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 """ nargs = 1 is_real = True @classmethod def eval(cls, arg): if arg is S.NaN: return S.NaN elif arg.is_real: return arg elif arg.is_Function and arg.func == conjugate: return re(arg.args[0]) else: included, reverted, excluded = [], [], [] arg = Add.make_args(arg) for term in arg: 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: included.append(term) if len(arg) != len(included): a, b, c = map(lambda xs: Add(*xs), [included, reverted, excluded]) return cls(a) - im(b) + c def _eval_conjugate(self): return self def as_real_imag(self, deep=True): return (self, S.Zero) def _eval_expand_complex(self, deep=True, **hints): # if deep: # return self.args[0].expand(deep, **hints).as_real_imag()[0] # else: return self.args[0].as_real_imag()[0] def _eval_derivative(self, x): return re(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. >>> 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) """ nargs = 1 is_real = True @classmethod def eval(cls, arg): if arg is S.NaN: return S.NaN elif arg.is_real: return S.Zero elif arg.is_Function and arg.func == conjugate: return -im(arg.args[0]) else: included, reverted, excluded = [], [], [] arg = Add.make_args(arg) for term in arg: 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: included.append(term) if len(arg) != len(included): a, b, c = map(lambda xs: Add(*xs), [included, reverted, excluded]) return cls(a) + re(b) + c def _eval_conjugate(self): return self def as_real_imag(self, deep=True): return (self, S.Zero) def _eval_expand_complex(self, deep=True, **hints): # if deep: # return self.args[0].expand(deep, **hints).as_real_imag()[1] return self.args[0].as_real_imag()[1] def _eval_derivative(self, x): return im(Derivative(self.args[0], x, **{'evaluate': True})) ############################################################################### ############### SIGN, ABSOLUTE VALUE, ARGUMENT and CONJUGATION ################ ###############################################################################
[docs]class sign(Function): """Return the sign of an expression, that is: -1 if expr < 0 0 if expr == 0 1 if expr > 0 """ nargs = 1 @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_Mul: c, args = arg.as_coeff_mul() unk = [] is_neg = c.is_negative for ai in args: if ai.is_negative is None: unk.append(ai) elif ai.is_negative: is_neg = not is_neg if c is S.One and len(unk) == len(args): return None return (S.NegativeOne if is_neg else S.One) * cls(arg._new_rawargs(*unk)) is_bounded = True def _eval_derivative(self, x): return S.Zero def _eval_conjugate(self): return self def _eval_is_zero(self): return (self.args[0] is S.Zero) 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)) <type 'int'> >>> type(abs(S.NegativeOne)) <class 'sympy.core.numbers.One'> Abs will always return a sympy object. """ nargs = 1 is_real = True is_negative = False def fdiff(self, argindex=1): if argindex == 1: return sign(self.args[0]) else: raise ArgumentIndexError(self, argindex) @classmethod def eval(cls, arg): if arg is S.NaN: return S.NaN if arg.is_zero: return arg if arg.is_positive: return arg if arg.is_negative: return -arg coeff, terms = arg.as_coeff_mul() if coeff is not S.One: return cls(coeff) * cls(Mul(*terms)) if arg.is_real is False: return sqrt( (arg * arg.conjugate()).expand() ) if arg.is_Pow: base, exponent = arg.as_base_exp() if exponent.is_even and base.is_real: return arg return def _eval_is_nonzero(self): return self._args[0].is_nonzero def _eval_is_positive(self): return self.is_nonzero def _eval_conjugate(self): return self def _eval_power(self,other): if self.args[0].is_real and other.is_integer: if other.is_even: return self.args[0]**other 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: return Derivative(self.args[0], x, **{'evaluate': True}) * sign(self.args[0]) return (re(self.args[0]) * re(Derivative(self.args[0], x, **{'evaluate': True})) + im(self.args[0]) * im(Derivative(self.args[0], x, **{'evaluate': True}))) / Abs(self.args[0])
[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_conjugate(self): return self 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. >>> from sympy import conjugate, I >>> conjugate(1 + I) 1 - I """ nargs = 1 @classmethod def eval(cls, arg): obj = arg._eval_conjugate() if obj is not None: return obj def _eval_conjugate(self): return self.args[0] def _eval_derivative(self, x): return conjugate(Derivative(self.args[0], x, **{'evaluate': True})) # /cyclic/
from sympy.core import basic as _ _.abs_ = Abs del _