def as_real_imag(self, deep=True, **hints): from sympy.polys.polytools import poly if self.exp.is_Integer: exp = self.exp re, im = self.base.as_real_imag(deep=deep) if not im: return self, S.Zero a, b = symbols('a b', cls=Dummy) if exp >= 0: if re.is_Number and im.is_Number: # We can be more efficient in this case expr = expand_multinomial(self.base**exp) return expr.as_real_imag() expr = poly( (a + b)**exp) # a = re, b = im; expr = (a + b*I)**exp else: mag = re**2 + im**2 re, im = re/mag, -im/mag if re.is_Number and im.is_Number: # We can be more efficient in this case expr = expand_multinomial((re + im*S.ImaginaryUnit)**-exp) return expr.as_real_imag() expr = poly((a + b)**-exp) # Terms with even b powers will be real r = [i for i in expr.terms() if not i[0][1] % 2] re_part = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r]) # Terms with odd b powers will be imaginary r = [i for i in expr.terms() if i[0][1] % 4 == 1] im_part1 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r]) r = [i for i in expr.terms() if i[0][1] % 4 == 3] im_part3 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r]) return (re_part.subs({a: re, b: S.ImaginaryUnit*im}), im_part1.subs({a: re, b: im}) + im_part3.subs({a: re, b: -im})) elif self.exp.is_Rational: # NOTE: This is not totally correct since for x**(p/q) with # x being imaginary there are actually q roots, but # only a single one is returned from here. re, im = self.base.as_real_imag(deep=deep) r = Pow(Pow(re, 2) + Pow(im, 2), S.Half) t = C.atan2(im, re) rp, tp = Pow(r, self.exp), t*self.exp return (rp*C.cos(tp), rp*C.sin(tp)) else: if deep: hints['complex'] = False expanded = self.expand(deep, **hints) if hints.get('ignore') == expanded: return None else: return (C.re(expanded), C.im(expanded)) else: return (C.re(self), C.im(self))