# Arbitrary precision complex balls#

This is an incomplete interface to the acb module of FLINT; it may be useful to refer to its documentation for more details.

Parts of the documentation for this module are copied or adapted from Arb’s (now FLINT’s) own documentation, licenced at the time under the GNU General Public License version 2, or later.

## Data Structure#

A ComplexBall represents a complex number with error bounds. It wraps an object of type acb_t, which consists of a pair of real number balls representing the real and imaginary part with separate error bounds. (See the documentation of sage.rings.real_arb for more information.)

A ComplexBall thus represents a rectangle $$[m_1-r_1, m_1+r_1] + [m_2-r_2, m_2+r_2] i$$ in the complex plane. This is used instead of a disk or square representation (consisting of a complex floating-point midpoint with a single radius), since it allows implementing many operations more conveniently by splitting into ball operations on the real and imaginary parts. It also allows tracking when complex numbers have an exact (for example exactly zero) real part and an inexact imaginary part, or vice versa.

The parents of complex balls are instances of ComplexBallField. The name CBF is bound to the complex ball field with the default precision of 53 bits:

sage: CBF is ComplexBallField() is ComplexBallField(53)
True

>>> from sage.all import *
>>> CBF is ComplexBallField() is ComplexBallField(Integer(53))
True


## Comparison#

Warning

In accordance with the semantics of FLINT/Arb, identical ComplexBall objects are understood to give permission for algebraic simplification. This assumption is made to improve performance. For example, setting z = x*x sets $$z$$ to a ball enclosing the set $$\{t^2 : t \in x\}$$ and not the (generally larger) set $$\{tu : t \in x, u \in x\}$$.

Two elements are equal if and only if they are exact and equal (in spite of the above warning, inexact balls are not considered equal to themselves):

sage: a = CBF(1, 2)
sage: b = CBF(1, 2)
sage: a is b
False
sage: a == a
True
sage: a == b
True

>>> from sage.all import *
>>> a = CBF(Integer(1), Integer(2))
>>> b = CBF(Integer(1), Integer(2))
>>> a is b
False
>>> a == a
True
>>> a == b
True

sage: a = CBF(1/3, 1/5)
sage: b = CBF(1/3, 1/5)
sage: a.is_exact()
False
sage: b.is_exact()
False
sage: a is b
False
sage: a == a
False
sage: a == b
False

>>> from sage.all import *
>>> a = CBF(Integer(1)/Integer(3), Integer(1)/Integer(5))
>>> b = CBF(Integer(1)/Integer(3), Integer(1)/Integer(5))
>>> a.is_exact()
False
>>> b.is_exact()
False
>>> a is b
False
>>> a == a
False
>>> a == b
False


A ball is non-zero in the sense of usual comparison if and only if it does not contain zero:

sage: a = CBF(RIF(-0.5, 0.5))
sage: a != 0
False
sage: b = CBF(1/3, 1/5)
sage: b != 0
True

>>> from sage.all import *
>>> a = CBF(RIF(-RealNumber('0.5'), RealNumber('0.5')))
>>> a != Integer(0)
False
>>> b = CBF(Integer(1)/Integer(3), Integer(1)/Integer(5))
>>> b != Integer(0)
True


However, bool(b) returns False for a ball b only if b is exactly zero:

sage: bool(a)
True
sage: bool(b)
True
sage: bool(CBF.zero())
False

>>> from sage.all import *
>>> bool(a)
True
>>> bool(b)
True
>>> bool(CBF.zero())
False


## Coercion#

Automatic coercions work as expected:

sage: # needs sage.symbolic
sage: bpol = 1/3*CBF(i) + AA(sqrt(2))
sage: bpol += polygen(RealBallField(20), 'x') + QQbar(i)
sage: bpol
x + [1.41421 +/- ...e-6] + [1.33333 +/- ...e-6]*I
sage: bpol.parent()
Univariate Polynomial Ring in x over Complex ball field with 20 bits of precision
sage: bpol/3
([0.333333 +/- ...e-7])*x + [0.47140 +/- ...e-6] + [0.44444 +/- ...e-6]*I

>>> from sage.all import *
>>> # needs sage.symbolic
>>> bpol = Integer(1)/Integer(3)*CBF(i) + AA(sqrt(Integer(2)))
>>> bpol += polygen(RealBallField(Integer(20)), 'x') + QQbar(i)
>>> bpol
x + [1.41421 +/- ...e-6] + [1.33333 +/- ...e-6]*I
>>> bpol.parent()
Univariate Polynomial Ring in x over Complex ball field with 20 bits of precision
>>> bpol/Integer(3)
([0.333333 +/- ...e-7])*x + [0.47140 +/- ...e-6] + [0.44444 +/- ...e-6]*I


## Classes and Methods#

class sage.rings.complex_arb.ComplexBall[source]#

Bases: RingElement

Hold one acb_t of the FLINT library

EXAMPLES:

sage: a = ComplexBallField()(1, 1)
sage: a
1.000000000000000 + 1.000000000000000*I

>>> from sage.all import *
>>> a = ComplexBallField()(Integer(1), Integer(1))
>>> a
1.000000000000000 + 1.000000000000000*I

Chi()[source]#

Return the hyperbolic cosine integral with argument self.

EXAMPLES:

sage: CBF(1, 1).Chi()
[0.882172180555936 +/- ...e-16] + [1.28354719327494 +/- ...e-15]*I
sage: CBF(0).Chi()
nan + nan*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).Chi()
[0.882172180555936 +/- ...e-16] + [1.28354719327494 +/- ...e-15]*I
>>> CBF(Integer(0)).Chi()
nan + nan*I

Ci()[source]#

Return the cosine integral with argument self.

EXAMPLES:

sage: CBF(1, 1).Ci()
[0.882172180555936 +/- ...e-16] + [0.287249133519956 +/- ...e-16]*I
sage: CBF(0).Ci()
nan + nan*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).Ci()
[0.882172180555936 +/- ...e-16] + [0.287249133519956 +/- ...e-16]*I
>>> CBF(Integer(0)).Ci()
nan + nan*I

Ei()[source]#

Return the exponential integral with argument self.

EXAMPLES:

sage: CBF(1, 1).Ei()
[1.76462598556385 +/- ...e-15] + [2.38776985151052 +/- ...e-15]*I
sage: CBF(0).Ei()
nan

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).Ei()
[1.76462598556385 +/- ...e-15] + [2.38776985151052 +/- ...e-15]*I
>>> CBF(Integer(0)).Ei()
nan

Li()[source]#

Offset logarithmic integral.

EXAMPLES:

sage: CBF(0).Li()
[-1.045163780117493 +/- ...e-16]
sage: li(0).n()                                                             # needs sage.symbolic
0.000000000000000
sage: Li(0).n()                                                             # needs sage.symbolic
-1.04516378011749

>>> from sage.all import *
>>> CBF(Integer(0)).Li()
[-1.045163780117493 +/- ...e-16]
>>> li(Integer(0)).n()                                                             # needs sage.symbolic
0.000000000000000
>>> Li(Integer(0)).n()                                                             # needs sage.symbolic
-1.04516378011749

Shi()[source]#

Return the hyperbolic sine integral with argument self.

EXAMPLES:

sage: CBF(1, 1).Shi()
[0.88245380500792 +/- ...e-15] + [1.10422265823558 +/- ...e-15]*I
sage: CBF(0).Shi()
0

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).Shi()
[0.88245380500792 +/- ...e-15] + [1.10422265823558 +/- ...e-15]*I
>>> CBF(Integer(0)).Shi()
0

Si()[source]#

Return the sine integral with argument self.

EXAMPLES:

sage: CBF(1, 1).Si()
[1.10422265823558 +/- ...e-15] + [0.88245380500792 +/- ...e-15]*I
sage: CBF(0).Si()
0

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).Si()
[1.10422265823558 +/- ...e-15] + [0.88245380500792 +/- ...e-15]*I
>>> CBF(Integer(0)).Si()
0

above_abs()[source]#

Return an upper bound for the absolute value of this complex ball.

OUTPUT:

EXAMPLES:

sage: b = ComplexBallField(8)(1+i).above_abs()
sage: b
[1.4 +/- 0.0219]
sage: b.is_exact()
True
sage: QQ(b)*128
182

>>> from sage.all import *
>>> b = ComplexBallField(Integer(8))(Integer(1)+i).above_abs()
>>> b
[1.4 +/- 0.0219]
>>> b.is_exact()
True
>>> QQ(b)*Integer(128)
182

accuracy()[source]#

Return the effective relative accuracy of this ball measured in bits.

This is computed as if calling accuracy() on the real ball whose midpoint is the larger out of the real and imaginary midpoints of this complex ball, and whose radius is the larger out of the real and imaginary radii of this complex ball.

EXAMPLES:

sage: CBF(exp(I*pi/3)).accuracy()                                           # needs sage.symbolic
51
sage: CBF(I/2).accuracy() == CBF.base().maximal_accuracy()
True
sage: CBF('nan', 'inf').accuracy() == -CBF.base().maximal_accuracy()
True

>>> from sage.all import *
>>> CBF(exp(I*pi/Integer(3))).accuracy()                                           # needs sage.symbolic
51
>>> CBF(I/Integer(2)).accuracy() == CBF.base().maximal_accuracy()
True
>>> CBF('nan', 'inf').accuracy() == -CBF.base().maximal_accuracy()
True


Increase the radii of the real and imaginary parts by (an upper bound on) ampl.

If ampl is negative, the radii remain unchanged.

INPUT:

• ampl – A real ball (or an object that can be coerced to a real ball).

OUTPUT:

A new complex ball.

EXAMPLES:

sage: CBF(1+i).add_error(10^-16)
[1.000000000000000 +/- ...e-16] + [1.000000000000000 +/- ...e-16]*I

>>> from sage.all import *
[1.000000000000000 +/- ...e-16] + [1.000000000000000 +/- ...e-16]*I

agm1()[source]#

Return the arithmetic-geometric mean of 1 and self.

The arithmetic-geometric mean is defined such that the function is continuous in the complex plane except for a branch cut along the negative half axis (where it is continuous from above). This corresponds to always choosing an “optimal” branch for the square root in the arithmetic-geometric mean iteration.

EXAMPLES:

sage: CBF(0, -1).agm1()
[0.599070117367796 +/- 3.9...e-16] + [-0.599070117367796 +/- 5.5...e-16]*I

>>> from sage.all import *
>>> CBF(Integer(0), -Integer(1)).agm1()
[0.599070117367796 +/- 3.9...e-16] + [-0.599070117367796 +/- 5.5...e-16]*I

airy()[source]#

Return the Airy functions Ai, Ai’, Bi, Bi’ with argument self, evaluated simultaneously.

EXAMPLES:

sage: CBF(10*pi).airy()                                                     # needs sage.symbolic
([1.2408955946101e-52 +/- ...e-66],
[-6.965048886977e-52 +/- ...e-65],
[2.2882956833435e+50 +/- ...e+36],
[1.2807602335816e+51 +/- ...e+37])
sage: ai, aip, bi, bip = CBF(1,2).airy()
sage: (ai * bip - bi * aip) * CBF(pi)                                       # needs sage.symbolic
[1.0000000000000 +/- ...e-15] + [+/- ...e-16]*I

>>> from sage.all import *
>>> CBF(Integer(10)*pi).airy()                                                     # needs sage.symbolic
([1.2408955946101e-52 +/- ...e-66],
[-6.965048886977e-52 +/- ...e-65],
[2.2882956833435e+50 +/- ...e+36],
[1.2807602335816e+51 +/- ...e+37])
>>> ai, aip, bi, bip = CBF(Integer(1),Integer(2)).airy()
>>> (ai * bip - bi * aip) * CBF(pi)                                       # needs sage.symbolic
[1.0000000000000 +/- ...e-15] + [+/- ...e-16]*I

airy_ai()[source]#

Return the Airy function Ai with argument self.

EXAMPLES:

sage: CBF(1,2).airy_ai()
[-0.2193862549814276 +/- ...e-17] + [-0.1753859114081094 +/- ...e-17]*I

>>> from sage.all import *
>>> CBF(Integer(1),Integer(2)).airy_ai()
[-0.2193862549814276 +/- ...e-17] + [-0.1753859114081094 +/- ...e-17]*I

airy_ai_prime()[source]#

Return the Airy function derivative Ai’ with argument self.

EXAMPLES:

sage: CBF(1,2).airy_ai_prime()
[0.1704449781789148 +/- ...e-17] + [0.387622439413295 +/- ...e-16]*I

>>> from sage.all import *
>>> CBF(Integer(1),Integer(2)).airy_ai_prime()
[0.1704449781789148 +/- ...e-17] + [0.387622439413295 +/- ...e-16]*I

airy_bi()[source]#

Return the Airy function Bi with argument self.

EXAMPLES:

sage: CBF(1,2).airy_bi()
[0.0488220324530612 +/- ...e-17] + [0.1332740579917484 +/- ...e-17]*I

>>> from sage.all import *
>>> CBF(Integer(1),Integer(2)).airy_bi()
[0.0488220324530612 +/- ...e-17] + [0.1332740579917484 +/- ...e-17]*I

airy_bi_prime()[source]#

Return the Airy function derivative Bi’ with argument self.

EXAMPLES:

sage: CBF(1,2).airy_bi_prime()
[-0.857239258605362 +/- ...e-16] + [0.4955063363095674 +/- ...e-17]*I

>>> from sage.all import *
>>> CBF(Integer(1),Integer(2)).airy_bi_prime()
[-0.857239258605362 +/- ...e-16] + [0.4955063363095674 +/- ...e-17]*I

arccos(analytic=False)[source]#

Return the arccosine of this ball.

INPUT:

• analytic (optional, boolean) – if True, return an indeterminate (not-a-number) value when the input ball touches the branch cut

EXAMPLES:

sage: CBF(1+i).arccos()
[0.90455689430238 +/- ...e-15] + [-1.06127506190504 +/- ...e-15]*I
sage: CBF(-1).arccos()
[3.141592653589793 +/- ...e-16]
sage: CBF(-1).arccos(analytic=True)
nan + nan*I

>>> from sage.all import *
>>> CBF(Integer(1)+i).arccos()
[0.90455689430238 +/- ...e-15] + [-1.06127506190504 +/- ...e-15]*I
>>> CBF(-Integer(1)).arccos()
[3.141592653589793 +/- ...e-16]
>>> CBF(-Integer(1)).arccos(analytic=True)
nan + nan*I

arccosh(analytic=False)[source]#

Return the hyperbolic arccosine of this ball.

