"""
Arbitrary Precision Complex Numbers
AUTHORS:
- William Stein (2006-01-26): complete rewrite
- Joel B. Mohler (2006-12-16): naive rewrite into pyrex
- William Stein(2007-01): rewrite of Mohler's rewrite
- Vincent Delecroix (2010-01): plot function
"""
import math
import operator
from sage.structure.element cimport FieldElement, RingElement, Element, ModuleElement
from sage.categories.map cimport Map
from complex_double cimport ComplexDoubleElement
from real_mpfr cimport RealNumber
import complex_field
import sage.misc.misc
import integer
import infinity
from sage.libs.mpmath.utils cimport mpfr_to_mpfval
include "../ext/stdsage.pxi"
cdef object numpy_complex_interface = {'typestr': '=c16'}
cdef object numpy_object_interface = {'typestr': '|O'}
cdef mp_rnd_t rnd
rnd = GMP_RNDN
cdef double LOG_TEN_TWO_PLUS_EPSILON = 3.321928094887363
def set_global_complex_round_mode(n):
global rnd
rnd = n
def is_ComplexNumber(x):
r"""
Returns True if x is a complex number. In particular, if x is of
the ``ComplexNumber`` type.
EXAMPLES::
sage: from sage.rings.complex_number import is_ComplexNumber
sage: a = ComplexNumber(1,2); a
1.00000000000000 + 2.00000000000000*I
sage: is_ComplexNumber(a)
True
sage: b = ComplexNumber(1); b
1.00000000000000
sage: is_ComplexNumber(b)
True
Note that the global element I is of type
``SymbolicConstant``. However, elements of the class
``ComplexField_class`` are of type
``ComplexNumber``::
sage: c = 1 + 2*I
sage: is_ComplexNumber(c)
False
sage: d = CC(1 + 2*I)
sage: is_ComplexNumber(d)
True
"""
return isinstance(x, ComplexNumber)
cdef class ComplexNumber(sage.structure.element.FieldElement):
"""
A floating point approximation to a complex number using any
specified precision. Answers derived from calculations with such
approximations may differ from what they would be if those
calculations were performed with true complex numbers. This is due
to the rounding errors inherent to finite precision calculations.
EXAMPLES::
sage: I = CC.0
sage: b = 1.5 + 2.5*I
sage: loads(b.dumps()) == b
True
"""
cdef ComplexNumber _new(self):
"""
Quickly creates a new initialized complex number with the same
parent as self.
"""
cdef ComplexNumber x
x = PY_NEW(ComplexNumber)
x._parent = self._parent
x._prec = self._prec
x._multiplicative_order = None
mpfr_init2(x.__re, self._prec)
mpfr_init2(x.__im, self._prec)
return x
def __cinit__(self, parent=None, real=None, imag=None):
self._prec = -1
def __init__(self, parent, real, imag=None):
r"""
Initialize ``ComplexNumber`` instance.
EXAMPLES::
sage: a = ComplexNumber(2,1)
sage: a.__init__(CC,2,1)
sage: a
2.00000000000000 + 1.00000000000000*I
sage: parent(a)
Complex Field with 53 bits of precision
sage: real(a)
2.00000000000000
sage: imag(a)
1.00000000000000
"""
cdef real_mpfr.RealNumber rr, ii
self._parent = parent
self._prec = self._parent._prec
self._multiplicative_order = None
mpfr_init2(self.__re, self._prec)
mpfr_init2(self.__im, self._prec)
if imag is None:
if real is None: return
if PY_TYPE_CHECK(real, ComplexNumber):
real, imag = (<ComplexNumber>real).real(), (<ComplexNumber>real).imag()
elif isinstance(real, sage.libs.pari.all.pari_gen):
real, imag = real.real(), real.imag()
elif isinstance(real, list) or isinstance(real, tuple):
re, imag = real
real = re
elif isinstance(real, complex):
real, imag = real.real, real.imag
else:
imag = 0
try:
R = parent._real_field()
rr = R(real)
ii = R(imag)
mpfr_set(self.__re, rr.value, rnd)
mpfr_set(self.__im, ii.value, rnd)
except TypeError:
raise TypeError, "unable to coerce to a ComplexNumber: %s" % type(real)
def __dealloc__(self):
"""
TESTS:
Check that #12038 is resolved::
sage: from sage.rings.complex_number import ComplexNumber as CN
sage: coerce(CN, 1+I)
Traceback (most recent call last):
...
TypeError: __init__() takes at least 2 positional arguments (1 given)
"""
if self._prec != -1:
mpfr_clear(self.__re)
mpfr_clear(self.__im)
def _interface_init_(self, I=None):
"""
Returns self formatted as a string, suitable as input to another
computer algebra system. (This is the default function used for
exporting to other computer algebra systems.)
EXAMPLES::
sage: s1 = CC(exp(I)); s1
0.540302305868140 + 0.841470984807897*I
sage: s1._interface_init_()
'0.54030230586813977 + 0.84147098480789650*I'
sage: s1 == CC(gp(s1))
True
"""
return self.str(truncate=False)
property __array_interface__:
def __get__(self):
"""
Used for NumPy conversion.
EXAMPLES::
sage: import numpy
sage: numpy.array([1.0, 2.5j]).dtype
dtype('complex128')
sage: numpy.array([1.000000000000000000000000000000000000j]).dtype
dtype('object')
"""
if self._prec <= 57:
return numpy_complex_interface
else:
return numpy_object_interface
def _sage_input_(self, sib, coerced):
r"""
Produce an expression which will reproduce this value when evaluated.
EXAMPLES:
sage: for prec in (2, 53, 200):
... fld = ComplexField(prec)
... var = polygen(fld)
... ins = [-20, 0, 1, -2^4000, 2^-4000] + [fld._real_field().random_element() for _ in range(3)]
... for v1 in ins:
... for v2 in ins:
... v = fld(v1, v2)
... _ = sage_input(fld(v), verify=True)
... _ = sage_input(fld(v) * var, verify=True)
sage: x = polygen(CC)
sage: for v1 in [-2, 0, 2]:
... for v2 in [-2, -1, 0, 1, 2]:
... print str(sage_input(x + CC(v1, v2))).splitlines()[1]
x + CC(-2 - RR(2)*I)
x + CC(-2 - RR(1)*I)
x - 2
x + CC(-2 + RR(1)*I)
x + CC(-2 + RR(2)*I)
x - CC(RR(2)*I)
x - CC(RR(1)*I)
x
x + CC(RR(1)*I)
x + CC(RR(2)*I)
x + CC(2 - RR(2)*I)
x + CC(2 - RR(1)*I)
x + 2
x + CC(2 + RR(1)*I)
x + CC(2 + RR(2)*I)
sage: from sage.misc.sage_input import SageInputBuilder
sage: sib = SageInputBuilder()
sage: sib_np = SageInputBuilder(preparse=False)
sage: CC(-infinity)._sage_input_(sib, True)
{unop:- {call: {atomic:RR}({atomic:Infinity})}}
sage: CC(0, infinity)._sage_input_(sib, True)
{call: {atomic:CC}({call: {atomic:RR}({atomic:0})}, {call: {atomic:RR}({atomic:Infinity})})}
sage: CC(NaN, 5)._sage_input_(sib, True)
{call: {atomic:CC}({call: {atomic:RR}({atomic:NaN})}, {call: {atomic:RR}({atomic:5})})}
sage: CC(5, NaN)._sage_input_(sib, True)
{call: {atomic:CC}({call: {atomic:RR}({atomic:5})}, {call: {atomic:RR}({atomic:NaN})})}
sage: CC(12345)._sage_input_(sib, True)
{atomic:12345}
sage: CC(-12345)._sage_input_(sib, False)
{call: {atomic:CC}({binop:+ {unop:- {atomic:12345}} {binop:* {call: {atomic:RR}({atomic:0})} {atomic:I}}})}
sage: CC(0, 12345)._sage_input_(sib, True)
{call: {atomic:CC}({binop:* {call: {atomic:RR}({atomic:12345})} {atomic:I}})}
sage: CC(0, -12345)._sage_input_(sib, False)
{unop:- {call: {atomic:CC}({binop:* {call: {atomic:RR}({atomic:12345})} {atomic:I}})}}
sage: CC(1.579)._sage_input_(sib, True)
{atomic:1.579}
sage: CC(1.579)._sage_input_(sib_np, True)
{atomic:1.579}
sage: ComplexField(150).zeta(37)._sage_input_(sib, True)
{call: {call: {atomic:ComplexField}({atomic:150})}({binop:+ {atomic:0.98561591034770846226477029397621845736859851519} {binop:* {call: {call: {atomic:RealField}({atomic:150})}({atomic:0.16900082032184907409303555538443060626072476297})} {atomic:I}}})}
sage: ComplexField(150).zeta(37)._sage_input_(sib_np, True)
{call: {call: {atomic:ComplexField}({atomic:150})}({binop:+ {call: {call: {atomic:RealField}({atomic:150})}({atomic:'0.98561591034770846226477029397621845736859851519'})} {binop:* {call: {call: {atomic:RealField}({atomic:150})}({atomic:'0.16900082032184907409303555538443060626072476297'})} {atomic:I}}})}
"""
if coerced and self.imag() == 0:
return sib(self.real(), True)
if not (mpfr_number_p(self.__re) and mpfr_number_p(self.__im)):
return sib(self.parent())(self.real(), self.imag())
real_part = sib(self.real(), 2)
imag_part = sib.prod([sib(self.imag()), sib.name('I')],
simplify=True)
sum = sib.sum([real_part, imag_part], simplify=True)
if sum._sie_is_negation():
return -sib(self.parent())(sum._sie_operand)
else:
return sib(self.parent())(sum)
def _repr_(self):
r"""
Returns self formatted as a string.
