def _element_constructor_(self, x):
"""
EXAMPLES::
sage: CC((1,2))
1.00000000000000 + 2.00000000000000*I
"""
if not isinstance(x, (real_mpfr.RealNumber, tuple)):
if isinstance(x, complex_double.ComplexDoubleElement):
return complex_number.ComplexNumber(self, x.real(), x.imag())
elif isinstance(x, str):
# TODO: this is probably not the best and most
# efficient way to do this. -- Martin Albrecht
return complex_number.ComplexNumber(self,
sage_eval(x.replace(' ',''), locals={"I":self.gen(),"i":self.gen()}))
late_import()
if isinstance(x, NumberFieldElement_quadratic) and list(x.parent().polynomial()) == [1, 0, 1]:
(re, im) = list(x)
return complex_number.ComplexNumber(self, re, im)
try:
return self(x.sage())
except:
pass
try:
return x._complex_mpfr_field_( self )
except AttributeError:
pass
return complex_number.ComplexNumber(self, x)
def zeta(self, n=2):
"""
Return a primitive `n`-th root of unity.
INPUT:
- ``n`` - an integer (default: 2)
OUTPUT: a complex `n`-th root of unity.
EXAMPLES::
sage: C = ComplexField()
sage: C.zeta(2)
-1.00000000000000
sage: C.zeta(5)
0.309016994374947 + 0.951056516295154*I
"""
from integer import Integer
n = Integer(n)
if n == 1:
x = self(1)
elif n == 2:
x = self(-1)
elif n >= 3:
# Use De Moivre
# e^(2*pi*i/n) = cos(2pi/n) + i *sin(2pi/n)
RR = self._real_field()
pi = RR.pi()
z = 2 * pi / n
x = complex_number.ComplexNumber(self, z.cos(), z.sin())
x._set_multiplicative_order(n)
return x
def zeta(self, n=2):
"""
Return a primitive `n`-th root of unity.
INPUT:
- ``n`` - an integer (default: 2)
OUTPUT: a complex n-th root of unity.
"""
from integer import Integer
n = Integer(n)
if n == 1:
x = self(1)
elif n == 2:
x = self(-1)
elif n >= 3:
# Use De Moivre
# e^(2*pi*i/n) = cos(2pi/n) + i *sin(2pi/n)
RR = self._real_field()
pi = RR.pi()
z = 2*pi/n
x = complex_number.ComplexNumber(self, z.cos(), z.sin())
x._set_multiplicative_order( n )
return x
def _element_constructor_(self, x):
"""
Construct a complex number.
EXAMPLES::
sage: CC((1,2)) # indirect doctest
1.00000000000000 + 2.00000000000000*I
Check that :trac:`14989` is fixed::
sage: QQi = NumberField(x^2+1, 'i', embedding=CC(0,1))
sage: i = QQi.order(QQi.gen()).gen(1)
sage: CC(i)
1.00000000000000*I
"""
if not isinstance(x, (real_mpfr.RealNumber, tuple)):
if isinstance(x, complex_double.ComplexDoubleElement):
return complex_number.ComplexNumber(self, x.real(), x.imag())
elif isinstance(x, str):
# TODO: this is probably not the best and most
# efficient way to do this. -- Martin Albrecht
return complex_number.ComplexNumber(
self,
sage_eval(x.replace(' ', ''),
locals={
"I": self.gen(),
"i": self.gen()
}))
late_import()
if isinstance(x, NumberFieldElement_quadratic):
if isinstance(x.parent(), NumberField_quadratic) and list(
x.parent().polynomial()) == [1, 0, 1]:
(re, im) = list(x)
return complex_number.ComplexNumber(self, re, im)
try:
return self(x.sage())
except (AttributeError, TypeError):
pass
try:
return x._complex_mpfr_field_(self)
except AttributeError:
pass
return complex_number.ComplexNumber(self, x)
def gen(self, n=0):
"""
Return the generator of the complex field.
EXAMPLES::
sage: ComplexField().gen(0)
1.00000000000000*I
"""
if n != 0:
raise IndexError("n must be 0")
return complex_number.ComplexNumber(self, 0, 1)
def gen(self, n=0):
if n != 0:
raise IndexError, "n must be 0"
return complex_number.ComplexNumber(self, 0, 1)