INPUT:

• analytic (optional, boolean) – if True, return an indeterminate (not-a-number) value when the input ball touches the branch cut

EXAMPLES:

sage: CBF(1+i).arccosh()
[1.061275061905035 +/- ...e-16] + [0.904556894302381 +/- ...e-16]*I
sage: CBF(-2).arccosh()
[1.316957896924817 +/- ...e-16] + [3.141592653589793 +/- ...e-16]*I
sage: CBF(-2).arccosh(analytic=True)
nan + nan*I

>>> from sage.all import *
>>> CBF(Integer(1)+i).arccosh()
[1.061275061905035 +/- ...e-16] + [0.904556894302381 +/- ...e-16]*I
>>> CBF(-Integer(2)).arccosh()
[1.316957896924817 +/- ...e-16] + [3.141592653589793 +/- ...e-16]*I
>>> CBF(-Integer(2)).arccosh(analytic=True)
nan + nan*I

arcsin(analytic=False)[source]#

Return the arcsine of this ball.

INPUT:

• analytic (optional, boolean) – if True, return an indeterminate (not-a-number) value when the input ball touches the branch cut

EXAMPLES:

sage: CBF(1+i).arcsin()
[0.66623943249252 +/- ...e-15] + [1.06127506190504 +/- ...e-15]*I
sage: CBF(1, RIF(0,1/1000)).arcsin()
[1.6 +/- 0.0619] + [+/- 0.0322]*I
sage: CBF(1, RIF(0,1/1000)).arcsin(analytic=True)
nan + nan*I

>>> from sage.all import *
>>> CBF(Integer(1)+i).arcsin()
[0.66623943249252 +/- ...e-15] + [1.06127506190504 +/- ...e-15]*I
>>> CBF(Integer(1), RIF(Integer(0),Integer(1)/Integer(1000))).arcsin()
[1.6 +/- 0.0619] + [+/- 0.0322]*I
>>> CBF(Integer(1), RIF(Integer(0),Integer(1)/Integer(1000))).arcsin(analytic=True)
nan + nan*I

arcsinh(analytic=False)[source]#

Return the hyperbolic arcsine of this ball.

INPUT:

• analytic (optional, boolean) – if True, return an indeterminate (not-a-number) value when the input ball touches the branch cut

EXAMPLES:

sage: CBF(1+i).arcsinh()
[1.06127506190504 +/- ...e-15] + [0.66623943249252 +/- ...e-15]*I
sage: CBF(2*i).arcsinh()
[1.31695789692482 +/- ...e-15] + [1.570796326794897 +/- ...e-16]*I
sage: CBF(2*i).arcsinh(analytic=True)
nan + nan*I

>>> from sage.all import *
>>> CBF(Integer(1)+i).arcsinh()
[1.06127506190504 +/- ...e-15] + [0.66623943249252 +/- ...e-15]*I
>>> CBF(Integer(2)*i).arcsinh()
[1.31695789692482 +/- ...e-15] + [1.570796326794897 +/- ...e-16]*I
>>> CBF(Integer(2)*i).arcsinh(analytic=True)
nan + nan*I

arctan(analytic=False)[source]#

Return the arctangent of this ball.

INPUT:

• analytic (optional, boolean) – if True, return an indeterminate (not-a-number) value when the input ball touches the branch cut

EXAMPLES:

sage: CBF(1+i).arctan()
[1.017221967897851 +/- ...e-16] + [0.4023594781085251 +/- ...e-17]*I
sage: CBF(i).arctan()
nan + nan*I
sage: CBF(2*i).arctan()
[1.570796326794897 +/- ...e-16] + [0.549306144334055 +/- ...e-16]*I
sage: CBF(2*i).arctan(analytic=True)
nan + nan*I

>>> from sage.all import *
>>> CBF(Integer(1)+i).arctan()
[1.017221967897851 +/- ...e-16] + [0.4023594781085251 +/- ...e-17]*I
>>> CBF(i).arctan()
nan + nan*I
>>> CBF(Integer(2)*i).arctan()
[1.570796326794897 +/- ...e-16] + [0.549306144334055 +/- ...e-16]*I
>>> CBF(Integer(2)*i).arctan(analytic=True)
nan + nan*I

arctanh(analytic=False)[source]#

Return the hyperbolic arctangent of this ball.

INPUT:

• analytic (optional, boolean) – if True, return an indeterminate (not-a-number) value when the input ball touches the branch cut

EXAMPLES:

sage: CBF(1+i).arctanh()
[0.4023594781085251 +/- ...e-17] + [1.017221967897851 +/- ...e-16]*I
sage: CBF(-2).arctanh()
[-0.549306144334055 +/- ...e-16] + [1.570796326794897 +/- ...e-16]*I
sage: CBF(-2).arctanh(analytic=True)
nan + nan*I

>>> from sage.all import *
>>> CBF(Integer(1)+i).arctanh()
[0.4023594781085251 +/- ...e-17] + [1.017221967897851 +/- ...e-16]*I
>>> CBF(-Integer(2)).arctanh()
[-0.549306144334055 +/- ...e-16] + [1.570796326794897 +/- ...e-16]*I
>>> CBF(-Integer(2)).arctanh(analytic=True)
nan + nan*I

arg()[source]#

Return the argument of this complex ball.

EXAMPLES:

sage: CBF(1 + i).arg()
[0.7853981633974483 +/- ...e-17]
sage: CBF(-1).arg()
[3.141592653589793 +/- ...e-16]
sage: CBF(-1).arg().parent()
Real ball field with 53 bits of precision

>>> from sage.all import *
>>> CBF(Integer(1) + i).arg()
[0.7853981633974483 +/- ...e-17]
>>> CBF(-Integer(1)).arg()
[3.141592653589793 +/- ...e-16]
>>> CBF(-Integer(1)).arg().parent()
Real ball field with 53 bits of precision

barnes_g()[source]#

Return the Barnes G-function of self.

EXAMPLES:

sage: CBF(-4).barnes_g()
0
sage: CBF(8).barnes_g()
24883200.00000000
sage: CBF(500,10).barnes_g()
[4.54078781e+254873 +/- ...e+254864] + [8.65835455e+254873 +/- ...e+254864]*I

>>> from sage.all import *
>>> CBF(-Integer(4)).barnes_g()
0
>>> CBF(Integer(8)).barnes_g()
24883200.00000000
>>> CBF(Integer(500),Integer(10)).barnes_g()
[4.54078781e+254873 +/- ...e+254864] + [8.65835455e+254873 +/- ...e+254864]*I

below_abs(test_zero=False)[source]#

Return a lower bound for the absolute value of this complex ball.

INPUT:

• test_zero (boolean, default False) – if True, make sure that the returned lower bound is positive, raising an error if the ball contains zero.

OUTPUT:

EXAMPLES:

sage: b = ComplexBallField(8)(1+i).below_abs()
sage: b
[1.4 +/- 0.0141]
sage: b.is_exact()
True
sage: QQ(b)*128
181
sage: (CBF(1/3) - 1/3).below_abs()
0
sage: (CBF(1/3) - 1/3).below_abs(test_zero=True)
Traceback (most recent call last):
...
ValueError: ball contains zero

>>> from sage.all import *
>>> b = ComplexBallField(Integer(8))(Integer(1)+i).below_abs()
>>> b
[1.4 +/- 0.0141]
>>> b.is_exact()
True
>>> QQ(b)*Integer(128)
181
>>> (CBF(Integer(1)/Integer(3)) - Integer(1)/Integer(3)).below_abs()
0
>>> (CBF(Integer(1)/Integer(3)) - Integer(1)/Integer(3)).below_abs(test_zero=True)
Traceback (most recent call last):
...
ValueError: ball contains zero

bessel_I(nu)[source]#

Return the modified Bessel function of the first kind with argument self and index nu.

EXAMPLES:

sage: CBF(1, 1).bessel_I(1)
[0.365028028827088 +/- ...e-16] + [0.614160334922903 +/- ...e-16]*I
sage: CBF(100, -100).bessel_I(1/3)
[5.4362189595644e+41 +/- ...e+27] + [7.1989436985321e+41 +/- ...e+27]*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).bessel_I(Integer(1))
[0.365028028827088 +/- ...e-16] + [0.614160334922903 +/- ...e-16]*I
>>> CBF(Integer(100), -Integer(100)).bessel_I(Integer(1)/Integer(3))
[5.4362189595644e+41 +/- ...e+27] + [7.1989436985321e+41 +/- ...e+27]*I

bessel_J(nu)[source]#

Return the Bessel function of the first kind with argument self and index nu.

EXAMPLES:

sage: CBF(1, 1).bessel_J(1)
[0.614160334922903 +/- ...e-16] + [0.365028028827088 +/- ...e-16]*I
sage: CBF(100, -100).bessel_J(1/3)
[1.108431870251e+41 +/- ...e+28] + [-8.952577603125e+41 +/- ...e+28]*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).bessel_J(Integer(1))
[0.614160334922903 +/- ...e-16] + [0.365028028827088 +/- ...e-16]*I
>>> CBF(Integer(100), -Integer(100)).bessel_J(Integer(1)/Integer(3))
[1.108431870251e+41 +/- ...e+28] + [-8.952577603125e+41 +/- ...e+28]*I

bessel_J_Y(nu)[source]#

Return the Bessel function of the first and second kind with argument self and index nu, computed simultaneously.

EXAMPLES:

sage: J, Y = CBF(1, 1).bessel_J_Y(1)
sage: J - CBF(1, 1).bessel_J(1)
[+/- ...e-16] + [+/- ...e-16]*I
sage: Y - CBF(1, 1).bessel_Y(1)
[+/- ...e-14] + [+/- ...e-14]*I

>>> from sage.all import *
>>> J, Y = CBF(Integer(1), Integer(1)).bessel_J_Y(Integer(1))
>>> J - CBF(Integer(1), Integer(1)).bessel_J(Integer(1))
[+/- ...e-16] + [+/- ...e-16]*I
>>> Y - CBF(Integer(1), Integer(1)).bessel_Y(Integer(1))
[+/- ...e-14] + [+/- ...e-14]*I

bessel_K(nu)[source]#

Return the modified Bessel function of the second kind with argument self and index nu.

EXAMPLES:

sage: CBF(1, 1).bessel_K(0)
[0.08019772694652 +/- ...e-15] + [-0.357277459285330 +/- ...e-16]*I
sage: CBF(1, 1).bessel_K(1)
[0.02456830552374 +/- ...e-15] + [-0.45971947380119 +/- ...e-15]*I
sage: CBF(100, 100).bessel_K(QQbar(i))
[3.8693896656383e-45 +/- ...e-59] + [5.507100423418e-46 +/- ...e-59]*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).bessel_K(Integer(0))
[0.08019772694652 +/- ...e-15] + [-0.357277459285330 +/- ...e-16]*I
>>> CBF(Integer(1), Integer(1)).bessel_K(Integer(1))
[0.02456830552374 +/- ...e-15] + [-0.45971947380119 +/- ...e-15]*I
>>> CBF(Integer(100), Integer(100)).bessel_K(QQbar(i))
[3.8693896656383e-45 +/- ...e-59] + [5.507100423418e-46 +/- ...e-59]*I

bessel_Y(nu)[source]#

Return the Bessel function of the second kind with argument self and index nu.

EXAMPLES:

sage: CBF(1, 1).bessel_Y(1)
[-0.6576945355913 +/- ...e-14] + [0.6298010039929 +/- ...e-14]*I
sage: CBF(100, -100).bessel_Y(1/3)
[-8.952577603125e+41 +/- ...e+28] + [-1.108431870251e+41 +/- ...e+28]*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).bessel_Y(Integer(1))
[-0.6576945355913 +/- ...e-14] + [0.6298010039929 +/- ...e-14]*I
>>> CBF(Integer(100), -Integer(100)).bessel_Y(Integer(1)/Integer(3))
[-8.952577603125e+41 +/- ...e+28] + [-1.108431870251e+41 +/- ...e+28]*I

chebyshev_T(n)[source]#

Return the Chebyshev function of the first kind of order n evaluated at self.

EXAMPLES:

sage: CBF(1/3).chebyshev_T(20)
[0.8710045668809 +/- ...e-14]
sage: CBF(1/3).chebyshev_T(CBF(5,1))
[1.84296854518763 +/- ...e-15] + [0.20053614301799 +/- ...e-15]*I

>>> from sage.all import *
>>> CBF(Integer(1)/Integer(3)).chebyshev_T(Integer(20))
[0.8710045668809 +/- ...e-14]
>>> CBF(Integer(1)/Integer(3)).chebyshev_T(CBF(Integer(5),Integer(1)))
[1.84296854518763 +/- ...e-15] + [0.20053614301799 +/- ...e-15]*I

chebyshev_U(n)[source]#

Return the Chebyshev function of the second kind of order n evaluated at self.

EXAMPLES:

sage: CBF(1/3).chebyshev_U(20)
[0.6973126541184 +/- ...e-14]
sage: CBF(1/3).chebyshev_U(CBF(5,1))
[1.75884964893425 +/- ...e-15] + [0.7497317165104 +/- ...e-14]*I

>>> from sage.all import *
>>> CBF(Integer(1)/Integer(3)).chebyshev_U(Integer(20))
[0.6973126541184 +/- ...e-14]
>>> CBF(Integer(1)/Integer(3)).chebyshev_U(CBF(Integer(5),Integer(1)))
[1.75884964893425 +/- ...e-15] + [0.7497317165104 +/- ...e-14]*I

conjugate()[source]#

Return the complex conjugate of this ball.

EXAMPLES:

sage: CBF(-2 + I/3).conjugate()
-2.000000000000000 + [-0.3333333333333333 +/- ...e-17]*I

>>> from sage.all import *
>>> CBF(-Integer(2) + I/Integer(3)).conjugate()
-2.000000000000000 + [-0.3333333333333333 +/- ...e-17]*I

contains_exact(other)[source]#

Return True iff other is contained in self.

Use other in self for a test that works for a wider range of inputs but may return false negatives.

INPUT:

EXAMPLES:

sage: CBF(RealBallField(100)(1/3), 0).contains_exact(1/3)
True
sage: CBF(1).contains_exact(1)
True
sage: CBF(1).contains_exact(CBF(1))
True

sage: CBF(sqrt(2)).contains_exact(sqrt(2))                                  # needs sage.symbolic
Traceback (most recent call last):
...
TypeError: unsupported type: <class 'sage.symbolic.expression.Expression'>

>>> from sage.all import *
>>> CBF(RealBallField(Integer(100))(Integer(1)/Integer(3)), Integer(0)).contains_exact(Integer(1)/Integer(3))
True
>>> CBF(Integer(1)).contains_exact(Integer(1))
True
>>> CBF(Integer(1)).contains_exact(CBF(Integer(1)))
True

>>> CBF(sqrt(Integer(2))).contains_exact(sqrt(Integer(2)))                                  # needs sage.symbolic
Traceback (most recent call last):
...
TypeError: unsupported type: <class 'sage.symbolic.expression.Expression'>

contains_integer()[source]#

Return True iff this ball contains any integer.

EXAMPLES:

sage: CBF(3, RBF(0.1)).contains_integer()
False
sage: CBF(3, RBF(0.1,0.1)).contains_integer()
True

>>> from sage.all import *
>>> CBF(Integer(3), RBF(RealNumber('0.1'))).contains_integer()
False
>>> CBF(Integer(3), RBF(RealNumber('0.1'),RealNumber('0.1'))).contains_integer()
True

contains_zero()[source]#

Return True iff this ball contains zero.