EXAMPLES::
sage: a = ComplexNumber(2,1); a
2.00000000000000 + 1.00000000000000*I
sage: a._repr_()
'2.00000000000000 + 1.00000000000000*I'
"""
return self.str(10)
def __hash__(self):
"""
Returns the hash of self, which coincides with the python complex
and float (and often int) types.
This has the drawback that two very close high precision numbers
will have the same hash, but allows them to play nicely with other
real types.
EXAMPLE::
sage: hash(CC(1.2, 33)) == hash(complex(1.2, 33))
True
"""
return hash(complex(self))
def __getitem__(self, i):
r"""
Returns either the real or imaginary component of self depending on
the choice of i: real (i=0), imaginary (i=1)
INPUTS:
- ``i`` - 0 or 1
- ``0`` - will return the real component of self
- ``1`` - will return the imaginary component of self
EXAMPLES::
sage: a = ComplexNumber(2,1)
sage: a.__getitem__(0)
2.00000000000000
sage: a.__getitem__(1)
1.00000000000000
::
sage: b = CC(42,0)
sage: b
42.0000000000000
sage: b.__getitem__(1)
0.000000000000000
"""
if i == 0:
return self.real()
elif i == 1:
return self.imag()
raise IndexError, "i must be between 0 and 1."
def __reduce__( self ):
"""
Pickling support
EXAMPLES::
sage: a = CC(1 + I)
sage: loads(dumps(a)) == a
True
"""
return (make_ComplexNumber0, (self._parent, self._multiplicative_order, self.real(), self.imag()))
def _set_multiplicative_order(self, n):
r"""
Function for setting the ``ComplexNumber`` class
attribute ``multiplicative_order`` of self.
INPUTS: n - an integer which will define the multiplicative order
of self
EXAMPLES: Note that it is not advisable to explicitly call
``_set_multiplicative_order`` for explicitly declared
complex numbers.
::
sage: a = ComplexNumber(2,1)
sage: a.multiplicative_order()
+Infinity
sage: a._set_multiplicative_order(5)
sage: a.multiplicative_order()
5
sage: a^5
-38.0000000000000 + 41.0000000000000*I
"""
self._multiplicative_order = integer.Integer(n)
def str(self, base=10, truncate=True):
r"""
Return a string representation of this number.
INPUTS:
- ``base`` - The base to use for printing (default 10)
- ``truncate`` - (default: ``True``) Whether to print fewer
digits than are available, to mask errors in the last bits.
EXAMPLES::
sage: a = CC(pi + I*e)
sage: a.str()
'3.14159265358979 + 2.71828182845905*I'
sage: a.str(truncate=False)
'3.1415926535897931 + 2.7182818284590451*I'
sage: a.str(base=2)
'11.001001000011111101101010100010001000010110100011000 + 10.101101111110000101010001011000101000101011101101001*I'
sage: CC(0.5 + 0.625*I).str(base=2)
'0.10000000000000000000000000000000000000000000000000000 + 0.10100000000000000000000000000000000000000000000000000*I'
sage: a.str(base=16)
'3.243f6a8885a30 + 2.b7e151628aed2*I'
sage: a.str(base=36)
'3.53i5ab8p5fc + 2.puw5nggjf8f*I'
sage: CC(0)
0.000000000000000
"""
s = ""
if self.real() != 0:
s = self.real().str(base, truncate=truncate)
if self.imag() != 0:
y = self.imag()
if s!="":
if y < 0:
s = s+" - "
y = -y
else:
s = s+" + "
s = s+"%s*I"%y.str(base, truncate=truncate)
if len(s) == 0:
s = self.real().str(base, truncate=truncate)
return s
def _latex_(self):
r"""
Method for converting self to a string with latex formatting.
Called by the global function ``latex``.
EXAMPLES::
sage: a = ComplexNumber(2,1)
sage: a
2.00000000000000 + 1.00000000000000*I
sage: latex(a)
2.00000000000000 + 1.00000000000000i
sage: a._latex_()
'2.00000000000000 + 1.00000000000000i'
::
sage: b = ComplexNumber(7,4,min_prec=16)
sage: b
7.000 + 4.000*I
sage: latex(b)
7.000 + 4.000i
sage: b._latex_()
'7.000 + 4.000i'
::
sage: ComplexNumber(0).log()._latex_()
'-\\infty'
"""
import re
s = self.str().replace('*I', 'i').replace('infinity','\\infty')
return re.sub(r"e(-?\d+)", r" \\times 10^{\1}", s)
def _pari_(self):
r"""
Coerces self into a Pari ``complex`` object.
EXAMPLES: Coerce the object using the ``pari``
function::
sage: a = ComplexNumber(2,1)
sage: pari(a)
2.00000000000000 + 1.00000000000000*I
sage: type(pari(a))
<type 'sage.libs.pari.gen.gen'>
sage: a._pari_()
2.00000000000000 + 1.00000000000000*I
sage: type(a._pari_())
<type 'sage.libs.pari.gen.gen'>
"""
return sage.libs.pari.all.pari.complex(self.real()._pari_(), self.imag()._pari_())
def _mpmath_(self, prec=None, rounding=None):
"""
Returns an mpmath version of this ComplexNumber.
.. note::
Currently, the rounding mode is ignored.
EXAMPLES::
sage: CC(1,2)._mpmath_()
mpc(real='1.0', imag='2.0')
"""
if prec is not None:
return self.n(prec=prec)._mpmath_()
from sage.libs.mpmath.all import make_mpc
re = mpfr_to_mpfval(self.__re)
im = mpfr_to_mpfval(self.__im)
return make_mpc((re, im))
cpdef ModuleElement _add_(self, ModuleElement right):
cdef ComplexNumber x
x = self._new()
mpfr_add(x.__re, self.__re, (<ComplexNumber>right).__re, rnd)
mpfr_add(x.__im, self.__im, (<ComplexNumber>right).__im, rnd)
return x
cpdef ModuleElement _sub_(self, ModuleElement right):
cdef ComplexNumber x
x = self._new()
mpfr_sub(x.__re, self.__re, (<ComplexNumber>right).__re, rnd)
mpfr_sub(x.__im, self.__im, (<ComplexNumber>right).__im, rnd)
return x
cpdef RingElement _mul_(self, RingElement right):
cdef ComplexNumber x
x = self._new()
cdef mpfr_t t0, t1
mpfr_init2(t0, self._prec)
mpfr_init2(t1, self._prec)
mpfr_mul(t0, self.__re, (<ComplexNumber>right).__re, rnd)
mpfr_mul(t1, self.__im, (<ComplexNumber>right).__im, rnd)
mpfr_sub(x.__re, t0, t1, rnd)
mpfr_mul(t0, self.__re, (<ComplexNumber>right).__im, rnd)
mpfr_mul(t1, self.__im, (<ComplexNumber>right).__re, rnd)
mpfr_add(x.__im, t0, t1, rnd)
mpfr_clear(t0)
mpfr_clear(t1)
return x
def norm(self):
r"""
Returns the norm of this complex number. If `c = a + bi` is a
complex number, then the norm of `c` is defined as the product of
`c` and its complex conjugate
.. MATH::
\text{norm}(c)
=
\text{norm}(a + bi)
=
c \cdot \overline{c}
=
a^2 + b^2.
The norm of a complex number is different from its absolute value.
