc import z >>> 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 ======== sympy.functions.elementary.exponential.exp_polar polar_lift : Lift argument to the riemann surface of the logarithm periodic_argument """ nargs = 2 is_polar = True is_comparable = False # cannot always be evalf'd @classmethod def eval(self, x, period): from sympy import oo, exp_polar, I, Mul, polar_lift, Symbol 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 None 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 True or arg == period/2) \ and m == (): return exp_polar(arg*I)*abs(c) def _eval_evalf(self, prec): from sympy import exp, pi, I z, period = self.args p = periodic_argument(z, period)._eval_evalf(prec) if abs(p) > pi or p == -pi: return self # Cannot evalf for this argument. return (abs(z)*exp(I*p))._eval_evalf(prec) # /cyclic/ from sympy.core import basic as _ _.abs_ = Abs del _