EXAMPLES:

sage: CBF(0).contains_zero()
True
sage: CBF(RIF(-1,1)).contains_zero()
True
sage: CBF(i).contains_zero()
False

>>> from sage.all import *
>>> CBF(Integer(0)).contains_zero()
True
>>> CBF(RIF(-Integer(1),Integer(1))).contains_zero()
True
>>> CBF(i).contains_zero()
False

cos()[source]#

Return the cosine of this ball.

EXAMPLES:

sage: CBF(i*pi).cos()                                                       # needs sage.symbolic
[11.59195327552152 +/- ...e-15]

>>> from sage.all import *
>>> CBF(i*pi).cos()                                                       # needs sage.symbolic
[11.59195327552152 +/- ...e-15]

cos_integral()[source]#

Return the cosine integral with argument self.

EXAMPLES:

sage: CBF(1, 1).Ci()
[0.882172180555936 +/- ...e-16] + [0.287249133519956 +/- ...e-16]*I
sage: CBF(0).Ci()
nan + nan*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).Ci()
[0.882172180555936 +/- ...e-16] + [0.287249133519956 +/- ...e-16]*I
>>> CBF(Integer(0)).Ci()
nan + nan*I

cosh()[source]#

Return the hyperbolic cosine of this ball.

EXAMPLES:

sage: CBF(1, 1).cosh()
[0.833730025131149 +/- ...e-16] + [0.988897705762865 +/- ...e-16]*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).cosh()
[0.833730025131149 +/- ...e-16] + [0.988897705762865 +/- ...e-16]*I

cosh_integral()[source]#

Return the hyperbolic cosine integral with argument self.

EXAMPLES:

sage: CBF(1, 1).Chi()
[0.882172180555936 +/- ...e-16] + [1.28354719327494 +/- ...e-15]*I
sage: CBF(0).Chi()
nan + nan*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).Chi()
[0.882172180555936 +/- ...e-16] + [1.28354719327494 +/- ...e-15]*I
>>> CBF(Integer(0)).Chi()
nan + nan*I

cot()[source]#

Return the cotangent of this ball.

EXAMPLES:

sage: CBF(pi, 1/10).cot()                                                   # needs sage.symbolic
[+/- ...e-14] + [-10.03331113225399 +/- ...e-15]*I
sage: CBF(pi).cot()                                                         # needs sage.symbolic
nan

>>> from sage.all import *
>>> CBF(pi, Integer(1)/Integer(10)).cot()                                                   # needs sage.symbolic
[+/- ...e-14] + [-10.03331113225399 +/- ...e-15]*I
>>> CBF(pi).cot()                                                         # needs sage.symbolic
nan

coth()[source]#

Return the hyperbolic cotangent of this ball.

EXAMPLES:

sage: CBF(1, 1).coth()
[0.868014142895925 +/- ...e-16] + [-0.2176215618544027 +/- ...e-17]*I
sage: CBF(0, pi).coth()                                                     # needs sage.symbolic
nan*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).coth()
[0.868014142895925 +/- ...e-16] + [-0.2176215618544027 +/- ...e-17]*I
>>> CBF(Integer(0), pi).coth()                                                     # needs sage.symbolic
nan*I

csc()[source]#

Return the cosecant of this ball.

EXAMPLES:

sage: CBF(1, 1).csc()
[0.621518017170428 +/- ...e-16] + [-0.303931001628426 +/- ...e-16]*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).csc()
[0.621518017170428 +/- ...e-16] + [-0.303931001628426 +/- ...e-16]*I

csch()[source]#

Return the hyperbolic cosecant of this ball.

EXAMPLES:

sage: CBF(1, 1).csch()
[0.303931001628426 +/- ...e-16] + [-0.621518017170428 +/- ...e-16]*I
sage: CBF(i*pi).csch()                                                      # needs sage.symbolic
nan*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).csch()
[0.303931001628426 +/- ...e-16] + [-0.621518017170428 +/- ...e-16]*I
>>> CBF(i*pi).csch()                                                      # needs sage.symbolic
nan*I

cube()[source]#

Return the cube of this ball.

The result is computed efficiently using two real squarings, two real multiplications, and scalar operations.

EXAMPLES:

sage: CBF(1, 1).cube()
-2.000000000000000 + 2.000000000000000*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).cube()
-2.000000000000000 + 2.000000000000000*I

diameter()[source]#

Return the diameter of this ball.

EXAMPLES:

sage: CBF(1 + i).diameter()
0.00000000
sage: CBF(i/3).diameter()
2.2204460e-16
sage: CBF(i/3).diameter().parent()
Real Field with 30 bits of precision
sage: CBF(CIF(RIF(1.02, 1.04), RIF(2.1, 2.2))).diameter()
0.20000000

>>> from sage.all import *
>>> CBF(Integer(1) + i).diameter()
0.00000000
>>> CBF(i/Integer(3)).diameter()
2.2204460e-16
>>> CBF(i/Integer(3)).diameter().parent()
Real Field with 30 bits of precision
>>> CBF(CIF(RIF(RealNumber('1.02'), RealNumber('1.04')), RIF(RealNumber('2.1'), RealNumber('2.2')))).diameter()
0.20000000


eisenstein(n)[source]#

Return the first n entries in the sequence of Eisenstein series $$G_4(\tau), G_6(\tau), G_8(\tau), \ldots$$ where tau is given by self. The output is a list.

EXAMPLES:

sage: a, b, c, d = 2, 5, 1, 3
sage: tau = CBF(1,3)
sage: tau.eisenstein(4)
[[2.1646498507193 +/- ...e-14],
[2.0346794456073 +/- ...e-14],
[2.0081609898081 +/- ...e-14],
[2.0019857082706 +/- ...e-14]]
sage: ((a*tau+b)/(c*tau+d)).eisenstein(3)[2]
[331011.2004330 +/- ...e-8] + [-711178.1655746 +/- ...e-8]*I
sage: (c*tau+d)^8 * tau.eisenstein(3)[2]
[331011.20043304 +/- ...e-9] + [-711178.1655746 +/- ...e-8]*I

>>> from sage.all import *
>>> a, b, c, d = Integer(2), Integer(5), Integer(1), Integer(3)
>>> tau = CBF(Integer(1),Integer(3))
>>> tau.eisenstein(Integer(4))
[[2.1646498507193 +/- ...e-14],
[2.0346794456073 +/- ...e-14],
[2.0081609898081 +/- ...e-14],
[2.0019857082706 +/- ...e-14]]
>>> ((a*tau+b)/(c*tau+d)).eisenstein(Integer(3))[Integer(2)]
[331011.2004330 +/- ...e-8] + [-711178.1655746 +/- ...e-8]*I
>>> (c*tau+d)**Integer(8) * tau.eisenstein(Integer(3))[Integer(2)]
[331011.20043304 +/- ...e-9] + [-711178.1655746 +/- ...e-8]*I

elliptic_e()[source]#

Return the complete elliptic integral of the second kind evaluated at m given by self.

EXAMPLES:

sage: CBF(2,3).elliptic_e()
[1.472797144959 +/- ...e-13] + [-1.231604783936 +/- ...e-14]*I

>>> from sage.all import *
>>> CBF(Integer(2),Integer(3)).elliptic_e()
[1.472797144959 +/- ...e-13] + [-1.231604783936 +/- ...e-14]*I

elliptic_e_inc(m)[source]#

Return the incomplete elliptic integral of the second kind evaluated at m.

See elliptic_e() for the corresponding complete integral

INPUT:

• m – complex ball

EXAMPLES:

sage: CBF(1,2).elliptic_e_inc(CBF(0,1))
[1.906576998914 +/- ...e-13] + [3.6896645289411 +/- ...e-14]*I

>>> from sage.all import *
>>> CBF(Integer(1),Integer(2)).elliptic_e_inc(CBF(Integer(0),Integer(1)))
[1.906576998914 +/- ...e-13] + [3.6896645289411 +/- ...e-14]*I


At parameter $$\pi/2$$ it is a complete integral:

sage: phi = CBF(1,1)
sage: (CBF.pi()/2).elliptic_e_inc(phi)
[1.2838409578982 +/- ...e-14] + [-0.5317843366915 +/- ...e-14]*I
sage: phi.elliptic_e()
[1.2838409578982 +/- 5...e-14] + [-0.5317843366915 +/- 3...e-14]*I

sage: phi = CBF(2, 3/7)
sage: (CBF.pi()/2).elliptic_e_inc(phi)
[0.787564350925 +/- ...e-13] + [-0.686896129145 +/- ...e-13]*I
sage: phi.elliptic_e()
[0.7875643509254 +/- ...e-14] + [-0.686896129145 +/- ...e-13]*I

>>> from sage.all import *
>>> phi = CBF(Integer(1),Integer(1))
>>> (CBF.pi()/Integer(2)).elliptic_e_inc(phi)
[1.2838409578982 +/- ...e-14] + [-0.5317843366915 +/- ...e-14]*I
>>> phi.elliptic_e()
[1.2838409578982 +/- 5...e-14] + [-0.5317843366915 +/- 3...e-14]*I

>>> phi = CBF(Integer(2), Integer(3)/Integer(7))
>>> (CBF.pi()/Integer(2)).elliptic_e_inc(phi)
[0.787564350925 +/- ...e-13] + [-0.686896129145 +/- ...e-13]*I
>>> phi.elliptic_e()
[0.7875643509254 +/- ...e-14] + [-0.686896129145 +/- ...e-13]*I

elliptic_f(m)[source]#

Return the incomplete elliptic integral of the first kind evaluated at m.

See elliptic_k() for the corresponding complete integral

INPUT:

• m – complex ball

EXAMPLES:

sage: CBF(1,2).elliptic_f(CBF(0,1))
[0.6821522911854 +/- ...e-14] + [1.2482780628143 +/- ...e-14]*I

>>> from sage.all import *
>>> CBF(Integer(1),Integer(2)).elliptic_f(CBF(Integer(0),Integer(1)))
[0.6821522911854 +/- ...e-14] + [1.2482780628143 +/- ...e-14]*I


At parameter $$\pi/2$$ it is a complete integral:

sage: phi = CBF(1,1)
sage: (CBF.pi()/2).elliptic_f(phi)
[1.5092369540513 +/- ...e-14] + [0.6251464152027 +/- ...e-15]*I
sage: phi.elliptic_k()
[1.50923695405127 +/- ...e-15] + [0.62514641520270 +/- ...e-15]*I

sage: phi = CBF(2, 3/7)
sage: (CBF.pi()/2).elliptic_f(phi)
[1.3393589639094 +/- ...e-14] + [1.1104369690719 +/- ...e-14]*I
sage: phi.elliptic_k()
[1.33935896390938 +/- ...e-15] + [1.11043696907194 +/- ...e-15]*I

>>> from sage.all import *
>>> phi = CBF(Integer(1),Integer(1))
>>> (CBF.pi()/Integer(2)).elliptic_f(phi)
[1.5092369540513 +/- ...e-14] + [0.6251464152027 +/- ...e-15]*I
>>> phi.elliptic_k()
[1.50923695405127 +/- ...e-15] + [0.62514641520270 +/- ...e-15]*I

>>> phi = CBF(Integer(2), Integer(3)/Integer(7))
>>> (CBF.pi()/Integer(2)).elliptic_f(phi)
[1.3393589639094 +/- ...e-14] + [1.1104369690719 +/- ...e-14]*I
>>> phi.elliptic_k()
[1.33935896390938 +/- ...e-15] + [1.11043696907194 +/- ...e-15]*I

elliptic_invariants()[source]#

Return the lattice invariants (g2, g3).

EXAMPLES:

sage: CBF(0,1).elliptic_invariants()
([189.07272012923 +/- ...e-12], [+/- ...e-12])
sage: CBF(sqrt(2)/2, sqrt(2)/2).elliptic_invariants()                       # needs sage.symbolic
([+/- ...e-12] + [-332.5338031465...]*I,
[1254.46842157...] + [1254.46842157...]*I)

>>> from sage.all import *
>>> CBF(Integer(0),Integer(1)).elliptic_invariants()
([189.07272012923 +/- ...e-12], [+/- ...e-12])
>>> CBF(sqrt(Integer(2))/Integer(2), sqrt(Integer(2))/Integer(2)).elliptic_invariants()                       # needs sage.symbolic
([+/- ...e-12] + [-332.5338031465...]*I,
[1254.46842157...] + [1254.46842157...]*I)

elliptic_k()[source]#

Return the complete elliptic integral of the first kind evaluated at m given by self.

EXAMPLES:

sage: CBF(2,3).elliptic_k()
[1.04291329192852 +/- ...e-15] + [0.62968247230864 +/- ...e-15]*I

>>> from sage.all import *
>>> CBF(Integer(2),Integer(3)).elliptic_k()
[1.04291329192852 +/- ...e-15] + [0.62968247230864 +/- ...e-15]*I

elliptic_p(tau, n=None)[source]#

Return the Weierstrass elliptic function with lattice parameter tau, evaluated at self. The function is doubly periodic in self with periods 1 and tau, which should lie in the upper half plane.

If n is given, return a list containing the first n terms in the Taylor expansion at self. In particular, with n = 2, compute the Weierstrass elliptic function together with its derivative, which generate the field of elliptic functions with periods 1 and tau.