The absolute value of a complex number is defined to be the square
root of its norm. A typical use of the complex norm is in the
integral domain `\ZZ[i]` of Gaussian integers, where the norm of
each Gaussian integer `c = a + bi` is defined as its complex norm.
.. SEEALSO::
- :func:`sage.misc.functional.norm`
- :meth:`sage.rings.complex_double.ComplexDoubleElement.norm`
EXAMPLES:
This indeed acts as the square function when the
imaginary component of self is equal to zero::
sage: a = ComplexNumber(2,1)
sage: a.norm()
5.00000000000000
sage: b = ComplexNumber(4.2,0)
sage: b.norm()
17.6400000000000
sage: b^2
17.6400000000000
"""
return self.norm_c()
cdef real_mpfr.RealNumber norm_c(ComplexNumber self):
cdef real_mpfr.RealNumber x
x = real_mpfr.RealNumber(self._parent._real_field(), None)
cdef mpfr_t t0, t1
mpfr_init2(t0, self._prec)
mpfr_init2(t1, self._prec)
mpfr_mul(t0, self.__re, self.__re, rnd)
mpfr_mul(t1, self.__im, self.__im, rnd)
mpfr_add(x.value, t0, t1, rnd)
mpfr_clear(t0)
mpfr_clear(t1)
return x
cdef real_mpfr.RealNumber abs_c(ComplexNumber self):
cdef real_mpfr.RealNumber x
x = real_mpfr.RealNumber(self._parent._real_field(), None)
cdef mpfr_t t0, t1
mpfr_init2(t0, self._prec)
mpfr_init2(t1, self._prec)
mpfr_mul(t0, self.__re, self.__re, rnd)
mpfr_mul(t1, self.__im, self.__im, rnd)
mpfr_add(x.value, t0, t1, rnd)
mpfr_sqrt(x.value, x.value, rnd)
mpfr_clear(t0)
mpfr_clear(t1)
return x
cpdef RingElement _div_(self, RingElement right):
cdef ComplexNumber x
x = self._new()
cdef mpfr_t a, b, t0, t1, right_nm
mpfr_init2(t0, self._prec)
mpfr_init2(t1, self._prec)
mpfr_init2(a, self._prec)
mpfr_init2(b, self._prec)
mpfr_init2(right_nm, self._prec)
mpfr_mul(t0, (<ComplexNumber>right).__re, (<ComplexNumber>right).__re, rnd)
mpfr_mul(t1, (<ComplexNumber>right).__im, (<ComplexNumber>right).__im, rnd)
mpfr_add(right_nm, t0, t1, rnd)
mpfr_div(a, (<ComplexNumber>right).__re, right_nm, rnd)
mpfr_div(b, (<ComplexNumber>right).__im, right_nm, rnd)
mpfr_mul(t0, a, self.__re, rnd)
mpfr_mul(t1, b, self.__im, rnd)
mpfr_add(x.__re, t0, t1, rnd)
mpfr_mul(t0, a, self.__im, rnd)
mpfr_mul(t1, b, self.__re, rnd)
mpfr_sub(x.__im, t0, t1, rnd)
mpfr_clear(t0)
mpfr_clear(t1)
mpfr_clear(a)
mpfr_clear(b)
mpfr_clear(right_nm)
return x
def __rdiv__(self, left):
r"""
Returns the quotient of left with self, that is:
left/self
as a complex number.
INPUTS:
- ``left`` - a complex number to divide by self
EXAMPLES::
sage: a = ComplexNumber(2,0)
sage: a.__rdiv__(CC(1))
0.500000000000000
sage: CC(1)/a
0.500000000000000
"""
return ComplexNumber(self._parent, left)/self
def __pow__(self, right, modulus):
"""
EXAMPLES::
sage: C.<i> = ComplexField(20)
sage: a = i^2; a
-1.0000
sage: a.parent()
Complex Field with 20 bits of precision
sage: a = (1+i)^i; a
0.42883 + 0.15487*I
sage: (1+i)^(1+i)
0.27396 + 0.58370*I
sage: a.parent()
Complex Field with 20 bits of precision
sage: i^i
0.20788
sage: (2+i)^(0.5)
1.4553 + 0.34356*I
"""
if isinstance(right, (int, long, integer.Integer)):
return sage.rings.ring_element.RingElement.__pow__(self, right)
try:
return (self.log()*right).exp()
except TypeError:
pass
try:
self = right.parent()(self)
return self**right
except AttributeError:
raise TypeError
def _magma_init_(self, magma):
r"""
EXAMPLES::
sage: magma(CC([1, 2])) # optional - magma
1.00000000000000 + 2.00000000000000*$.1
sage: v = magma(CC([1, 2])).sage(); v # optional - magma
1.00000000000000 + 2.00000000000000*I
sage: v.parent() # optional - magma
Complex Field with 53 bits of precision
"""
return "%s![%s, %s]" % (self.parent()._magma_init_(magma), self.real(), self.imag())
def __nonzero__(self):
"""
Return True if self is not zero. This is an internal function; use
self.is_zero() instead.
EXAMPLES::
sage: z = 1 + CC(I)
sage: z.is_zero()
False
"""
return not (mpfr_zero_p(self.__re) and mpfr_zero_p(self.__im))
def prec(self):
"""
Return precision of this complex number.
EXAMPLES::
sage: i = ComplexField(2000).0
sage: i.prec()
2000
"""
return self._parent.prec()
def real(self):
"""
Return real part of self.
EXAMPLES::
sage: i = ComplexField(100).0
sage: z = 2 + 3*i
sage: x = z.real(); x
2.0000000000000000000000000000
sage: x.parent()
Real Field with 100 bits of precision
sage: z.real_part()
2.0000000000000000000000000000
"""
cdef real_mpfr.RealNumber x
x = real_mpfr.RealNumber(self._parent._real_field(), None)
mpfr_set(x.value, self.__re, rnd)
return x
real_part = real
def imag(self):
"""
Return imaginary part of self.
EXAMPLES::
sage: i = ComplexField(100).0
sage: z = 2 + 3*i
sage: x = z.imag(); x
3.0000000000000000000000000000
sage: x.parent()
Real Field with 100 bits of precision
sage: z.imag_part()
3.0000000000000000000000000000
"""
cdef real_mpfr.RealNumber x
x = real_mpfr.RealNumber(self._parent._real_field(), None)
mpfr_set(x.value, self.__im, rnd)
return x
imag_part = imag
def __neg__(self):
r"""
Method for computing the negative of self.
-(a + bi) = -a - bi
EXAMPLES::
sage: a = ComplexNumber(2,1)
sage: -a
-2.00000000000000 - 1.00000000000000*I
sage: a.__neg__()
-2.00000000000000 - 1.00000000000000*I
"""
cdef ComplexNumber x
x = self._new()
mpfr_neg(x.__re, self.__re, rnd)
mpfr_neg(x.__im, self.__im, rnd)
return x
def __pos__(self):
r"""
Method for computing the "positive" of self.
EXAMPLES::
sage: a = ComplexNumber(2,1)
sage: +a
2.00000000000000 + 1.00000000000000*I
sage: a.__pos__()
2.00000000000000 + 1.00000000000000*I
"""
return self
def __abs__(self):
r"""
Method for computing the absolute value or modulus of self
`|a + bi| = sqrt(a^2 + b^2)`
EXAMPLES: Note that the absolute value of a complex number with
imaginary component equal to zero is the absolute value of the real
component.
::
sage: a = ComplexNumber(2,1)
sage: abs(a)
2.23606797749979
sage: a.__abs__()
2.23606797749979
sage: float(sqrt(2^2 + 1^1))
2.23606797749979
::
sage: b = ComplexNumber(42,0)
sage: abs(b)
42.0000000000000
sage: b.__abs__()
42.0000000000000
sage: b
42.0000000000000
"""
return self.abs_c()
def __invert__(self):
"""
Return the multiplicative inverse.
EXAMPLES::
sage: I = CC.0
sage: a = ~(5+I)
sage: a * (5+I)
1.00000000000000
"""
cdef ComplexNumber x
x = self._new()
cdef mpfr_t t0, t1
mpfr_init2(t0, self._prec)
mpfr_init2(t1, self._prec)
mpfr_mul(t0, self.__re, self.__re, rnd)
mpfr_mul(t1, self.__im, self.__im, rnd)
mpfr_add(t0, t0, t1, rnd)
mpfr_div(x.__re, self.__re, t0, rnd)
mpfr_neg(t1, self.__im, rnd)
mpfr_div(x.__im, t1, t0, rnd)
mpfr_clear(t0)
mpfr_clear(t1)
return x
def __int__(self):
r"""
Method for converting self to type int. Called by the
``int`` function. Note that calling this method returns
an error since, in general, complex numbers cannot be coerced into
integers.