EXAMPLES:

sage: tau = CBF(1,4)
sage: z = CBF(sqrt(2), sqrt(3))                                             # needs sage.symbolic
sage: z.elliptic_p(tau)                                                     # needs sage.symbolic
[-3.28920996772709 +/- ...e-15] + [-0.0003673767302933 +/- ...e-17]*I
sage: (z + tau).elliptic_p(tau)                                             # needs sage.symbolic
[-3.28920996772709 +/- ...e-15] + [-0.000367376730293 +/- ...e-16]*I
sage: (z + 1).elliptic_p(tau)                                               # needs sage.symbolic
[-3.28920996772709 +/- ...e-15] + [-0.0003673767302933 +/- ...e-17]*I

sage: z.elliptic_p(tau, 3)                                                  # needs sage.symbolic
[[-3.28920996772709 +/- ...e-15] + [-0.0003673767302933 +/- ...e-17]*I,
[0.002473055794309 +/- ...e-16] + [0.003859554040267 +/- ...e-16]*I,
[-0.01299087561709 +/- ...e-15] + [0.00725027521915 +/- ...e-15]*I]
sage: (z + 3 + 4*tau).elliptic_p(tau, 3)                                    # needs sage.symbolic
[[-3.28920996772709 +/- ...e-15] + [-0.00036737673029 +/- ...e-15]*I,
[0.0024730557943 +/- ...e-14] + [0.0038595540403 +/- ...e-14]*I,
[-0.01299087562 +/- ...e-12] + [0.00725027522 +/- ...e-12]*I]

>>> from sage.all import *
>>> tau = CBF(Integer(1),Integer(4))
>>> z = CBF(sqrt(Integer(2)), sqrt(Integer(3)))                                             # needs sage.symbolic
>>> z.elliptic_p(tau)                                                     # needs sage.symbolic
[-3.28920996772709 +/- ...e-15] + [-0.0003673767302933 +/- ...e-17]*I
>>> (z + tau).elliptic_p(tau)                                             # needs sage.symbolic
[-3.28920996772709 +/- ...e-15] + [-0.000367376730293 +/- ...e-16]*I
>>> (z + Integer(1)).elliptic_p(tau)                                               # needs sage.symbolic
[-3.28920996772709 +/- ...e-15] + [-0.0003673767302933 +/- ...e-17]*I

>>> z.elliptic_p(tau, Integer(3))                                                  # needs sage.symbolic
[[-3.28920996772709 +/- ...e-15] + [-0.0003673767302933 +/- ...e-17]*I,
[0.002473055794309 +/- ...e-16] + [0.003859554040267 +/- ...e-16]*I,
[-0.01299087561709 +/- ...e-15] + [0.00725027521915 +/- ...e-15]*I]
>>> (z + Integer(3) + Integer(4)*tau).elliptic_p(tau, Integer(3))                                    # needs sage.symbolic
[[-3.28920996772709 +/- ...e-15] + [-0.00036737673029 +/- ...e-15]*I,
[0.0024730557943 +/- ...e-14] + [0.0038595540403 +/- ...e-14]*I,
[-0.01299087562 +/- ...e-12] + [0.00725027522 +/- ...e-12]*I]

elliptic_pi(m)[source]#

Return the complete elliptic integral of the third kind evaluated at m given by self.

EXAMPLES:

sage: CBF(2,3).elliptic_pi(CBF(1,1))
[0.2702999736198...] + [0.715676058329...]*I

>>> from sage.all import *
>>> CBF(Integer(2),Integer(3)).elliptic_pi(CBF(Integer(1),Integer(1)))
[0.2702999736198...] + [0.715676058329...]*I

elliptic_pi_inc(phi, m)[source]#

Return the Legendre incomplete elliptic integral of the third kind.

See: elliptic_pi() for the complete integral.

INPUT:

• phi – complex ball

• m – complex ball

EXAMPLES:

sage: CBF(1,2).elliptic_pi_inc(CBF(0,1), CBF(2,-3))
[0.05738864021418 +/- ...e-15] + [0.55557494549951 +/- ...e-15]*I

>>> from sage.all import *
>>> CBF(Integer(1),Integer(2)).elliptic_pi_inc(CBF(Integer(0),Integer(1)), CBF(Integer(2),-Integer(3)))
[0.05738864021418 +/- ...e-15] + [0.55557494549951 +/- ...e-15]*I


At parameter $$\pi/2$$ it is a complete integral:

sage: n = CBF(1,1)
sage: m = CBF(-2/3, 3/5)
sage: n.elliptic_pi_inc(CBF.pi()/2, m) # this is a regression, see :issue:28623
nan + nan*I
sage: n.elliptic_pi(m)
[0.8934793755173...] + [0.957078687107...]*I

sage: n = CBF(2, 3/7)
sage: m = CBF(-1/3, 2/9)
sage: n.elliptic_pi_inc(CBF.pi()/2, m) # this is a regression, see :issue:28623
nan + nan*I
sage: n.elliptic_pi(m)
[0.296958874641...] + [1.318879533273...]*I

>>> from sage.all import *
>>> n = CBF(Integer(1),Integer(1))
>>> m = CBF(-Integer(2)/Integer(3), Integer(3)/Integer(5))
>>> n.elliptic_pi_inc(CBF.pi()/Integer(2), m) # this is a regression, see :issue:28623
nan + nan*I
>>> n.elliptic_pi(m)
[0.8934793755173...] + [0.957078687107...]*I

>>> n = CBF(Integer(2), Integer(3)/Integer(7))
>>> m = CBF(-Integer(1)/Integer(3), Integer(2)/Integer(9))
>>> n.elliptic_pi_inc(CBF.pi()/Integer(2), m) # this is a regression, see :issue:28623
nan + nan*I
>>> n.elliptic_pi(m)
[0.296958874641...] + [1.318879533273...]*I

elliptic_rf(y, z)[source]#

Return the Carlson symmetric elliptic integral of the first kind evaluated at (self, y, z).

INPUT:

• y – complex ball

• z – complex ball

EXAMPLES:

sage: CBF(0,1).elliptic_rf(CBF(-1/2,1), CBF(-1,-1))
[1.469800396738515 +/- ...e-16] + [-0.2358791199824196 +/- ...e-17]*I

>>> from sage.all import *
>>> CBF(Integer(0),Integer(1)).elliptic_rf(CBF(-Integer(1)/Integer(2),Integer(1)), CBF(-Integer(1),-Integer(1)))
[1.469800396738515 +/- ...e-16] + [-0.2358791199824196 +/- ...e-17]*I

elliptic_rg(y, z)[source]#

Return the Carlson symmetric elliptic integral of the second kind evaluated at (self, y, z).

INPUT:

• y – complex ball

• z – complex ball

EXAMPLES:

sage: CBF(0,1).elliptic_rg(CBF(-1/2,1), CBF(-1,-1))
[0.1586786770922370 +/- ...e-17] + [0.2239733128130531 +/- ...e-17]*I

>>> from sage.all import *
>>> CBF(Integer(0),Integer(1)).elliptic_rg(CBF(-Integer(1)/Integer(2),Integer(1)), CBF(-Integer(1),-Integer(1)))
[0.1586786770922370 +/- ...e-17] + [0.2239733128130531 +/- ...e-17]*I

elliptic_rj(y, z, p)[source]#

Return the Carlson symmetric elliptic integral of the third kind evaluated at (self, y, z).

INPUT:

• y – complex ball

• z – complex ball

• p – complex bamm

EXAMPLES:

sage: CBF(0,1).elliptic_rj(CBF(-1/2,1), CBF(-1,-1), CBF(2))
[1.00438675628573...] + [-0.24516268343916...]*I

>>> from sage.all import *
>>> CBF(Integer(0),Integer(1)).elliptic_rj(CBF(-Integer(1)/Integer(2),Integer(1)), CBF(-Integer(1),-Integer(1)), CBF(Integer(2)))
[1.00438675628573...] + [-0.24516268343916...]*I

elliptic_roots()[source]#

Return the lattice roots (e1, e2, e3) of $$4 z^3 - g_2 z - g_3$$.

EXAMPLES:

sage: e1, e2, e3 = CBF(0,1).elliptic_roots()
sage: e1, e2, e3
([6.8751858180204 +/- ...e-14],
[+/- ...e-14],
[-6.8751858180204 +/- ...e-14])
sage: g2, g3 = CBF(0,1).elliptic_invariants()
sage: 4 * e1^3 - g2 * e1 - g3
[+/- ...e-11]

>>> from sage.all import *
>>> e1, e2, e3 = CBF(Integer(0),Integer(1)).elliptic_roots()
>>> e1, e2, e3
([6.8751858180204 +/- ...e-14],
[+/- ...e-14],
[-6.8751858180204 +/- ...e-14])
>>> g2, g3 = CBF(Integer(0),Integer(1)).elliptic_invariants()
>>> Integer(4) * e1**Integer(3) - g2 * e1 - g3
[+/- ...e-11]

elliptic_sigma(tau)[source]#

Return the value of the Weierstrass sigma function at (self, tau)

EXAMPLES:

- tau -- a complex ball with positive imaginary part


EXAMPLES:

sage: CBF(1,1).elliptic_sigma(CBF(1,3))
[-0.543073363596 +/- ...e-13] + [3.6357291186244 +/- ...e-14]*I

>>> from sage.all import *
>>> CBF(Integer(1),Integer(1)).elliptic_sigma(CBF(Integer(1),Integer(3)))
[-0.543073363596 +/- ...e-13] + [3.6357291186244 +/- ...e-14]*I

elliptic_zeta(tau)[source]#

Return the value of the Weierstrass zeta function at (self, tau)

EXAMPLES:

- tau -- a complex ball with positive imaginary part


EXAMPLES:

sage: CBF(1,1).elliptic_zeta(CBF(1,3))
[3.2898676194970 +/- ...e-14] + [0.1365414361782 +/- ...e-14]*I

>>> from sage.all import *
>>> CBF(Integer(1),Integer(1)).elliptic_zeta(CBF(Integer(1),Integer(3)))
[3.2898676194970 +/- ...e-14] + [0.1365414361782 +/- ...e-14]*I

erf()[source]#

Return the error function with argument self.

EXAMPLES:

sage: CBF(1, 1).erf()
[1.316151281697947 +/- ...e-16] + [0.1904534692378347 +/- ...e-17]*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).erf()
[1.316151281697947 +/- ...e-16] + [0.1904534692378347 +/- ...e-17]*I

erfc()[source]#

Compute the complementary error function with argument self.

EXAMPLES:

sage: CBF(20).erfc() # abs tol 1e-190
[5.39586561160790e-176 +/- 6.73e-191]
sage: CBF(100, 100).erfc()
[0.00065234366376858 +/- ...e-18] + [-0.00393572636292141 +/- ...e-18]*I

>>> from sage.all import *
>>> CBF(Integer(20)).erfc() # abs tol 1e-190
[5.39586561160790e-176 +/- 6.73e-191]
>>> CBF(Integer(100), Integer(100)).erfc()
[0.00065234366376858 +/- ...e-18] + [-0.00393572636292141 +/- ...e-18]*I

exp()[source]#

Return the exponential of this ball.

EXAMPLES:

sage: CBF(i*pi).exp()                                                       # needs sage.symbolic
[-1.00000000000000 +/- ...e-16] + [+/- ...e-16]*I

>>> from sage.all import *
>>> CBF(i*pi).exp()                                                       # needs sage.symbolic
[-1.00000000000000 +/- ...e-16] + [+/- ...e-16]*I

exp_integral_e(s)[source]#

Return the image of this ball by the generalized exponential integral with index s.

EXAMPLES:

sage: CBF(1+i).exp_integral_e(1)
[0.00028162445198 +/- ...e-15] + [-0.17932453503936 +/- ...e-15]*I
sage: CBF(1+i).exp_integral_e(QQbar(i))
[-0.10396361883964 +/- ...e-15] + [-0.16268401277783 +/- ...e-15]*I

>>> from sage.all import *
>>> CBF(Integer(1)+i).exp_integral_e(Integer(1))
[0.00028162445198 +/- ...e-15] + [-0.17932453503936 +/- ...e-15]*I
>>> CBF(Integer(1)+i).exp_integral_e(QQbar(i))
[-0.10396361883964 +/- ...e-15] + [-0.16268401277783 +/- ...e-15]*I

exppii()[source]#

Return exp(pi*i*self).

EXAMPLES:

sage: CBF(1/2).exppii()
1.000000000000000*I
sage: CBF(0, -1/pi).exppii()                                                # needs sage.symbolic
[2.71828182845904 +/- ...e-15]

>>> from sage.all import *
>>> CBF(Integer(1)/Integer(2)).exppii()
1.000000000000000*I
>>> CBF(Integer(0), -Integer(1)/pi).exppii()                                                # needs sage.symbolic
[2.71828182845904 +/- ...e-15]

gamma(z=None)[source]#

Return the image of this ball by the Euler Gamma function (if z = None) or the incomplete Gamma function (otherwise).

EXAMPLES:

sage: CBF(1, 1).gamma() # abs tol 1e-15
[0.498015668118356 +/- 1.26e-16] + [-0.1549498283018107 +/- 8.43e-17]*I
sage: CBF(-1).gamma()
nan
sage: CBF(1, 1).gamma(0) # abs tol 1e-15
[0.498015668118356 +/- 1.26e-16] + [-0.1549498283018107 +/- 8.43e-17]*I
sage: CBF(1, 1).gamma(100)
[-3.6143867454139e-45 +/- ...e-59] + [-3.7022961377791e-44 +/- ...e-58]*I
sage: CBF(1, 1).gamma(CLF(i)) # abs tol 1e-14
[0.328866841935004 +/- 7.07e-16] + [-0.189749450456210 +/- 9.05e-16]*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).gamma() # abs tol 1e-15
[0.498015668118356 +/- 1.26e-16] + [-0.1549498283018107 +/- 8.43e-17]*I
>>> CBF(-Integer(1)).gamma()
nan
>>> CBF(Integer(1), Integer(1)).gamma(Integer(0)) # abs tol 1e-15
[0.498015668118356 +/- 1.26e-16] + [-0.1549498283018107 +/- 8.43e-17]*I
>>> CBF(Integer(1), Integer(1)).gamma(Integer(100))
[-3.6143867454139e-45 +/- ...e-59] + [-3.7022961377791e-44 +/- ...e-58]*I
>>> CBF(Integer(1), Integer(1)).gamma(CLF(i)) # abs tol 1e-14
[0.328866841935004 +/- 7.07e-16] + [-0.189749450456210 +/- 9.05e-16]*I

gamma_inc(z=None)[source]#

Return the image of this ball by the Euler Gamma function (if z = None) or the incomplete Gamma function (otherwise).

EXAMPLES:

sage: CBF(1, 1).gamma() # abs tol 1e-15
[0.498015668118356 +/- 1.26e-16] + [-0.1549498283018107 +/- 8.43e-17]*I
sage: CBF(-1).gamma()
nan
sage: CBF(1, 1).gamma(0) # abs tol 1e-15
[0.498015668118356 +/- 1.26e-16] + [-0.1549498283018107 +/- 8.43e-17]*I
sage: CBF(1, 1).gamma(100)
[-3.6143867454139e-45 +/- ...e-59] + [-3.7022961377791e-44 +/- ...e-58]*I
sage: CBF(1, 1).gamma(CLF(i)) # abs tol 1e-14
[0.328866841935004 +/- 7.07e-16] + [-0.189749450456210 +/- 9.05e-16]*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).gamma() # abs tol 1e-15
[0.498015668118356 +/- 1.26e-16] + [-0.1549498283018107 +/- 8.43e-17]*I
>>> CBF(-Integer(1)).gamma()
nan
>>> CBF(Integer(1), Integer(1)).gamma(Integer(0)) # abs tol 1e-15
[0.498015668118356 +/- 1.26e-16] + [-0.1549498283018107 +/- 8.43e-17]*I
>>> CBF(Integer(1), Integer(1)).gamma(Integer(100))
[-3.6143867454139e-45 +/- ...e-59] + [-3.7022961377791e-44 +/- ...e-58]*I
>>> CBF(Integer(1), Integer(1)).gamma(CLF(i)) # abs tol 1e-14
[0.328866841935004 +/- 7.07e-16] + [-0.189749450456210 +/- 9.05e-16]*I

gegenbauer_C(n, m)[source]#

Return the Gegenbauer polynomial (or function) $$C_n^m(z)$$ evaluated at self.

EXAMPLES:

sage: CBF(-10).gegenbauer_C(7, 1/2)
[-263813415.6250000 +/- ...e-8]

>>> from sage.all import *
>>> CBF(-Integer(10)).gegenbauer_C(Integer(7), Integer(1)/Integer(2))
[-263813415.6250000 +/- ...e-8]

hermite_H(n)[source]#

Return the Hermite function (or polynomial) of order n evaluated at self.

EXAMPLES:

sage: CBF(10).hermite_H(1)
20.00000000000000
sage: CBF(10).hermite_H(30)
[8.0574670961707e+37 +/- ...e+23]

>>> from sage.all import *
>>> CBF(Integer(10)).hermite_H(Integer(1))
20.00000000000000
>>> CBF(Integer(10)).hermite_H(Integer(30))
[8.0574670961707e+37 +/- ...e+23]

hypergeometric(a, b, regularized=False)[source]#

Return the generalized hypergeometric function of self.

INPUT:

• a – upper parameters, list of complex numbers that coerce into this ball’s parent;

• b – lower parameters, list of complex numbers that coerce into this ball’s parent.

• regularized – if True, the regularized generalized hypergeometric function is computed.

OUTPUT:

The generalized hypergeometric function defined by

${}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{k=0}^\infty \frac{(a_1)_k\dots(a_p)_k}{(b_1)_k\dots(b_q)_k} \frac {z^k} {k!}$

extended using analytic continuation or regularization when the sum does not converge.

The regularized generalized hypergeometric function

${}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{k=0}^\infty \frac{(a_1)_k\dots(a_p)_k}{\Gamma(b_1+k)\dots\Gamma(b_q+k)} \frac {z^k} {k!}$

is well-defined even when the lower parameters are nonpositive integers. Currently, this is only supported for some $$p$$ and $$q$$.

EXAMPLES:

sage: CBF(1, pi/2).hypergeometric([], [])                                   # needs sage.symbolic
[+/- ...e-16] + [2.71828182845904 +/- ...e-15]*I

sage: CBF(1, pi).hypergeometric([1/4], [1/4])                               # needs sage.symbolic
[-2.7182818284590 +/- ...e-14] + [+/- ...e-14]*I

sage: CBF(1000, 1000).hypergeometric([10], [AA(sqrt(2))])                   # needs sage.symbolic
[9.79300951360e+454 +/- ...e+442] + [5.522579106816e+455 +/- ...e+442]*I
sage: CBF(1000, 1000).hypergeometric([100], [AA(sqrt(2))])                  # needs sage.symbolic
[1.27967355557e+590 +/- ...e+578] + [-9.32333491987e+590 +/- ...e+578]*I

sage: CBF(0, 1).hypergeometric([], [1/2, 1/3, 1/4])
[-3.7991962344383 +/- ...e-14] + [23.878097177805 +/- ...e-13]*I

sage: CBF(0).hypergeometric([1], [])
1.000000000000000
sage: CBF(1, 1).hypergeometric([1], [])
1.000000000000000*I

sage: CBF(2+3*I).hypergeometric([1/4,1/3],[1/2]) # abs tol 1e-14
[0.7871684267473 +/- 6.79e-14] + [0.2749254173721 +/- 8.82e-14]*I
sage: CBF(2+3*I).hypergeometric([1/4,1/3],[1/2],regularized=True)
[0.4441122268685 +/- 3...e-14] + [0.1551100567338 +/- 5...e-14]*I

sage: CBF(5).hypergeometric([2,3], [-5])
nan + nan*I
sage: CBF(5).hypergeometric([2,3], [-5], regularized=True)
[5106.925964355 +/- ...e-10]

sage: CBF(2016).hypergeometric([], [2/3]) # abs tol 1e+26
[2.0256426923278e+38 +/- 9.59e+24]
sage: CBF(-2016).hypergeometric([], [2/3], regularized=True)
[-0.0005428550847 +/- ...e-14]

sage: CBF(-7).hypergeometric([4], [])
0.0002441406250000000

sage: CBF(0, 3).hypergeometric([CBF(1,1)], [-4], regularized=True)
[239.514000752841 +/- ...e-13] + [105.175157349015 +/- ...e-13]*I

>>> from sage.all import *
>>> CBF(Integer(1), pi/Integer(2)).hypergeometric([], [])                                   # needs sage.symbolic
[+/- ...e-16] + [2.71828182845904 +/- ...e-15]*I

>>> CBF(Integer(1), pi).hypergeometric([Integer(1)/Integer(4)], [Integer(1)/Integer(4)])                               # needs sage.symbolic
[-2.7182818284590 +/- ...e-14] + [+/- ...e-14]*I

>>> CBF(Integer(1000), Integer(1000)).hypergeometric([Integer(10)], [AA(sqrt(Integer(2)))])                   # needs sage.symbolic
[9.79300951360e+454 +/- ...e+442] + [5.522579106816e+455 +/- ...e+442]*I
>>> CBF(Integer(1000), Integer(1000)).hypergeometric([Integer(100)], [AA(sqrt(Integer(2)))])                  # needs sage.symbolic
[1.27967355557e+590 +/- ...e+578] + [-9.32333491987e+590 +/- ...e+578]*I

>>> CBF(Integer(0), Integer(1)).hypergeometric([], [Integer(1)/Integer(2), Integer(1)/Integer(3), Integer(1)/Integer(4)])
[-3.7991962344383 +/- ...e-14] + [23.878097177805 +/- ...e-13]*I

>>> CBF(Integer(0)).hypergeometric([Integer(1)], [])
1.000000000000000
>>> CBF(Integer(1), Integer(1)).hypergeometric([Integer(1)], [])
1.000000000000000*I

>>> CBF(Integer(2)+Integer(3)*I).hypergeometric([Integer(1)/Integer(4),Integer(1)/Integer(3)],[Integer(1)/Integer(2)]) # abs tol 1e-14
[0.7871684267473 +/- 6.79e-14] + [0.2749254173721 +/- 8.82e-14]*I
>>> CBF(Integer(2)+Integer(3)*I).hypergeometric([Integer(1)/Integer(4),Integer(1)/Integer(3)],[Integer(1)/Integer(2)],regularized=True)
[0.4441122268685 +/- 3...e-14] + [0.1551100567338 +/- 5...e-14]*I

>>> CBF(Integer(5)).hypergeometric([Integer(2),Integer(3)], [-Integer(5)])
nan + nan*I
>>> CBF(Integer(5)).hypergeometric([Integer(2),Integer(3)], [-Integer(5)], regularized=True)
[5106.925964355 +/- ...e-10]

>>> CBF(Integer(2016)).hypergeometric([], [Integer(2)/Integer(3)]) # abs tol 1e+26
[2.0256426923278e+38 +/- 9.59e+24]
>>> CBF(-Integer(2016)).hypergeometric([], [Integer(2)/Integer(3)], regularized=True)
[-0.0005428550847 +/- ...e-14]

>>> CBF(-Integer(7)).hypergeometric([Integer(4)], [])
0.0002441406250000000

>>> CBF(Integer(0), Integer(3)).hypergeometric([CBF(Integer(1),Integer(1))], [-Integer(4)], regularized=True)
[239.514000752841 +/- ...e-13] + [105.175157349015 +/- ...e-13]*I

hypergeometric_U(a, b)[source]#

Return the Tricomi confluent hypergeometric function U(a, b, self) of this ball.

EXAMPLES:

sage: CBF(1000, 1000).hypergeometric_U(RLF(pi), -100)                       # needs sage.symbolic
[-7.261605907166e-11 +/- ...e-24] + [-7.928136216391e-11 +/- ...e-24]*I
sage: CBF(1000, 1000).hypergeometric_U(0, -100)
1.000000000000000

>>> from sage.all import *
>>> CBF(Integer(1000), Integer(1000)).hypergeometric_U(RLF(pi), -Integer(100))                       # needs sage.symbolic
[-7.261605907166e-11 +/- ...e-24] + [-7.928136216391e-11 +/- ...e-24]*I
>>> CBF(Integer(1000), Integer(1000)).hypergeometric_U(Integer(0), -Integer(100))
1.000000000000000

identical(other)[source]#

Return whether self and other represent the same ball.

INPUT:

OUTPUT:

Return True iff self and other are equal as sets, i.e. if their real and imaginary parts each have the same midpoint and radius.

Note that this is not the same thing as testing whether both self and other certainly represent the complex real number, unless either self or other is exact (and neither contains NaN). To test whether both operands might represent the same mathematical quantity, use overlaps() or in, depending on the circumstance.

EXAMPLES:

sage: CBF(1, 1/3).identical(1 + CBF(0, 1)/3)
True
sage: CBF(1, 1).identical(1 + CBF(0, 1/3)*3)
False

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)/Integer(3)).identical(Integer(1) + CBF(Integer(0), Integer(1))/Integer(3))
True
>>> CBF(Integer(1), Integer(1)).identical(Integer(1) + CBF(Integer(0), Integer(1)/Integer(3))*Integer(3))
False

imag()[source]#

Return the imaginary part of this ball.

OUTPUT:

EXAMPLES:

sage: a = CBF(1/3, 1/5)
sage: a.imag()
[0.2000000000000000 +/- ...e-17]
sage: a.imag().parent()
Real ball field with 53 bits of precision

>>> from sage.all import *
>>> a = CBF(Integer(1)/Integer(3), Integer(1)/Integer(5))
>>> a.imag()
[0.2000000000000000 +/- ...e-17]
>>> a.imag().parent()
Real ball field with 53 bits of precision

is_NaN()[source]#

Return True iff either the real or the imaginary part is not-a-number.

EXAMPLES:

sage: CBF(NaN).is_NaN()                                                     # needs sage.symbolic
True
sage: CBF(-5).gamma().is_NaN()
True
sage: CBF(oo).is_NaN()
False
sage: CBF(42+I).is_NaN()
False

>>> from sage.all import *
>>> CBF(NaN).is_NaN()                                                     # needs sage.symbolic
True
>>> CBF(-Integer(5)).gamma().is_NaN()
True
>>> CBF(oo).is_NaN()
False
>>> CBF(Integer(42)+I).is_NaN()
False

is_exact()[source]#

Return True iff the radius of this ball is zero.

EXAMPLES:

sage: CBF(1).is_exact()
True
sage: CBF(1/3, 1/3).is_exact()
False

>>> from sage.all import *
>>> CBF(Integer(1)).is_exact()
True
>>> CBF(Integer(1)/Integer(3), Integer(1)/Integer(3)).is_exact()
False

is_nonzero()[source]#

Return True iff zero is not contained in the interval represented by this ball.

Note

This method is not the negation of is_zero(): it only returns True if zero is known not to be contained in the ball.

Use bool(b) (or, equivalently, not b.is_zero()) to check if a ball b may represent a nonzero number (for instance, to determine the “degree” of a polynomial with ball coefficients).

EXAMPLES:

sage: CBF(pi, 1/3).is_nonzero()                                             # needs sage.symbolic
True
sage: CBF(RIF(-0.5, 0.5), 1/3).is_nonzero()
True
sage: CBF(1/3, RIF(-0.5, 0.5)).is_nonzero()
True
sage: CBF(RIF(-0.5, 0.5), RIF(-0.5, 0.5)).is_nonzero()
False

>>> from sage.all import *
>>> CBF(pi, Integer(1)/Integer(3)).is_nonzero()                                             # needs sage.symbolic
True
>>> CBF(RIF(-RealNumber('0.5'), RealNumber('0.5')), Integer(1)/Integer(3)).is_nonzero()
True
>>> CBF(Integer(1)/Integer(3), RIF(-RealNumber('0.5'), RealNumber('0.5'))).is_nonzero()
True
>>> CBF(RIF(-RealNumber('0.5'), RealNumber('0.5')), RIF(-RealNumber('0.5'), RealNumber('0.5'))).is_nonzero()
False

is_real()[source]#

Return True iff the imaginary part of this ball is exactly zero.

EXAMPLES:

sage: CBF(1/3, 0).is_real()
True
sage: (CBF(i/3) - CBF(1, 1/3)).is_real()
False
sage: CBF('inf').is_real()
True

>>> from sage.all import *
>>> CBF(Integer(1)/Integer(3), Integer(0)).is_real()
True
>>> (CBF(i/Integer(3)) - CBF(Integer(1), Integer(1)/Integer(3))).is_real()
False
>>> CBF('inf').is_real()
True

is_zero()[source]#

Return True iff the midpoint and radius of this ball are both zero.

EXAMPLES:

sage: CBF(0).is_zero()
True
sage: CBF(RIF(-0.5, 0.5)).is_zero()
False

>>> from sage.all import *
>>> CBF(Integer(0)).is_zero()
True
>>> CBF(RIF(-RealNumber('0.5'), RealNumber('0.5'))).is_zero()
False

jacobi_P(n, a, b)[source]#

Return the Jacobi polynomial (or function) $$P_n^{(a,b)}(z)$$ evaluated at self.

EXAMPLES:

sage: CBF(5,-6).jacobi_P(8, CBF(1,2), CBF(2,3))
[-920983000.45982 +/- ...e-6] + [6069919969.92857 +/- ...e-6]*I

>>> from sage.all import *
>>> CBF(Integer(5),-Integer(6)).jacobi_P(Integer(8), CBF(Integer(1),Integer(2)), CBF(Integer(2),Integer(3)))
[-920983000.45982 +/- ...e-6] + [6069919969.92857 +/- ...e-6]*I

jacobi_theta(tau)[source]#

Return the four Jacobi theta functions evaluated at the argument self (representing $$z$$) and the parameter tau which should lie in the upper half plane.

The following definitions are used:

\begin{align}\begin{aligned}\theta_1(z,\tau) = 2 q_{1/4} \sum_{n=0}^{\infty} (-1)^n q^{n(n+1)} \sin((2n+1) \pi z)\\\theta_2(z,\tau) = 2 q_{1/4} \sum_{n=0}^{\infty} q^{n(n+1)} \cos((2n+1) \pi z)\\\theta_3(z,\tau) = 1 + 2 \sum_{n=1}^{\infty} q^{n^2} \cos(2n \pi z)\\\theta_4(z,\tau) = 1 + 2 \sum_{n=1}^{\infty} (-1)^n q^{n^2} \cos(2n \pi z)\end{aligned}\end{align}

where $$q = \exp(\pi i \tau)$$ and $$q_{1/4} = \exp(\pi i \tau / 4)$$. Note that $$z$$ is multiplied by $$\pi$$; some authors omit this factor.