EXAMPLES::
sage: a = ComplexNumber(2,1)
sage: int(a)
Traceback (most recent call last):
...
TypeError: can't convert complex to int; use int(abs(z))
sage: a.__int__()
Traceback (most recent call last):
...
TypeError: can't convert complex to int; use int(abs(z))
"""
raise TypeError, "can't convert complex to int; use int(abs(z))"
def __long__(self):
r"""
Method for converting self to type long. Called by the
``long`` function. Note that calling this method
returns an error since, in general, complex numbers cannot be
coerced into integers.
EXAMPLES::
sage: a = ComplexNumber(2,1)
sage: long(a)
Traceback (most recent call last):
...
TypeError: can't convert complex to long; use long(abs(z))
sage: a.__long__()
Traceback (most recent call last):
...
TypeError: can't convert complex to long; use long(abs(z))
"""
raise TypeError, "can't convert complex to long; use long(abs(z))"
def __float__(self):
r"""
Method for converting self to type float. Called by the
``float`` function. This conversion will throw an error if
the number has a nonzero imaginary part.
EXAMPLES::
sage: a = ComplexNumber(1, 0)
sage: float(a)
1.0
sage: a = ComplexNumber(2,1)
sage: float(a)
Traceback (most recent call last):
...
TypeError: Unable to convert 2.00000000000000 + 1.00000000000000*I to float; use abs() or real_part() as desired
sage: a.__float__()
Traceback (most recent call last):
...
TypeError: Unable to convert 2.00000000000000 + 1.00000000000000*I to float; use abs() or real_part() as desired
sage: float(abs(ComplexNumber(1,1)))
1.4142135623730951
"""
if mpfr_zero_p(self.__im):
return mpfr_get_d(self.__re, rnd)
else:
raise TypeError, "Unable to convert %s to float; use abs() or real_part() as desired"%self
def __complex__(self):
r"""
Method for converting self to type complex. Called by the
``complex`` function.
EXAMPLES::
sage: a = ComplexNumber(2,1)
sage: complex(a)
(2+1j)
sage: type(complex(a))
<type 'complex'>
sage: a.__complex__()
(2+1j)
"""
return complex(mpfr_get_d(self.__re, rnd),
mpfr_get_d(self.__im, rnd))
def __richcmp__(left, right, int op):
return (<Element>left)._richcmp(right, op)
cdef int _cmp_c_impl(left, sage.structure.element.Element right) except -2:
cdef int a, b
a = mpfr_nan_p(left.__re)
b = mpfr_nan_p((<ComplexNumber>right).__re)
if a != b:
return -1
cdef int i
i = mpfr_cmp(left.__re, (<ComplexNumber>right).__re)
if i < 0:
return -1
elif i > 0:
return 1
i = mpfr_cmp(left.__im, (<ComplexNumber>right).__im)
if i < 0:
return -1
elif i > 0:
return 1
return 0
def multiplicative_order(self):
"""
Return the multiplicative order of this complex number, if known,
or raise a NotImplementedError.
EXAMPLES::
sage: C.<i> = ComplexField()
sage: i.multiplicative_order()
4
sage: C(1).multiplicative_order()
1
sage: C(-1).multiplicative_order()
2
sage: C(i^2).multiplicative_order()
2
sage: C(-i).multiplicative_order()
4
sage: C(2).multiplicative_order()
+Infinity
sage: w = (1+sqrt(-3.0))/2; w
0.500000000000000 + 0.866025403784439*I
sage: abs(w)
1.00000000000000
sage: w.multiplicative_order()
Traceback (most recent call last):
...
NotImplementedError: order of element not known
"""
if self == 1:
return integer.Integer(1)
elif self == -1:
return integer.Integer(2)
elif self == self._parent.gen():
return integer.Integer(4)
elif self == -self._parent.gen():
return integer.Integer(4)
elif not self._multiplicative_order is None:
return integer.Integer(self._multiplicative_order)
elif abs(abs(self) - 1) > 0.1:
return infinity.infinity
raise NotImplementedError, "order of element not known"
def plot(self, **kargs):
"""
Plots this complex number as a point in the plane
The accepted options are the ones of :meth:`~sage.plot.point.point2d`.
Type ``point2d.options`` to see all options.
.. note::
Just wraps the sage.plot.point.point2d method
EXAMPLES:
You can either use the indirect::
sage: z = CC(0,1)
sage: plot(z)
or the more direct::
sage: z = CC(0,1)
sage: z.plot()
"""
return sage.plot.point.point2d((self.real(), self.imag()), **kargs)
def arccos(self):
"""
EXAMPLES::
sage: (1+CC(I)).arccos()
0.904556894302381 - 1.06127506190504*I
"""
return self._parent(self._pari_().acos())
def arccosh(self):
"""
EXAMPLES::
sage: (1+CC(I)).arccosh()
1.06127506190504 + 0.904556894302381*I
"""
return self._parent(self._pari_().acosh())
def arcsin(self):
"""
EXAMPLES::
sage: (1+CC(I)).arcsin()
0.666239432492515 + 1.06127506190504*I
"""
return self._parent(self._pari_().asin())
def arcsinh(self):
"""
EXAMPLES::
sage: (1+CC(I)).arcsinh()
1.06127506190504 + 0.666239432492515*I
"""
return self._parent(self._pari_().asinh())
def arctan(self):
"""
EXAMPLES::
sage: (1+CC(I)).arctan()
1.01722196789785 + 0.402359478108525*I
"""
return self._parent(self._pari_().atan())
def arctanh(self):
"""
EXAMPLES::
sage: (1+CC(I)).arctanh()
0.402359478108525 + 1.01722196789785*I
"""
return self._parent(self._pari_().atanh())
def coth(self):
"""
EXAMPLES::
sage: ComplexField(100)(1,1).coth()
0.86801414289592494863584920892 - 0.21762156185440268136513424361*I
"""
return ~(self.tanh())
def arccoth(self):
"""
EXAMPLES::
sage: ComplexField(100)(1,1).arccoth()
0.40235947810852509365018983331 - 0.55357435889704525150853273009*I
"""
return (~self).arctanh()
def csc(self):
"""
EXAMPLES::
sage: ComplexField(100)(1,1).csc()
0.62151801717042842123490780586 - 0.30393100162842645033448560451*I
"""
return ~(self.sin())
def csch(self):
"""
EXAMPLES::
sage: ComplexField(100)(1,1).csch()
0.30393100162842645033448560451 - 0.62151801717042842123490780586*I
"""
return ~(self.sinh())
def arccsch(self):
"""
EXAMPLES::
sage: ComplexField(100)(1,1).arccsch()
0.53063753095251782601650945811 - 0.45227844715119068206365839783*I
"""
return (~self).arcsinh()
def sec(self):
"""
EXAMPLES::
sage: ComplexField(100)(1,1).sec()
0.49833703055518678521380589177 + 0.59108384172104504805039169297*I
"""
return ~(self.cos())
def sech(self):
"""
EXAMPLES::
sage: ComplexField(100)(1,1).sech()
0.49833703055518678521380589177 - 0.59108384172104504805039169297*I
"""
return ~(self.cosh())
def arcsech(self):
"""
EXAMPLES::
sage: ComplexField(100)(1,1).arcsech()
0.53063753095251782601650945811 - 1.1185178796437059371676632938*I
"""
return (~self).arccosh()
def cotan(self):
"""
EXAMPLES::
sage: (1+CC(I)).cotan()
0.217621561854403 - 0.868014142895925*I
sage: i = ComplexField(200).0
sage: (1+i).cotan()
0.21762156185440268136513424360523807352075436916785404091068 - 0.86801414289592494863584920891627388827343874994609327121115*I
sage: i = ComplexField(220).0
sage: (1+i).cotan()
0.21762156185440268136513424360523807352075436916785404091068124239 - 0.86801414289592494863584920891627388827343874994609327121115071646*I
"""
return ~(self.tan())
def cos(self):
"""
EXAMPLES::
sage: (1+CC(I)).cos()
0.833730025131149 - 0.988897705762865*I
"""
cdef ComplexNumber z
z = self._new()
cdef mpfr_t ch, sh
mpfr_init2(sh, self._prec)
mpfr_sinh(sh, self.__im, rnd)
mpfr_init2(ch, self._prec)
mpfr_sqr(ch, sh, rnd)
mpfr_add_ui(ch, ch, 1, rnd)
mpfr_sqrt(ch, ch, rnd)
mpfr_neg(sh, sh, rnd)
mpfr_sin_cos(z.__im, z.__re, self.__re, rnd)
mpfr_mul(z.__re, z.__re, ch, rnd)
mpfr_mul(z.__im, z.__im, sh, rnd)
mpfr_clear(sh)
mpfr_clear(ch)
return z
def cosh(self):
"""
EXAMPLES::
sage: (1+CC(I)).cosh()
0.833730025131149 + 0.988897705762865*I
"""
cdef ComplexNumber z
z = self._new()
cdef mpfr_t ch, sh
mpfr_init2(sh, self._prec)
mpfr_sinh(sh, self.__re, rnd)
mpfr_init2(ch, self._prec)
mpfr_sqr(ch, sh, rnd)
mpfr_add_ui(ch, ch, 1, rnd)
mpfr_sqrt(ch, ch, rnd)
mpfr_sin_cos(z.__im, z.__re, self.__im, rnd)
mpfr_mul(z.__re, z.__re, ch, rnd)
mpfr_mul(z.__im, z.__im, sh, rnd)
mpfr_clear(sh)
mpfr_clear(ch)
return z
def eta(self, omit_frac=False):
r"""
Return the value of the Dedekind `\eta` function on self,
intelligently computed using `\mathbb{SL}(2,\ZZ)`
transformations.
INPUT:
- ``self`` - element of the upper half plane (if not,
raises a ValueError).
- ``omit_frac`` - (bool, default: False), if True,
omit the `e^{\pi i z / 12}` factor.
OUTPUT: a complex number
The `\eta` function is
.. math::
\eta(z) = e^{\pi i z / 12} \prod_{n=1}^{\infty}(1-e^{2\pi inz})
ALGORITHM: Uses the PARI C library.
EXAMPLES:
First we compute `\eta(1+i)`::
sage: i = CC.0
sage: z = 1+i; z.eta()
0.742048775836565 + 0.198831370229911*I
We compute eta to low precision directly from the definition.
::
sage: z = 1 + i; z.eta()
0.742048775836565 + 0.198831370229911*I
sage: pi = CC(pi) # otherwise we will get a symbolic result.
sage: exp(pi * i * z / 12) * prod([1-exp(2*pi*i*n*z) for n in range(1,10)])
0.742048775836565 + 0.198831370229911*I
The optional argument allows us to omit the fractional part::
sage: z = 1 + i
sage: z.eta(omit_frac=True)
0.998129069925959 - 8.12769318...e-22*I
sage: prod([1-exp(2*pi*i*n*z) for n in range(1,10)])
0.998129069925958 + 4.59099857829247e-19*I
We illustrate what happens when `z` is not in the upper
half plane.
::
sage: z = CC(1)
sage: z.eta()
Traceback (most recent call last):
...
ValueError: value must be in the upper half plane
You can also use functional notation.
::
sage: eta(1+CC(I))
0.742048775836565 + 0.198831370229911*I
"""
try:
return self._parent(self._pari_().eta(not omit_frac))
except sage.libs.pari.all.PariError:
raise ValueError, "value must be in the upper half plane"
def sin(self):
"""
EXAMPLES::
sage: (1+CC(I)).sin()
1.29845758141598 + 0.634963914784736*I
"""
cdef ComplexNumber z
z = self._new()
cdef mpfr_t ch, sh
mpfr_init2(sh, self._prec)
mpfr_sinh(sh, self.__im, rnd)
mpfr_init2(ch, self._prec)
mpfr_sqr(ch, sh, rnd)
mpfr_add_ui(ch, ch, 1, rnd)
mpfr_sqrt(ch, ch, rnd)
mpfr_sin_cos(z.__re, z.__im, self.__re, rnd)
mpfr_mul(z.__re, z.__re, ch, rnd)
mpfr_mul(z.__im, z.__im, sh, rnd)
mpfr_clear(sh)
mpfr_clear(ch)
return z
def sinh(self):
"""
EXAMPLES::
sage: (1+CC(I)).sinh()
0.634963914784736 + 1.29845758141598*I
"""
cdef ComplexNumber z
z = self._new()
cdef mpfr_t ch, sh
mpfr_init2(sh, self._prec)
mpfr_sinh(sh, self.__re, rnd)
mpfr_init2(ch, self._prec)
mpfr_sqr(ch, sh, rnd)
mpfr_add_ui(ch, ch, 1, rnd)
mpfr_sqrt(ch, ch, rnd)
mpfr_sin_cos(z.__im, z.__re, self.__im, rnd)
mpfr_mul(z.__re, z.__re, sh, rnd)
mpfr_mul(z.__im, z.__im, ch, rnd)
mpfr_clear(sh)
mpfr_clear(ch)
return z
def tan(self):
"""
EXAMPLES::
sage: (1+CC(I)).tan()
0.271752585319512 + 1.08392332733869*I
"""
cdef ComplexNumber z
z = self._new()
cdef mpfr_t ch, sh, c, s, a, b
mpfr_init2(sh, self._prec)
mpfr_sinh(sh, self.__im, rnd)
mpfr_init2(ch, self._prec)
mpfr_init2(a, self._prec)
mpfr_sqr(a, sh, rnd)
mpfr_add_ui(ch, a, 1, rnd)
mpfr_sqrt(ch, ch, rnd)
mpfr_init2(c, self._prec)
mpfr_init2(s, self._prec)
mpfr_sin_cos(s, c, self.__re, rnd)
mpfr_init2(b, self._prec)
mpfr_sqr(b, c, rnd)
mpfr_add(a, a, b, rnd)
mpfr_mul(z.__re, c, s, rnd)
mpfr_div(z.__re, z.__re, a, rnd)
mpfr_mul(z.__im, ch, sh, rnd)
mpfr_div(z.__im, z.__im, a, rnd)
mpfr_clear(sh)
mpfr_clear(ch)
mpfr_clear(c)
mpfr_clear(s)
mpfr_clear(b)
mpfr_clear(a)
return z
def tanh(self):
"""
EXAMPLES::
sage: (1+CC(I)).tanh()
1.08392332733869 + 0.271752585319512*I
"""
cdef ComplexNumber z
z = self._new()
cdef mpfr_t ch, sh, c, s, a, b
mpfr_init2(sh, self._prec)
mpfr_sinh(sh, self.__re, rnd)
mpfr_init2(ch, self._prec)
mpfr_init2(a, self._prec)
mpfr_sqr(a, sh, rnd)
mpfr_add_ui(ch, a, 1, rnd)
mpfr_sqrt(ch, ch, rnd)
mpfr_init2(c, self._prec)
mpfr_init2(s, self._prec)
mpfr_sin_cos(s, c, self.__im, rnd)
mpfr_init2(b, self._prec)
mpfr_sqr(b, c, rnd)
mpfr_add(a, a, b, rnd)
mpfr_mul(z.__im, c, s, rnd)
mpfr_div(z.__im, z.__im, a, rnd)
mpfr_mul(z.__re, ch, sh, rnd)
mpfr_div(z.__re, z.__re, a, rnd)
mpfr_clear(sh)
mpfr_clear(ch)
mpfr_clear(c)
mpfr_clear(s)
mpfr_clear(b)
mpfr_clear(a)
return z
def agm(self, right, algorithm="optimal"):
"""
Return the Arithmetic-Geometric Mean (AGM) of self and right.
INPUT:
- right (complex)-- another complex number
- algorithm (string, default "optimal")-- the algorithm to use
(see below).
OUTPUT:
(complex) A value of the AGM of self and right. Note that
this is a multi-valued function, and the algorithm used
affects the value returned, as follows:
- "pari": Call the sgm function from the pari library.
- "optimal": Use the AGM sequence such that at each stage
`(a,b)` is replaced by `(a_1,b_1)=((a+b)/2,\pm\sqrt{ab})`
where the sign is chosen so that `|a_1-b_1|\le|a_1+b_1|`, or
equivalently `\Re(b_1/a_1)\ge 0`. The resulting limit is
maximal among all possible values.
- "principal": Use the AGM sequence such that at each stage
`(a,b)` is replaced by `(a_1,b_1)=((a+b)/2,\pm\sqrt{ab})`
where the sign is chosen so that `\Re(b_1)\ge 0` (the
so-called principal branch of the square root).
The values `AGM(a,0)`, `AGM(0,a)`, and `AGM(a,-a)` are all taken to be 0.