EXAMPLES:

sage: CBF(3,-1/2).jacobi_theta(CBF(1/4,2))
([-0.186580562274757 +/- ...e-16] + [0.93841744788594 +/- ...e-15]*I,
[-1.02315311037951 +/- ...e-15] + [-0.203600094532010 +/- ...e-16]*I,
[1.030613911309632 +/- ...e-16] + [0.030613917822067 +/- ...e-16]*I,
[0.969386075665498 +/- ...e-16] + [-0.030613917822067 +/- ...e-16]*I)

sage: CBF(3,-1/2).jacobi_theta(CBF(1/4,-2))
(nan + nan*I, nan + nan*I, nan + nan*I, nan + nan*I)

sage: CBF(0).jacobi_theta(CBF(0,1))
(0,
[0.913579138156117 +/- ...e-16],
[1.086434811213308 +/- ...e-16],
[0.913579138156117 +/- ...e-16])

>>> from sage.all import *
>>> CBF(Integer(3),-Integer(1)/Integer(2)).jacobi_theta(CBF(Integer(1)/Integer(4),Integer(2)))
([-0.186580562274757 +/- ...e-16] + [0.93841744788594 +/- ...e-15]*I,
[-1.02315311037951 +/- ...e-15] + [-0.203600094532010 +/- ...e-16]*I,
[1.030613911309632 +/- ...e-16] + [0.030613917822067 +/- ...e-16]*I,
[0.969386075665498 +/- ...e-16] + [-0.030613917822067 +/- ...e-16]*I)

>>> CBF(Integer(3),-Integer(1)/Integer(2)).jacobi_theta(CBF(Integer(1)/Integer(4),-Integer(2)))
(nan + nan*I, nan + nan*I, nan + nan*I, nan + nan*I)

>>> CBF(Integer(0)).jacobi_theta(CBF(Integer(0),Integer(1)))
(0,
[0.913579138156117 +/- ...e-16],
[1.086434811213308 +/- ...e-16],
[0.913579138156117 +/- ...e-16])

laguerre_L(n, m=0)[source]#

Return the Laguerre polynomial (or function) $$L_n^m(z)$$ evaluated at self.

EXAMPLES:

sage: CBF(10).laguerre_L(3)
[-45.6666666666666 +/- ...e-14]
sage: CBF(10).laguerre_L(3, 2)
[-6.666666666667 +/- ...e-13]
sage: CBF(5,7).laguerre_L(CBF(2,3), CBF(1,-2)) # abs tol 1e-9
[5515.3150302713 +/- 5.02e-11] + [-12386.9428452714 +/- 6.21e-11]*I

>>> from sage.all import *
>>> CBF(Integer(10)).laguerre_L(Integer(3))
[-45.6666666666666 +/- ...e-14]
>>> CBF(Integer(10)).laguerre_L(Integer(3), Integer(2))
[-6.666666666667 +/- ...e-13]
>>> CBF(Integer(5),Integer(7)).laguerre_L(CBF(Integer(2),Integer(3)), CBF(Integer(1),-Integer(2))) # abs tol 1e-9
[5515.3150302713 +/- 5.02e-11] + [-12386.9428452714 +/- 6.21e-11]*I

lambert_w(branch=0)[source]#

Return the image of this ball by the specified branch of the Lambert W function.

EXAMPLES:

sage: CBF(1 + I).lambert_w()
[0.6569660692304...] + [0.3254503394134...]*I
sage: CBF(1 + I).lambert_w(2)
[-2.1208839379437...] + [11.600137110774...]*I
sage: CBF(1 + I).lambert_w(2^100)
[-70.806021532123...] + [7.9648836259913...]*I

>>> from sage.all import *
>>> CBF(Integer(1) + I).lambert_w()
[0.6569660692304...] + [0.3254503394134...]*I
>>> CBF(Integer(1) + I).lambert_w(Integer(2))
[-2.1208839379437...] + [11.600137110774...]*I
>>> CBF(Integer(1) + I).lambert_w(Integer(2)**Integer(100))
[-70.806021532123...] + [7.9648836259913...]*I

legendre_P(n, m=0, type=2)[source]#

Return the Legendre function of the first kind $$P_n^m(z)$$ evaluated at self.

The type parameter can be either 2 or 3. This selects between different branch cut conventions. The definitions of the “type 2” and “type 3” functions are the same as those used by Mathematica and mpmath.

EXAMPLES:

sage: CBF(1/2).legendre_P(5)
[0.0898437500000000 +/- 7...e-17]
sage: CBF(1,2).legendre_P(CBF(2,3), CBF(0,1))
[0.10996180744364 +/- ...e-15] + [0.14312767804055 +/- ...e-15]*I
sage: CBF(-10).legendre_P(5, 325/100)
[-22104403.487377 +/- ...e-7] + [53364750.687392 +/- ...e-7]*I
sage: CBF(-10).legendre_P(5, 325/100, type=3)
[-57761589.914581 +/- ...e-7] + [+/- ...e-7]*I

>>> from sage.all import *
>>> CBF(Integer(1)/Integer(2)).legendre_P(Integer(5))
[0.0898437500000000 +/- 7...e-17]
>>> CBF(Integer(1),Integer(2)).legendre_P(CBF(Integer(2),Integer(3)), CBF(Integer(0),Integer(1)))
[0.10996180744364 +/- ...e-15] + [0.14312767804055 +/- ...e-15]*I
>>> CBF(-Integer(10)).legendre_P(Integer(5), Integer(325)/Integer(100))
[-22104403.487377 +/- ...e-7] + [53364750.687392 +/- ...e-7]*I
>>> CBF(-Integer(10)).legendre_P(Integer(5), Integer(325)/Integer(100), type=Integer(3))
[-57761589.914581 +/- ...e-7] + [+/- ...e-7]*I

legendre_Q(n, m=0, type=2)[source]#

Return the Legendre function of the second kind $$Q_n^m(z)$$ evaluated at self.

The type parameter can be either 2 or 3. This selects between different branch cut conventions. The definitions of the “type 2” and “type 3” functions are the same as those used by Mathematica and mpmath.

EXAMPLES:

sage: CBF(1/2).legendre_Q(5)
[0.55508089057168 +/- ...e-15]
sage: CBF(1,2).legendre_Q(CBF(2,3), CBF(0,1))
[0.167678710 +/- ...e-10] + [-0.161558598 +/- ...e-10]*I
sage: CBF(-10).legendre_Q(5, 325/100)
[-83825154.36008 +/- ...e-6] + [-34721515.80396 +/- ...e-6]*I
sage: CBF(-10).legendre_Q(5, 325/100, type=3)
[-4.797306921692e-6 +/- ...e-19] + [-4.797306921692e-6 +/- ...e-19]*I

>>> from sage.all import *
>>> CBF(Integer(1)/Integer(2)).legendre_Q(Integer(5))
[0.55508089057168 +/- ...e-15]
>>> CBF(Integer(1),Integer(2)).legendre_Q(CBF(Integer(2),Integer(3)), CBF(Integer(0),Integer(1)))
[0.167678710 +/- ...e-10] + [-0.161558598 +/- ...e-10]*I
>>> CBF(-Integer(10)).legendre_Q(Integer(5), Integer(325)/Integer(100))
[-83825154.36008 +/- ...e-6] + [-34721515.80396 +/- ...e-6]*I
>>> CBF(-Integer(10)).legendre_Q(Integer(5), Integer(325)/Integer(100), type=Integer(3))
[-4.797306921692e-6 +/- ...e-19] + [-4.797306921692e-6 +/- ...e-19]*I

li(offset=False)[source]#

Return the logarithmic integral with argument self.

If offset is True, return the offset logarithmic integral.

EXAMPLES:

sage: CBF(1, 1).li()
[0.61391166922120 +/- ...e-15] + [2.05958421419258 +/- ...e-15]*I
sage: CBF(0).li()
0
sage: CBF(0).li(offset=True)
[-1.045163780117493 +/- ...e-16]
sage: li(0).n()                                                             # needs sage.symbolic
0.000000000000000
sage: Li(0).n()                                                             # needs sage.symbolic
-1.04516378011749

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).li()
[0.61391166922120 +/- ...e-15] + [2.05958421419258 +/- ...e-15]*I
>>> CBF(Integer(0)).li()
0
>>> CBF(Integer(0)).li(offset=True)
[-1.045163780117493 +/- ...e-16]
>>> li(Integer(0)).n()                                                             # needs sage.symbolic
0.000000000000000
>>> Li(Integer(0)).n()                                                             # needs sage.symbolic
-1.04516378011749

log(base=None, analytic=False)[source]#

General logarithm (principal branch).

INPUT:

• base (optional, complex ball or number) – if None, return the principal branch of the natural logarithm ln(self), otherwise, return the general logarithm ln(self)/ln(base)

• analytic (optional, boolean) – if True, return an indeterminate (not-a-number) value when the input ball touches the branch cut (with respect to self)

EXAMPLES:

sage: CBF(2*i).log()
[0.693147180559945 +/- ...e-16] + [1.570796326794897 +/- ...e-16]*I
sage: CBF(-1).log()
[3.141592653589793 +/- ...e-16]*I

sage: CBF(2*i).log(2)
[1.000000000000000 +/- ...e-16] + [2.26618007091360 +/- ...e-15]*I
sage: CBF(2*i).log(CBF(i))
[1.000000000000000 +/- ...e-16] + [-0.441271200305303 +/- ...e-16]*I

sage: CBF('inf').log()
[+/- inf]
sage: CBF(2).log(0)
nan + nan*I

sage: CBF(-1).log(2)
[4.53236014182719 +/- ...e-15]*I
sage: CBF(-1).log(2, analytic=True)
nan + nan*I
[+/- ...e-3] + [+/- 3.15]*I

>>> from sage.all import *
>>> CBF(Integer(2)*i).log()
[0.693147180559945 +/- ...e-16] + [1.570796326794897 +/- ...e-16]*I
>>> CBF(-Integer(1)).log()
[3.141592653589793 +/- ...e-16]*I

>>> CBF(Integer(2)*i).log(Integer(2))
[1.000000000000000 +/- ...e-16] + [2.26618007091360 +/- ...e-15]*I
>>> CBF(Integer(2)*i).log(CBF(i))
[1.000000000000000 +/- ...e-16] + [-0.441271200305303 +/- ...e-16]*I

>>> CBF('inf').log()
[+/- inf]
>>> CBF(Integer(2)).log(Integer(0))
nan + nan*I

>>> CBF(-Integer(1)).log(Integer(2))
[4.53236014182719 +/- ...e-15]*I
>>> CBF(-Integer(1)).log(Integer(2), analytic=True)
nan + nan*I
[+/- ...e-3] + [+/- 3.15]*I

log1p(analytic=False)[source]#

Return log(1 + self), computed accurately when self is close to zero.

INPUT:

• analytic (optional, boolean) – if True, return an indeterminate (not-a-number) value when the input ball touches the branch cut

EXAMPLES:

sage: eps = RBF(1e-50)
sage: CBF(1+eps, eps).log()
[+/- ...e-16] + [1.000000000000000e-50 +/- ...e-66]*I
sage: CBF(eps, eps).log1p()
[1.000000000000000e-50 +/- ...e-68] + [1.00000000000000e-50 +/- ...e-66]*I
sage: CBF(-3/2).log1p(analytic=True)
nan + nan*I

>>> from sage.all import *
>>> eps = RBF(RealNumber('1e-50'))
>>> CBF(Integer(1)+eps, eps).log()
[+/- ...e-16] + [1.000000000000000e-50 +/- ...e-66]*I
>>> CBF(eps, eps).log1p()
[1.000000000000000e-50 +/- ...e-68] + [1.00000000000000e-50 +/- ...e-66]*I
>>> CBF(-Integer(3)/Integer(2)).log1p(analytic=True)
nan + nan*I

log_barnes_g()[source]#

Return the logarithmic Barnes G-function of self.

EXAMPLES:

sage: CBF(10^100).log_barnes_g()
[1.14379254649702e+202 +/- ...e+187]
sage: CBF(0,1000).log_barnes_g()
[-2702305.04929258 +/- ...e-9] + [-790386.325561423 +/- ...e-10]*I

>>> from sage.all import *
>>> CBF(Integer(10)**Integer(100)).log_barnes_g()
[1.14379254649702e+202 +/- ...e+187]
>>> CBF(Integer(0),Integer(1000)).log_barnes_g()
[-2702305.04929258 +/- ...e-9] + [-790386.325561423 +/- ...e-10]*I

log_gamma(analytic=False)[source]#

Return the image of this ball by the logarithmic Gamma function.

The branch cut of the logarithmic gamma function is placed on the negative half-axis, which means that log_gamma(z) + log z = log_gamma(z+1) holds for all $$z$$, whereas log_gamma(z) != log(gamma(z)) in general.

INPUT:

• analytic (optional, boolean) – if True, return an indeterminate (not-a-number) value when the input ball touches the branch cut

EXAMPLES:

sage: CBF(1000, 1000).log_gamma()
[5466.22252162990 +/- ...e-12] + [7039.33429191119 +/- ...e-12]*I
sage: CBF(-1/2).log_gamma()
[1.265512123484645 +/- ...e-16] + [-3.141592653589793 +/- ...e-16]*I
sage: CBF(-1).log_gamma()
nan + ...*I
sage: CBF(-3/2).log_gamma() # abs tol 1e-14
[0.860047015376481 +/- 3.82e-16] + [-6.283185307179586 +/- 6.77e-16]*I
sage: CBF(-3/2).log_gamma(analytic=True)
nan + nan*I

>>> from sage.all import *
>>> CBF(Integer(1000), Integer(1000)).log_gamma()
[5466.22252162990 +/- ...e-12] + [7039.33429191119 +/- ...e-12]*I
>>> CBF(-Integer(1)/Integer(2)).log_gamma()
[1.265512123484645 +/- ...e-16] + [-3.141592653589793 +/- ...e-16]*I
>>> CBF(-Integer(1)).log_gamma()
nan + ...*I
>>> CBF(-Integer(3)/Integer(2)).log_gamma() # abs tol 1e-14
[0.860047015376481 +/- 3.82e-16] + [-6.283185307179586 +/- 6.77e-16]*I
>>> CBF(-Integer(3)/Integer(2)).log_gamma(analytic=True)
nan + nan*I

log_integral(offset=False)[source]#

Return the logarithmic integral with argument self.

If offset is True, return the offset logarithmic integral.

EXAMPLES:

sage: CBF(1, 1).li()
[0.61391166922120 +/- ...e-15] + [2.05958421419258 +/- ...e-15]*I
sage: CBF(0).li()
0
sage: CBF(0).li(offset=True)
[-1.045163780117493 +/- ...e-16]
sage: li(0).n()                                                             # needs sage.symbolic
0.000000000000000
sage: Li(0).n()                                                             # needs sage.symbolic
-1.04516378011749

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).li()
[0.61391166922120 +/- ...e-15] + [2.05958421419258 +/- ...e-15]*I
>>> CBF(Integer(0)).li()
0
>>> CBF(Integer(0)).li(offset=True)
[-1.045163780117493 +/- ...e-16]
>>> li(Integer(0)).n()                                                             # needs sage.symbolic
0.000000000000000
>>> Li(Integer(0)).n()                                                             # needs sage.symbolic
-1.04516378011749

log_integral_offset()[source]#

Offset logarithmic integral.