EXAMPLES::
sage: a = CC(1,1)
sage: b = CC(2,-1)
sage: a.agm(b)
1.62780548487271 + 0.136827548397369*I
sage: a.agm(b, algorithm="optimal")
1.62780548487271 + 0.136827548397369*I
sage: a.agm(b, algorithm="principal")
1.62780548487271 + 0.136827548397369*I
sage: a.agm(b, algorithm="pari")
1.62780548487271 + 0.136827548397369*I
An example to show that the returned value depends on the algorithm parameter::
sage: a = CC(-0.95,-0.65)
sage: b = CC(0.683,0.747)
sage: a.agm(b, algorithm="optimal")
-0.371591652351761 + 0.319894660206830*I
sage: a.agm(b, algorithm="principal")
0.338175462986180 - 0.0135326969565405*I
sage: a.agm(b, algorithm="pari")
0.0806891850759812 + 0.239036532685557*I
sage: a.agm(b, algorithm="optimal").abs()
0.490319232466314
sage: a.agm(b, algorithm="principal").abs()
0.338446122230459
sage: a.agm(b, algorithm="pari").abs()
0.252287947683910
TESTS:
An example which came up in testing::
sage: I = CC(I)
sage: a = 0.501648970493109 + 1.11877240294744*I
sage: b = 1.05946309435930 + 1.05946309435930*I
sage: a.agm(b)
0.774901870587681 + 1.10254945079875*I
sage: a = CC(-0.32599972608379413, 0.60395514542928641)
sage: b = CC( 0.6062314525690593, 0.1425693337776659)
sage: a.agm(b)
0.199246281325876 + 0.478401702759654*I
sage: a.agm(-a)
0.000000000000000
sage: a.agm(0)
0.000000000000000
sage: CC(0).agm(a)
0.000000000000000
Consistency::
sage: a = 1 + 0.5*I
sage: b = 2 - 0.25*I
sage: a.agm(b) - ComplexField(100)(a).agm(b)
0.000000000000000
sage: ComplexField(200)(a).agm(b) - ComplexField(500)(a).agm(b)
0.00000000000000000000000000000000000000000000000000000000000
sage: ComplexField(500)(a).agm(b) - ComplexField(1000)(a).agm(b)
0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
"""
if algorithm=="pari":
t = self._parent(right)._pari_()
return self._parent(self._pari_().agm(t))
cdef ComplexNumber a, b, a1, b1, d, e, res
cdef mp_exp_t rel_prec
cdef bint optimal = algorithm == "optimal"
if optimal or algorithm == "principal":
if not isinstance(right, ComplexNumber) or (<ComplexNumber>right)._parent is not self._parent:
right = self._parent(right)
res = self._new()
if mpfr_zero_p(self.__re) and mpfr_zero_p(self.__im):
return self
elif mpfr_zero_p((<ComplexNumber>right).__re) and mpfr_zero_p((<ComplexNumber>right).__im):
return right
elif (mpfr_cmpabs(self.__re, (<ComplexNumber>right).__re) == 0 and
mpfr_cmpabs(self.__im, (<ComplexNumber>right).__im) == 0 and
mpfr_cmp(self.__re, (<ComplexNumber>right).__re) != 0 and
mpfr_cmp(self.__im, (<ComplexNumber>right).__im) != 0):
mpfr_set_ui(res.__re, 0, rnd)
mpfr_set_ui(res.__im, 0, rnd)
return res
a = ComplexNumber(self._parent.to_prec(self._prec+5), None)
b = a._new()
a1 = a._new()
b1 = a._new()
d = a._new()
if optimal:
e = a._new()
mpfr_set(a.__re, self.__re, rnd)
mpfr_set(a.__im, self.__im, rnd)
mpfr_set(b.__re, (<ComplexNumber>right).__re, rnd)
mpfr_set(b.__im, (<ComplexNumber>right).__im, rnd)
if optimal:
mpfr_add(e.__re, a.__re, b.__re, rnd)
mpfr_add(e.__im, a.__im, b.__im, rnd)
while True:
if optimal:
mpfr_swap(a1.__re, e.__re)
mpfr_swap(a1.__im, e.__im)
else:
mpfr_add(a1.__re, a.__re, b.__re, rnd)
mpfr_add(a1.__im, a.__im, b.__im, rnd)
mpfr_mul_2si(a1.__re, a1.__re, -1, rnd)
mpfr_mul_2si(a1.__im, a1.__im, -1, rnd)
mpfr_mul(d.__re, a.__re, b.__re, rnd)
mpfr_mul(d.__im, a.__im, b.__im, rnd)
mpfr_sub(b1.__re, d.__re, d.__im, rnd)
mpfr_mul(d.__re, a.__re, b.__im, rnd)
mpfr_mul(d.__im, a.__im, b.__re, rnd)
mpfr_add(b1.__im, d.__re, d.__im, rnd)
b1 = b1.sqrt()
mpfr_sub(d.__re, a1.__re, b1.__re, rnd)
mpfr_sub(d.__im, a1.__im, b1.__im, rnd)
if mpfr_zero_p(d.__re) and mpfr_zero_p(d.__im):
mpfr_set(res.__re, a1.__re, rnd)
mpfr_set(res.__im, a1.__im, rnd)
return res
if optimal:
mpfr_add(e.__re, a1.__re, b1.__re, rnd)
mpfr_add(e.__im, a1.__im, b1.__im, rnd)
if mpfr_zero_p(e.__re) and mpfr_zero_p(e.__im):
mpfr_set(res.__re, a1.__re, rnd)
mpfr_set(res.__im, a1.__im, rnd)
return res
if cmp_abs(e, d) < 0:
mpfr_swap(d.__re, e.__re)
mpfr_swap(d.__im, e.__im)
mpfr_neg(b1.__re, b1.__re, rnd)
mpfr_neg(b1.__im, b1.__im, rnd)
rel_prec = min_exp_t(max_exp(a1), max_exp(b1)) - max_exp(d)
if rel_prec > self._prec:
mpfr_set(res.__re, a1.__re, rnd)
mpfr_set(res.__im, a1.__im, rnd)
return res
mpfr_swap(a.__re, a1.__re)
mpfr_swap(a.__im, a1.__im)
mpfr_swap(b.__re, b1.__re)
mpfr_swap(b.__im, b1.__im)
raise ValueError, "agm algorithm must be one of 'pari', 'optimal', 'principal'"
def argument(self):
r"""
The argument (angle) of the complex number, normalized so that
`-\pi < \theta \leq \pi`.
EXAMPLES::
sage: i = CC.0
sage: (i^2).argument()
3.14159265358979
sage: (1+i).argument()
0.785398163397448
sage: i.argument()
1.57079632679490
sage: (-i).argument()
-1.57079632679490
sage: (RR('-0.001') - i).argument()
-1.57179632646156
"""
cdef real_mpfr.RealNumber x
x = real_mpfr.RealNumber(self._parent._real_field(), None)
mpfr_atan2(x.value, self.__im, self.__re, rnd)
return x
def arg(self):
"""
Same as argument.
EXAMPLES::
sage: i = CC.0
sage: (i^2).arg()
3.14159265358979
"""
return self.argument()
def conjugate(self):
"""
Return the complex conjugate of this complex number.
EXAMPLES::
sage: i = CC.0
sage: (1+i).conjugate()
1.00000000000000 - 1.00000000000000*I
"""
cdef ComplexNumber x
x = self._new()
cdef mpfr_t i
mpfr_init2(i, self._prec)
mpfr_neg(i, self.__im, rnd)
mpfr_set(x.__re, self.__re, rnd)
mpfr_set(x.__im, i, rnd)
mpfr_clear(i)
return x
def dilog(self):
r"""
Returns the complex dilogarithm of self. The complex dilogarithm,
or Spence's function, is defined by
`Li_2(z) = - \int_0^z \frac{\log|1-\zeta|}{\zeta} d(\zeta)`
`= \sum_{k=1}^\infty \frac{z^k}{k}`
Note that the series definition can only be used for
`|z| < 1`
EXAMPLES::
sage: a = ComplexNumber(1,0)
sage: a.dilog()
1.64493406684823
sage: float(pi^2/6)
1.6449340668482262
::
sage: b = ComplexNumber(0,1)
sage: b.dilog()
-0.205616758356028 + 0.915965594177219*I
::
sage: c = ComplexNumber(0,0)
sage: c.dilog()
0.000000000000000
"""
return self._parent(self._pari_().dilog())
def exp(ComplexNumber self):
"""
Compute exp(z).