EXAMPLES:

sage: CBF(0).Li()
[-1.045163780117493 +/- ...e-16]
sage: li(0).n()                                                             # needs sage.symbolic
0.000000000000000
sage: Li(0).n()                                                             # needs sage.symbolic
-1.04516378011749

>>> from sage.all import *
>>> CBF(Integer(0)).Li()
[-1.045163780117493 +/- ...e-16]
>>> li(Integer(0)).n()                                                             # needs sage.symbolic
0.000000000000000
>>> Li(Integer(0)).n()                                                             # needs sage.symbolic
-1.04516378011749

mid()[source]#

Return the midpoint of this ball.

OUTPUT:

ComplexNumber, floating-point complex number formed by the centers of the real and imaginary parts of this ball.

EXAMPLES:

sage: CBF(1/3, 1).mid()
0.333333333333333 + 1.00000000000000*I
sage: CBF(1/3, 1).mid().parent()
Complex Field with 53 bits of precision
sage: CBF('inf', 'nan').mid()
+infinity + NaN*I
sage: CBF('nan', 'inf').mid()
NaN + +infinity*I
sage: CBF('nan').mid()
NaN
sage: CBF('inf').mid()
+infinity
sage: CBF(0, 'inf').mid()
+infinity*I

>>> from sage.all import *
>>> CBF(Integer(1)/Integer(3), Integer(1)).mid()
0.333333333333333 + 1.00000000000000*I
>>> CBF(Integer(1)/Integer(3), Integer(1)).mid().parent()
Complex Field with 53 bits of precision
>>> CBF('inf', 'nan').mid()
+infinity + NaN*I
>>> CBF('nan', 'inf').mid()
NaN + +infinity*I
>>> CBF('nan').mid()
NaN
>>> CBF('inf').mid()
+infinity
>>> CBF(Integer(0), 'inf').mid()
+infinity*I

modular_delta()[source]#

Return the modular discriminant with tau given by self.

EXAMPLES:

sage: CBF(0,1).modular_delta()
[0.0017853698506421 +/- ...e-17]
sage: a, b, c, d = 2, 5, 1, 3
sage: tau = CBF(1,3)
sage: ((a*tau+b)/(c*tau+d)).modular_delta()
[0.20921376655 +/- ...e-12] + [1.57611925523 +/- ...e-12]*I
sage: (c*tau+d)^12 * tau.modular_delta()
[0.20921376654986 +/- ...e-15] + [1.5761192552253 +/- ...e-14]*I

>>> from sage.all import *
>>> CBF(Integer(0),Integer(1)).modular_delta()
[0.0017853698506421 +/- ...e-17]
>>> a, b, c, d = Integer(2), Integer(5), Integer(1), Integer(3)
>>> tau = CBF(Integer(1),Integer(3))
>>> ((a*tau+b)/(c*tau+d)).modular_delta()
[0.20921376655 +/- ...e-12] + [1.57611925523 +/- ...e-12]*I
>>> (c*tau+d)**Integer(12) * tau.modular_delta()
[0.20921376654986 +/- ...e-15] + [1.5761192552253 +/- ...e-14]*I

modular_eta()[source]#

Return the Dedekind eta function with tau given by self.

EXAMPLES:

sage: CBF(0,1).modular_eta()
[0.768225422326057 +/- ...e-16]
sage: CBF(12,1).modular_eta()
[-0.768225422326057 +/- ...e-16]

>>> from sage.all import *
>>> CBF(Integer(0),Integer(1)).modular_eta()
[0.768225422326057 +/- ...e-16]
>>> CBF(Integer(12),Integer(1)).modular_eta()
[-0.768225422326057 +/- ...e-16]

modular_j()[source]#

Return the modular j-invariant with tau given by self.

EXAMPLES:

sage: CBF(0,1).modular_j()
[1728.0000000000 +/- ...e-11]

>>> from sage.all import *
>>> CBF(Integer(0),Integer(1)).modular_j()
[1728.0000000000 +/- ...e-11]

modular_lambda()[source]#

Return the modular lambda function with tau given by self.

EXAMPLES:

sage: # needs sage.symbolic
sage: tau = CBF(sqrt(2),pi)
sage: tau.modular_lambda()
[-0.00022005123884157 +/- ...e-18] + [-0.00079787346459944 +/- ...e-18]*I
sage: (tau + 2).modular_lambda()
[-0.00022005123884157 +/- ...e-18] + [-0.00079787346459944 +/- ...e-18]*I
sage: (tau / (1 - 2*tau)).modular_lambda()
[-0.00022005123884 +/- ...e-15] + [-0.00079787346460 +/- ...e-15]*I

>>> from sage.all import *
>>> # needs sage.symbolic
>>> tau = CBF(sqrt(Integer(2)),pi)
>>> tau.modular_lambda()
[-0.00022005123884157 +/- ...e-18] + [-0.00079787346459944 +/- ...e-18]*I
>>> (tau + Integer(2)).modular_lambda()
[-0.00022005123884157 +/- ...e-18] + [-0.00079787346459944 +/- ...e-18]*I
>>> (tau / (Integer(1) - Integer(2)*tau)).modular_lambda()
[-0.00022005123884 +/- ...e-15] + [-0.00079787346460 +/- ...e-15]*I

nbits()[source]#

Return the minimum precision sufficient to represent this ball exactly.

More precisely, the output is the number of bits needed to represent the absolute value of the mantissa of both the real and the imaginary part of the midpoint.

EXAMPLES:

sage: CBF(17, 1023).nbits()
10
sage: CBF(1/3, NaN).nbits()                                                 # needs sage.symbolic
53
sage: CBF(NaN).nbits()                                                      # needs sage.symbolic
0

>>> from sage.all import *
>>> CBF(Integer(17), Integer(1023)).nbits()
10
>>> CBF(Integer(1)/Integer(3), NaN).nbits()                                                 # needs sage.symbolic
53
>>> CBF(NaN).nbits()                                                      # needs sage.symbolic
0

overlaps(other)[source]#

Return True iff self and other have some point in common.

INPUT:

EXAMPLES:

sage: CBF(1, 1).overlaps(1 + CBF(0, 1/3)*3)
True
sage: CBF(1, 1).overlaps(CBF(1, 'nan'))
True
sage: CBF(1, 1).overlaps(CBF(0, 'nan'))
False

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).overlaps(Integer(1) + CBF(Integer(0), Integer(1)/Integer(3))*Integer(3))
True
>>> CBF(Integer(1), Integer(1)).overlaps(CBF(Integer(1), 'nan'))
True
>>> CBF(Integer(1), Integer(1)).overlaps(CBF(Integer(0), 'nan'))
False

polylog(s)[source]#

Return the polylogarithm $$\operatorname{Li}_s(\mathrm{self})$$.

EXAMPLES:

sage: CBF(2).polylog(1)
[+/- ...e-15] + [-3.14159265358979 +/- ...e-15]*I
sage: CBF(1, 1).polylog(CBF(1, 1))
[0.3708160030469 +/- ...e-14] + [2.7238016577979 +/- ...e-14]*I

>>> from sage.all import *
>>> CBF(Integer(2)).polylog(Integer(1))
[+/- ...e-15] + [-3.14159265358979 +/- ...e-15]*I
>>> CBF(Integer(1), Integer(1)).polylog(CBF(Integer(1), Integer(1)))
[0.3708160030469 +/- ...e-14] + [2.7238016577979 +/- ...e-14]*I

pow(expo, analytic=False)[source]#

Raise this ball to the power of expo.

INPUT:

• analytic (optional, boolean) – if True, return an indeterminate (not-a-number) value when the exponent is not an integer and the base ball touches the branch cut of the logarithm

EXAMPLES:

sage: CBF(-1).pow(CBF(i))
[0.0432139182637723 +/- ...e-17]
sage: CBF(-1).pow(CBF(i), analytic=True)
nan + nan*I
sage: CBF(-10).pow(-2)
[0.0100000000000000 +/- ...e-18]
sage: CBF(-10).pow(-2, analytic=True)
[0.0100000000000000 +/- ...e-18]

>>> from sage.all import *
>>> CBF(-Integer(1)).pow(CBF(i))
[0.0432139182637723 +/- ...e-17]
>>> CBF(-Integer(1)).pow(CBF(i), analytic=True)
nan + nan*I
>>> CBF(-Integer(10)).pow(-Integer(2))
[0.0100000000000000 +/- ...e-18]
>>> CBF(-Integer(10)).pow(-Integer(2), analytic=True)
[0.0100000000000000 +/- ...e-18]

psi(n=None)[source]#

Compute the digamma function with argument self.

If n is provided, compute the polygamma function of order n and argument self.

EXAMPLES:

sage: CBF(1, 1).psi()
[0.0946503206224770 +/- ...e-17] + [1.076674047468581 +/- ...e-16]*I
sage: CBF(-1).psi()
nan
sage: CBF(1,1).psi(10)
[56514.8269344249 +/- ...e-11] + [56215.1218005823 +/- ...e-11]*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).psi()
[0.0946503206224770 +/- ...e-17] + [1.076674047468581 +/- ...e-16]*I
>>> CBF(-Integer(1)).psi()
nan
>>> CBF(Integer(1),Integer(1)).psi(Integer(10))
[56514.8269344249 +/- ...e-11] + [56215.1218005823 +/- ...e-11]*I


Return an upper bound for the error radius of this ball.

OUTPUT:

A RealNumber of the same precision as the radii of real balls.

Warning

Unlike a RealBall, a ComplexBall is not defined by its midpoint and radius. (Instances of ComplexBall are actually rectangles, not balls.)

EXAMPLES:

sage: CBF(1 + i).rad()
0.00000000
1.1102230e-16
Real Field with 30 bits of precision

>>> from sage.all import *
0.00000000
1.1102230e-16
Real Field with 30 bits of precision


real()[source]#

Return the real part of this ball.

OUTPUT:

EXAMPLES:

sage: a = CBF(1/3, 1/5)
sage: a.real()
[0.3333333333333333 +/- ...e-17]
sage: a.real().parent()
Real ball field with 53 bits of precision

>>> from sage.all import *
>>> a = CBF(Integer(1)/Integer(3), Integer(1)/Integer(5))
>>> a.real()
[0.3333333333333333 +/- ...e-17]
>>> a.real().parent()
Real ball field with 53 bits of precision

rgamma()[source]#

Compute the reciprocal gamma function with argument self.

EXAMPLES:

sage: CBF(6).rgamma()
[0.00833333333333333 +/- ...e-18]
sage: CBF(-1).rgamma()
0

>>> from sage.all import *
>>> CBF(Integer(6)).rgamma()
[0.00833333333333333 +/- ...e-18]
>>> CBF(-Integer(1)).rgamma()
0

rising_factorial(n)[source]#

Return the n-th rising factorial of this ball.

The $$n$$-th rising factorial of $$x$$ is equal to $$x (x+1) \cdots (x+n-1)$$.

For complex $$n$$, it is a quotient of gamma functions.

EXAMPLES:

sage: CBF(1).rising_factorial(5)
120.0000000000000
sage: CBF(1/3, 1/2).rising_factorial(300)
[-3.87949484514e+612 +/- 5...e+600] + [-3.52042209763e+612 +/- 5...e+600]*I

sage: CBF(1).rising_factorial(-1)
nan
sage: CBF(1).rising_factorial(2**64)
[+/- ...e+347382171326740403407]
sage: ComplexBallField(128)(1).rising_factorial(2**64)
[2.34369112679686134...e+347382171305201285713 +/- ...]
sage: CBF(1/2).rising_factorial(CBF(2,3)) # abs tol 1e-15
[-0.123060451458124 +/- 3.06e-16] + [0.0406412631676552 +/- 7.57e-17]*I

>>> from sage.all import *
>>> CBF(Integer(1)).rising_factorial(Integer(5))
120.0000000000000
>>> CBF(Integer(1)/Integer(3), Integer(1)/Integer(2)).rising_factorial(Integer(300))
[-3.87949484514e+612 +/- 5...e+600] + [-3.52042209763e+612 +/- 5...e+600]*I

>>> CBF(Integer(1)).rising_factorial(-Integer(1))
nan
>>> CBF(Integer(1)).rising_factorial(Integer(2)**Integer(64))
[+/- ...e+347382171326740403407]
>>> ComplexBallField(Integer(128))(Integer(1)).rising_factorial(Integer(2)**Integer(64))
[2.34369112679686134...e+347382171305201285713 +/- ...]
>>> CBF(Integer(1)/Integer(2)).rising_factorial(CBF(Integer(2),Integer(3))) # abs tol 1e-15
[-0.123060451458124 +/- 3.06e-16] + [0.0406412631676552 +/- 7.57e-17]*I

round()[source]#

Return a copy of this ball rounded to the precision of the parent.

EXAMPLES:

It is possible to create balls whose midpoint is more precise that their parent’s nominal precision (see real_arb for more information):

sage: b = CBF(exp(I*pi/3).n(100))                                           # needs sage.symbolic
sage: b.mid()                                                               # needs sage.symbolic
0.50000000000000000000000000000 + 0.86602540378443864676372317075*I

>>> from sage.all import *
>>> b = CBF(exp(I*pi/Integer(3)).n(Integer(100)))                                           # needs sage.symbolic
>>> b.mid()                                                               # needs sage.symbolic
0.50000000000000000000000000000 + 0.86602540378443864676372317075*I


The round() method rounds such a ball to its parent’s precision:

sage: b.round().mid()                                                       # needs sage.symbolic
0.500000000000000 + 0.866025403784439*I

>>> from sage.all import *
>>> b.round().mid()                                                       # needs sage.symbolic
0.500000000000000 + 0.866025403784439*I

rsqrt(analytic=False)[source]#

Return the reciprocal square root of self.

If either the real or imaginary part is exactly zero, only a single real reciprocal square root is needed.

INPUT:

• analytic (optional, boolean) – if True, return an indeterminate (not-a-number) value when the input ball touches the branch cut

EXAMPLES:

sage: CBF(-2).rsqrt()
[-0.707106781186547 +/- ...e-16]*I
sage: CBF(-2).rsqrt(analytic=True)
nan + nan*I
sage: CBF(0, 1/2).rsqrt()
1.000000000000000 - 1.000000000000000*I
sage: CBF(0).rsqrt()
nan + nan*I

>>> from sage.all import *
>>> CBF(-Integer(2)).rsqrt()
[-0.707106781186547 +/- ...e-16]*I
>>> CBF(-Integer(2)).rsqrt(analytic=True)
nan + nan*I
>>> CBF(Integer(0), Integer(1)/Integer(2)).rsqrt()
1.000000000000000 - 1.000000000000000*I
>>> CBF(Integer(0)).rsqrt()
nan + nan*I

sec()[source]#

Return the secant of this ball.

EXAMPLES:

sage: CBF(1, 1).sec()
[0.498337030555187 +/- ...e-16] + [0.591083841721045 +/- ...e-16]*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).sec()
[0.498337030555187 +/- ...e-16] + [0.591083841721045 +/- ...e-16]*I

sech()[source]#

Return the hyperbolic secant of this ball.