EXAMPLES::
sage: i = ComplexField(300).0
sage: z = 1 + i
sage: z.exp()
1.46869393991588515713896759732660426132695673662900872279767567631093696585951213872272450 + 2.28735528717884239120817190670050180895558625666835568093865811410364716018934540926734485*I
"""
cdef ComplexNumber z
z = self._new()
cdef mpfr_t r
mpfr_init2(r, self._prec)
mpfr_exp(r, self.__re, rnd)
mpfr_sin_cos(z.__im, z.__re, self.__im, rnd)
mpfr_mul(z.__re, z.__re, r, rnd)
mpfr_mul(z.__im, z.__im, r, rnd)
mpfr_clear(r)
return z
def gamma(self):
"""
Return the Gamma function evaluated at this complex number.
EXAMPLES::
sage: i = ComplexField(30).0
sage: (1+i).gamma()
0.49801567 - 0.15494983*I
TESTS::
sage: CC(0).gamma()
Infinity
::
sage: CC(-1).gamma()
Infinity
"""
try:
return self._parent(self._pari_().gamma())
except sage.libs.pari.all.PariError:
from sage.rings.infinity import UnsignedInfinityRing
return UnsignedInfinityRing.gen()
def gamma_inc(self, t):
"""
Return the incomplete Gamma function evaluated at this complex
number.
EXAMPLES::
sage: C, i = ComplexField(30).objgen()
sage: (1+i).gamma_inc(2 + 3*i)
0.0020969149 - 0.059981914*I
sage: (1+i).gamma_inc(5)
-0.0013781309 + 0.0065198200*I
sage: C(2).gamma_inc(1 + i)
0.70709210 - 0.42035364*I
sage: CC(2).gamma_inc(5)
0.0404276819945128
"""
return self._parent(self._pari_().incgam(t))
def log(self,base=None):
r"""
Complex logarithm of z with branch chosen as follows: Write
`z = \rho e^{i \theta} with `-\pi < \theta <= pi`. Then
`\mathrm{log}(z) = \mathrm{log}(\rho) + i \theta`.
.. warning::
Currently the real log is computed using floats, so there
is potential precision loss.
EXAMPLES::
sage: a = ComplexNumber(2,1)
sage: a.log()
0.804718956217050 + 0.463647609000806*I
sage: log(a.abs())
0.804718956217050
sage: a.argument()
0.463647609000806
::
sage: b = ComplexNumber(float(exp(42)),0)
sage: b.log()
41.99999999999971
::
sage: c = ComplexNumber(-1,0)
sage: c.log()
3.14159265358979*I
The option of a base is included for compatibility with other logs::
sage: c = ComplexNumber(-1,0)
sage: c.log(2)
4.53236014182719*I
"""
theta = self.argument()
rho = abs(self)
if base is None:
return ComplexNumber(self._parent, rho.log(), theta)
else:
from real_mpfr import RealField
return ComplexNumber(self._parent, rho.log()/RealNumber(RealField(self.prec()),base).log(), theta/RealNumber(RealField(self.prec()),base).log())
def additive_order(self):
"""
EXAMPLES::
sage: CC(0).additive_order()
1
sage: CC.gen().additive_order()
+Infinity
"""
if self == 0:
return 1
else:
return infinity.infinity
def sqrt(self, all=False):
"""
The square root function, taking the branch cut to be the negative
real axis.
INPUT:
- ``all`` - bool (default: False); if True, return a
list of all square roots.
EXAMPLES::
sage: C.<i> = ComplexField(30)
sage: i.sqrt()
0.70710678 + 0.70710678*I
sage: (1+i).sqrt()
1.0986841 + 0.45508986*I
sage: (C(-1)).sqrt()
1.0000000*I
sage: (1 + 1e-100*i).sqrt()^2
1.0000000 + 1.0000000e-100*I
sage: i = ComplexField(200).0
sage: i.sqrt()
0.70710678118654752440084436210484903928483593768847403658834 + 0.70710678118654752440084436210484903928483593768847403658834*I
"""
cdef ComplexNumber z = self._new()
if mpfr_zero_p(self.__im):
if mpfr_sgn(self.__re) >= 0:
mpfr_set_ui(z.__im, 0, rnd)
mpfr_sqrt(z.__re, self.__re, rnd)
else:
mpfr_set_ui(z.__re, 0, rnd)
mpfr_neg(z.__im, self.__re, rnd)
mpfr_sqrt(z.__im, z.__im, rnd)
if all:
return [z, -z] if z else [z]
else:
return z
cdef bint avoid_branch = mpfr_sgn(self.__re) < 0 and mpfr_cmpabs(self.__im, self.__re) < 0
cdef mpfr_t a2
mpfr_init2(a2, self._prec)
mpfr_hypot(a2, self.__re, self.__im, rnd)
if avoid_branch:
mpfr_sub(a2, a2, self.__re, rnd)
else:
mpfr_add(a2, a2, self.__re, rnd)
mpfr_mul_2si(a2, a2, -1, rnd)
mpfr_sqrt(z.__re, a2, rnd)
mpfr_div(z.__im, self.__im, z.__re, rnd)
mpfr_mul_2si(z.__im, z.__im, -1, rnd)
mpfr_clear(a2)
if avoid_branch:
mpfr_swap(z.__re, z.__im)
if mpfr_sgn(self.__im) < 0:
mpfr_neg(z.__re, z.__re, rnd)
mpfr_neg(z.__im, z.__im, rnd)
if all:
return [z, -z]
else:
return z
def nth_root(self, n, all=False):
"""
The n-th root function.
INPUT:
- ``all`` - bool (default: False); if True, return a
list of all n-th roots.
EXAMPLES::
sage: a = CC(27)
sage: a.nth_root(3)
3.00000000000000
sage: a.nth_root(3, all=True)
[3.00000000000000, -1.50000000000000 + 2.59807621135332*I, -1.50000000000000 - 2.59807621135332*I]
sage: a = ComplexField(20)(2,1)
sage: [r^7 for r in a.nth_root(7, all=True)]
[2.0000 + 1.0000*I, 2.0000 + 1.0000*I, 2.0000 + 1.0000*I, 2.0000 + 1.0000*I, 2.0000 + 1.0000*I, 2.0000 + 1.0001*I, 2.0000 + 1.0001*I]
"""
if self.is_zero():
return [self] if all else self
cdef ComplexNumber z
z = self._new()
cdef real_mpfr.RealNumber arg, rho
cdef mpfr_t r
rho = abs(self)
arg = self.argument() / n
mpfr_init2(r, self._prec)
mpfr_root(r, rho.value, n, rnd)
mpfr_sin_cos(z.__im, z.__re, arg.value, rnd)
mpfr_mul(z.__re, z.__re, r, rnd)
mpfr_mul(z.__im, z.__im, r, rnd)
if not all:
mpfr_clear(r)
return z
R = self._parent._real_field()
cdef real_mpfr.RealNumber theta
theta = R.pi()*2/n
zlist = [z]
for k in range(1, n):
z = self._new()
arg += theta
mpfr_sin_cos(z.__im, z.__re, arg.value, rnd)
mpfr_mul(z.__re, z.__re, r, rnd)
mpfr_mul(z.__im, z.__im, r, rnd)
zlist.append(z)
mpfr_clear(r)
return zlist
def is_square(self):
"""
This function always returns true as `\CC` is
algebraically closed.
EXAMPLES::
sage: a = ComplexNumber(2,1)
sage: a.is_square()
True
`\CC` is algebraically closed, hence every element
is a square::
sage: b = ComplexNumber(5)
sage: b.is_square()
True
"""
return True
def is_real(self):
"""
Return True if self is real, i.e. has imaginary part zero.
EXAMPLES::
sage: CC(1.23).is_real()
True
sage: CC(1+i).is_real()
False
"""
return (mpfr_zero_p(self.__im) <> 0)
def is_imaginary(self):
"""
Return True if self is imaginary, i.e. has real part zero.
EXAMPLES::
sage: CC(1.23*i).is_imaginary()
True
sage: CC(1+i).is_imaginary()
False
"""
return (mpfr_zero_p(self.__re) <> 0)
def zeta(self):
"""
Return the Riemann zeta function evaluated at this complex number.
EXAMPLES::
sage: i = ComplexField(30).gen()
sage: z = 1 + i
sage: z.zeta()
0.58215806 - 0.92684856*I
sage: zeta(z)
0.58215806 - 0.92684856*I
sage: CC(1).zeta()
Infinity
"""
if mpfr_zero_p(self.__im) and mpfr_cmp_ui(self.__re, 1) == 0:
import infinity
return infinity.unsigned_infinity
return self._parent(self._pari_().zeta())
def algdep(self, n, **kwds):
"""
Returns a polynomial of degree at most `n` which is
approximately satisfied by this complex number. Note that the
returned polynomial need not be irreducible, and indeed usually
won't be if `z` is a good approximation to an algebraic
number of degree less than `n`.
ALGORITHM: Uses the PARI C-library algdep command.
INPUT: Type algdep? at the top level prompt. All additional
parameters are passed onto the top-level algdep command.
EXAMPLE::
sage: C = ComplexField()
sage: z = (1/2)*(1 + sqrt(3.0) *C.0); z
0.500000000000000 + 0.866025403784439*I
sage: p = z.algdep(5); p
x^3 + 1
sage: p.factor()
(x + 1) * (x^2 - x + 1)
sage: z^2 - z + 1
1.11022302462516e-16
"""
import sage.rings.arith
return sage.rings.arith.algdep(self,n, **kwds)
def algebraic_dependancy( self, n ):
"""
Returns a polynomial of degree at most `n` which is
approximately satisfied by this complex number. Note that the
returned polynomial need not be irreducible, and indeed usually
won't be if `z` is a good approximation to an algebraic
number of degree less than `n`.
ALGORITHM: Uses the PARI C-library algdep command.
INPUT: Type algdep? at the top level prompt. All additional
parameters are passed onto the top-level algdep command.
EXAMPLE::
sage: C = ComplexField()
sage: z = (1/2)*(1 + sqrt(3.0) *C.0); z
0.500000000000000 + 0.866025403784439*I
sage: p = z.algebraic_dependancy(5); p
x^3 + 1
sage: p.factor()
(x + 1) * (x^2 - x + 1)
sage: z^2 - z + 1
1.11022302462516e-16
"""
return self.algdep( n )
def make_ComplexNumber0( fld, mult_order, re, im ):
x = ComplexNumber( fld, re, im )
x._set_multiplicative_order( mult_order )
return x
def create_ComplexNumber(s_real, s_imag=None, int pad=0, min_prec=53):
r"""
Return the complex number defined by the strings s_real and
s_imag as an element of ``ComplexField(prec=n)``,
where n potentially has slightly more (controlled by pad) bits than
given by s.
INPUT:
- ``s_real`` - a string that defines a real number
(or something whose string representation defines a number)
- ``s_imag`` - a string that defines a real number
(or something whose string representation defines a number)
- ``pad`` - an integer >= 0.
- ``min_prec`` - number will have at least this many
bits of precision, no matter what.
EXAMPLES::
sage: ComplexNumber('2.3')
2.30000000000000
sage: ComplexNumber('2.3','1.1')
2.30000000000000 + 1.10000000000000*I
sage: ComplexNumber(10)
10.0000000000000
sage: ComplexNumber(10,10)
10.0000000000000 + 10.0000000000000*I
sage: ComplexNumber(1.000000000000000000000000000,2)
1.00000000000000000000000000 + 2.00000000000000000000000000*I
sage: ComplexNumber(1,2.000000000000000000000)
1.00000000000000000000 + 2.00000000000000000000*I
::
sage: sage.rings.complex_number.create_ComplexNumber(s_real=2,s_imag=1)
2.00000000000000 + 1.00000000000000*I
TESTS:
Make sure we've rounded up log(10,2) enough to guarantee
sufficient precision (trac #10164)::
sage: s = "1." + "0"*10**6 + "1"
sage: sage.rings.complex_number.create_ComplexNumber(s,0).real()-1 == 0
False
sage: sage.rings.complex_number.create_ComplexNumber(0,s).imag()-1 == 0
False
"""
if s_imag is None:
s_imag = 0
if not isinstance(s_real, str):
s_real = str(s_real).strip()
if not isinstance(s_imag, str):
s_imag = str(s_imag).strip()
bits = max(int(LOG_TEN_TWO_PLUS_EPSILON*len(s_real)),
int(LOG_TEN_TWO_PLUS_EPSILON*len(s_imag)))
C = complex_field.ComplexField(prec=max(bits+pad, min_prec))
return ComplexNumber(C, s_real, s_imag)
cdef class RRtoCC(Map):
cdef ComplexNumber _zero
def __init__(self, RR, CC):
"""
EXAMPLES::
sage: from sage.rings.complex_number import RRtoCC
sage: RRtoCC(RR, CC)
Natural map:
From: Real Field with 53 bits of precision
To: Complex Field with 53 bits of precision
"""
Map.__init__(self, RR, CC)
self._zero = ComplexNumber(CC, 0)
self._repr_type_str = "Natural"
cpdef Element _call_(self, x):
"""
EXAMPLES::
sage: from sage.rings.complex_number import RRtoCC
sage: f = RRtoCC(RealField(100), ComplexField(10))
sage: f(1/3)
0.33
"""
cdef ComplexNumber z = self._zero._new()
mpfr_set(z.__re, (<RealNumber>x).value, rnd)
mpfr_set_ui(z.__im, 0, rnd)
return z
cdef class CCtoCDF(Map):
cpdef Element _call_(self, x):
"""
EXAMPLES::
sage: from sage.rings.complex_number import CCtoCDF
sage: f = CCtoCDF(CC, CDF)
sage: f(CC.0)
1.0*I
sage: f(exp(pi*CC.0/4))
0.707106781187 + 0.707106781187*I
"""
cdef ComplexDoubleElement z = <ComplexDoubleElement>PY_NEW(ComplexDoubleElement)
z._complex.dat[0] = mpfr_get_d((<ComplexNumber>x).__re, GMP_RNDN)
z._complex.dat[1] = mpfr_get_d((<ComplexNumber>x).__im, GMP_RNDN)
return z
cdef inline mp_exp_t min_exp_t(mp_exp_t a, mp_exp_t b):
return a if a < b else b
cdef inline mp_exp_t max_exp_t(mp_exp_t a, mp_exp_t b):
return a if a > b else b
cdef inline mp_exp_t max_exp(ComplexNumber z):
"""
Quickly return the maximum exponent of the real and complex parts of z,
which is useful for estimating its magnitude.
"""
if mpfr_zero_p(z.__im):
return mpfr_get_exp(z.__re)
elif mpfr_zero_p(z.__re):
return mpfr_get_exp(z.__im)
return max_exp_t(mpfr_get_exp(z.__re), mpfr_get_exp(z.__im))
cpdef int cmp_abs(ComplexNumber a, ComplexNumber b):
"""
Returns -1, 0, or 1 according to whether `|a|` is less than, equal to, or greater than `|b|`.
Optimized for non-close numbers, where the ordering can be determined by examining exponents.
EXAMPLES::
sage: from sage.rings.complex_number import cmp_abs
sage: cmp_abs(CC(5), CC(1))
1
sage: cmp_abs(CC(5), CC(4))
1
sage: cmp_abs(CC(5), CC(5))
0
sage: cmp_abs(CC(5), CC(6))
-1
sage: cmp_abs(CC(5), CC(100))
-1
sage: cmp_abs(CC(-100), CC(1))
1
sage: cmp_abs(CC(-100), CC(100))
0
sage: cmp_abs(CC(-100), CC(1000))
-1
sage: cmp_abs(CC(1,1), CC(1))
1
sage: cmp_abs(CC(1,1), CC(2))
-1
sage: cmp_abs(CC(1,1), CC(1,0.99999))
1
sage: cmp_abs(CC(1,1), CC(1,-1))
0
sage: cmp_abs(CC(0), CC(1))
-1
sage: cmp_abs(CC(1), CC(0))
1
sage: cmp_abs(CC(0), CC(0))
0
sage: cmp_abs(CC(2,1), CC(1,2))
0
"""
if mpfr_zero_p(b.__re) and mpfr_zero_p(b.__im):
return not ((mpfr_zero_p(a.__re) and mpfr_zero_p(a.__im)))
elif (mpfr_zero_p(a.__re) and mpfr_zero_p(a.__im)):
return -1
cdef mp_exp_t exp_diff = max_exp(a) - max_exp(b)
if exp_diff <= -2:
return -1
elif exp_diff >= 2:
return 1
cdef int res
cdef mpfr_t abs_a, abs_b, tmp
mpfr_init2(abs_a, mpfr_get_prec(a.__re))
mpfr_init2(abs_b, mpfr_get_prec(b.__re))
mpfr_init2(tmp, mpfr_get_prec(a.__re))
mpfr_sqr(abs_a, a.__re, rnd)
mpfr_sqr(tmp, a.__im, rnd)
mpfr_add(abs_a, abs_a, tmp, rnd)
mpfr_sqr(abs_b, b.__re, rnd)
mpfr_sqr(tmp, b.__im, rnd)
mpfr_add(abs_b, abs_b, tmp, rnd)
res = mpfr_cmpabs(abs_a, abs_b)
mpfr_clear(abs_a)
mpfr_clear(abs_b)
mpfr_clear(tmp)
return res