EXAMPLES:

sage: CBF(pi/2, 1/10).sech()                                                # needs sage.symbolic
[0.397174529918189 +/- ...e-16] + [-0.0365488656274242 +/- ...e-17]*I

>>> from sage.all import *
>>> CBF(pi/Integer(2), Integer(1)/Integer(10)).sech()                                                # needs sage.symbolic
[0.397174529918189 +/- ...e-16] + [-0.0365488656274242 +/- ...e-17]*I

sin()[source]#

Return the sine of this ball.

EXAMPLES:

sage: CBF(i*pi).sin()                                                       # needs sage.symbolic
[11.54873935725775 +/- ...e-15]*I

>>> from sage.all import *
>>> CBF(i*pi).sin()                                                       # needs sage.symbolic
[11.54873935725775 +/- ...e-15]*I

sin_integral()[source]#

Return the sine integral with argument self.

EXAMPLES:

sage: CBF(1, 1).Si()
[1.10422265823558 +/- ...e-15] + [0.88245380500792 +/- ...e-15]*I
sage: CBF(0).Si()
0

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).Si()
[1.10422265823558 +/- ...e-15] + [0.88245380500792 +/- ...e-15]*I
>>> CBF(Integer(0)).Si()
0

sinh()[source]#

Return the hyperbolic sine of this ball.

EXAMPLES:

sage: CBF(1, 1).sinh()
[0.634963914784736 +/- ...e-16] + [1.298457581415977 +/- ...e-16]*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).sinh()
[0.634963914784736 +/- ...e-16] + [1.298457581415977 +/- ...e-16]*I

sinh_integral()[source]#

Return the hyperbolic sine integral with argument self.

EXAMPLES:

sage: CBF(1, 1).Shi()
[0.88245380500792 +/- ...e-15] + [1.10422265823558 +/- ...e-15]*I
sage: CBF(0).Shi()
0

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).Shi()
[0.88245380500792 +/- ...e-15] + [1.10422265823558 +/- ...e-15]*I
>>> CBF(Integer(0)).Shi()
0

spherical_harmonic(phi, n, m)[source]#

Return the spherical harmonic $$Y_n^m(\theta,\phi)$$ evaluated at $$\theta$$ given by self. In the current implementation, n and m must be small integers.

EXAMPLES:

sage: CBF(1+I).spherical_harmonic(1/2, -3, -2)
[0.80370071745224 +/- ...e-15] + [-0.07282031864711 +/- ...e-15]*I

>>> from sage.all import *
>>> CBF(Integer(1)+I).spherical_harmonic(Integer(1)/Integer(2), -Integer(3), -Integer(2))
[0.80370071745224 +/- ...e-15] + [-0.07282031864711 +/- ...e-15]*I

sqrt(analytic=False)[source]#

Return the square root of this ball.

If either the real or imaginary part is exactly zero, only a single real square root is needed.

INPUT:

• analytic (optional, boolean) – if True, return an indeterminate (not-a-number) value when the input ball touches the branch cut

EXAMPLES:

sage: CBF(-2).sqrt()
[1.414213562373095 +/- ...e-16]*I
sage: CBF(-2).sqrt(analytic=True)
nan + nan*I

>>> from sage.all import *
>>> CBF(-Integer(2)).sqrt()
[1.414213562373095 +/- ...e-16]*I
>>> CBF(-Integer(2)).sqrt(analytic=True)
nan + nan*I

squash()[source]#

Return an exact ball with the same midpoint as this ball.

OUTPUT:

EXAMPLES:

sage: mid = CBF(1/3, 1/10).squash()
sage: mid
[0.3333333333333333 +/- ...e-17] + [0.09999999999999999 +/- ...e-18]*I
sage: mid.parent()
Complex ball field with 53 bits of precision
sage: mid.is_exact()
True

>>> from sage.all import *
>>> mid = CBF(Integer(1)/Integer(3), Integer(1)/Integer(10)).squash()
>>> mid
[0.3333333333333333 +/- ...e-17] + [0.09999999999999999 +/- ...e-18]*I
>>> mid.parent()
Complex ball field with 53 bits of precision
>>> mid.is_exact()
True

tan()[source]#

Return the tangent of this ball.

EXAMPLES:

sage: CBF(pi/2, 1/10).tan()                                                 # needs sage.symbolic
[+/- ...e-14] + [10.03331113225399 +/- ...e-15]*I
sage: CBF(pi/2).tan()                                                       # needs sage.symbolic
nan

>>> from sage.all import *
>>> CBF(pi/Integer(2), Integer(1)/Integer(10)).tan()                                                 # needs sage.symbolic
[+/- ...e-14] + [10.03331113225399 +/- ...e-15]*I
>>> CBF(pi/Integer(2)).tan()                                                       # needs sage.symbolic
nan

tanh()[source]#

Return the hyperbolic tangent of this ball.

EXAMPLES:

sage: CBF(1, 1).tanh()
[1.083923327338694 +/- ...e-16] + [0.2717525853195117 +/- ...e-17]*I
sage: CBF(0, pi/2).tanh()                                                   # needs sage.symbolic
nan*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).tanh()
[1.083923327338694 +/- ...e-16] + [0.2717525853195117 +/- ...e-17]*I
>>> CBF(Integer(0), pi/Integer(2)).tanh()                                                   # needs sage.symbolic
nan*I

trim()[source]#

Return a trimmed copy of this ball.

Return a copy of this ball with both the real and imaginary parts trimmed (see trim()).

EXAMPLES:

sage: b = CBF(1/3, RBF(1/3, rad=.01))
sage: b.mid()
0.333333333333333 + 0.333333333333333*I
sage: b.trim().mid()
0.333333333333333 + 0.333333015441895*I

>>> from sage.all import *
>>> b = CBF(Integer(1)/Integer(3), RBF(Integer(1)/Integer(3), rad=RealNumber('.01')))
>>> b.mid()
0.333333333333333 + 0.333333333333333*I
>>> b.trim().mid()
0.333333333333333 + 0.333333015441895*I

union(other)[source]#

Return a ball containing the convex hull of self and other.

EXAMPLES:

sage: b = CBF(1 + i).union(0)
sage: b.real().endpoints()
(-9.31322574615479e-10, 1.00000000093133)

>>> from sage.all import *
>>> b = CBF(Integer(1) + i).union(Integer(0))
>>> b.real().endpoints()
(-9.31322574615479e-10, 1.00000000093133)

zeta(a=None)[source]#

Return the image of this ball by the Hurwitz zeta function.

For a = None, this computes the Riemann zeta function.

EXAMPLES:

sage: CBF(1, 1).zeta()
[0.5821580597520036 +/- ...e-17] + [-0.9268485643308071 +/- ...e-17]*I
sage: CBF(1, 1).zeta(1)
[0.5821580597520036 +/- ...e-17] + [-0.9268485643308071 +/- ...e-17]*I
sage: CBF(1, 1).zeta(1/2)
[1.497919876084167 +/- ...e-16] + [0.2448655353684164 +/- ...e-17]*I
sage: CBF(1, 1).zeta(CBF(1, 1))
[-0.3593983122202835 +/- ...e-17] + [-2.875283329756940 +/- ...e-16]*I
sage: CBF(1, 1).zeta(-1)
nan + nan*I

>>> from sage.all import *
>>> CBF(Integer(1), Integer(1)).zeta()
[0.5821580597520036 +/- ...e-17] + [-0.9268485643308071 +/- ...e-17]*I
>>> CBF(Integer(1), Integer(1)).zeta(Integer(1))
[0.5821580597520036 +/- ...e-17] + [-0.9268485643308071 +/- ...e-17]*I
>>> CBF(Integer(1), Integer(1)).zeta(Integer(1)/Integer(2))
[1.497919876084167 +/- ...e-16] + [0.2448655353684164 +/- ...e-17]*I
>>> CBF(Integer(1), Integer(1)).zeta(CBF(Integer(1), Integer(1)))
[-0.3593983122202835 +/- ...e-17] + [-2.875283329756940 +/- ...e-16]*I
>>> CBF(Integer(1), Integer(1)).zeta(-Integer(1))
nan + nan*I


Return the image of this ball by the k-th derivative of the Riemann zeta function.

For a more flexible interface, see the low-level method _zeta_series of polynomials with complex ball coefficients.

EXAMPLES:

sage: CBF(1/2, 3).zetaderiv(1)
[0.191759884092721...] + [-0.073135728865928...]*I
[-6.0001458028430...]

>>> from sage.all import *
[0.191759884092721...] + [-0.073135728865928...]*I
[-6.0001458028430...]

class sage.rings.complex_arb.ComplexBallField(precision=53)[source]#

An approximation of the field of complex numbers using pairs of mid-rad intervals.

INPUT:

• precision – an integer $$\ge 2$$.

EXAMPLES:

sage: CBF(1)
1.000000000000000

>>> from sage.all import *
>>> CBF(Integer(1))
1.000000000000000

Element[source]#

alias of ComplexBall

characteristic()[source]#

Complex ball fields have characteristic zero.

EXAMPLES:

sage: ComplexBallField().characteristic()
0

>>> from sage.all import *
>>> ComplexBallField().characteristic()
0

complex_field()[source]#

Return the complex ball field with the same precision, i.e. self

EXAMPLES:

sage: CBF.complex_field() is CBF
True

>>> from sage.all import *
>>> CBF.complex_field() is CBF
True

construction()[source]#

Return the construction of a complex ball field as the algebraic closure of the real ball field with the same precision.

EXAMPLES:

sage: functor, base = CBF.construction()
sage: functor, base
(AlgebraicClosureFunctor, Real ball field with 53 bits of precision)
sage: functor(base) is CBF
True

>>> from sage.all import *
>>> functor, base = CBF.construction()
>>> functor, base
(AlgebraicClosureFunctor, Real ball field with 53 bits of precision)
>>> functor(base) is CBF
True

gen(i)[source]#

For i = 0, return the imaginary unit in this complex ball field.

EXAMPLES:

sage: CBF.0
1.000000000000000*I
sage: CBF.gen(1)
Traceback (most recent call last):
...
ValueError: only one generator

>>> from sage.all import *
>>> CBF.gen(0)
1.000000000000000*I
>>> CBF.gen(Integer(1))
Traceback (most recent call last):
...
ValueError: only one generator

gens()[source]#

Return the tuple of generators of this complex ball field, i.e. (i,).

EXAMPLES:

sage: CBF.gens()
(1.000000000000000*I,)
sage: CBF.gens_dict()
{'1.000000000000000*I': 1.000000000000000*I}

>>> from sage.all import *
>>> CBF.gens()
(1.000000000000000*I,)
>>> CBF.gens_dict()
{'1.000000000000000*I': 1.000000000000000*I}

integral(func, a, b, params=None, rel_tol=None, abs_tol=None, deg_limit=None, eval_limit=None, depth_limit=None, use_heap=None, verbose=None)[source]#

Compute a rigorous enclosure of the integral of func on the interval [a, b].

INPUT:

• func – a callable object accepting two parameters, a complex ball x and a boolean flag analytic, and returning an element of this ball field (or some value that coerces into this ball field), such that:

• func(x, False) evaluates the integrand $$f$$ on the ball x. There are no restrictions on the behavior of $$f$$ on x; in particular, it can be discontinuous.

• func(x, True) evaluates $$f(x)$$ if $$f$$ is analytic on the whole x, and returns some non-finite ball (e.g., self(NaN)) otherwise.

(The analytic flag only needs to be checked for integrands that are non-analytic but bounded in some regions, typically complex functions with branch cuts, like $$\sqrt{z}$$. In particular, it can be ignored for meromorphic functions.)

• a, b – integration bounds. The bounds can be real or complex balls, or elements of any parent that coerces into this ball field, e.g. rational or algebraic numbers.

• rel_tol (default: $$2^{-p}$$ where $$p$$ is the precision of the ball field) – relative accuracy goal

• abs_tol (default: $$2^{-p}$$ where $$p$$ is the precision of the ball field) – absolute accuracy goal

Additionally, the following optional parameters can be used to control the integration algorithm. See the FLINT documentation for more information.

• deg_limit – maximum quadrature degree for each subinterval

• eval_limit – maximum number of function evaluations

• depth_limit – maximum search depth for adaptive subdivision

• use_heap (boolean, default False) – if True, use a priority queue instead of a stack to manage subintervals. This sometimes gives better results for integrals with slow convergence but may require more memory and increasing depth_limit.

• verbose (integer, default 0) – If set to 1, some information about the overall integration process is printed to standard output. If set to 2, information about each subinterval is printed.

EXAMPLES:

Some analytic integrands:

sage: CBF.integral(lambda x, _: x, 0, 1)
[0.500000000000000 +/- ...e-16]

sage: CBF.integral(lambda x, _: x.gamma(), 1 - CBF(i), 1 + CBF(i)) # abs tol 1e-13
[+/- 1.39e-15] + [1.57239266949806 +/- 8.33e-15]*I

sage: C = ComplexBallField(100)
sage: C.integral(lambda x, _: x.cos() * x.sin(), 0, 1)
[0.35403670913678559674939205737 +/- ...e-30]

sage: CBF.integral(lambda x, _: (x + x.exp()).sin(), 0, 8)
[0.34740017266 +/- ...e-12]

sage: C = ComplexBallField(2000)
sage: C.integral(lambda x, _: (x + x.exp()).sin(), 0, 8) # long time
[0.34740017...55347713 +/- ...e-598]

>>> from sage.all import *
>>> CBF.integral(lambda x, _: x, Integer(0), Integer(1))
[0.500000000000000 +/- ...e-16]

>>> CBF.integral(lambda x, _: x.gamma(), Integer(1) - CBF(i), Integer(1) + CBF(i)) # abs tol 1e-13
[+/- 1.39e-15] + [1.57239266949806 +/- 8.33e-15]*I

>>> C = ComplexBallField(Integer(100))
>>> C.integral(lambda x, _: x.cos() * x.sin(), Integer(0), Integer(1))
[0.35403670913678559674939205737 +/- ...e-30]

>>> CBF.integral(lambda x, _: (x + x.exp()).sin(), Integer(0), Integer(8))
[0.34740017266 +/- ...e-12]

>>> C = ComplexBallField(Integer(2000))
>>> C.integral(lambda x, _: (x + x.exp()).sin(), Integer(0), Integer(8)) # long time
[0.34740017...55347713 +/- ...e-598]


Here the integration path crosses the branch cut of the square root:

sage: def my_sqrt(z, analytic):
....:     if (analytic and not z.real() > 0
....:                  and z.imag().contains_zero()):
....:         return CBF(NaN)
....:     else:
....:         return z.sqrt()
sage: CBF.integral(my_sqrt, -1 + CBF(i), -1 - CBF(i))                       # needs sage.symbolic
[+/- ...e-14] + [-0.4752076627926 +/- 5...e-14]*I

>>> from sage.all import *
>>> def my_sqrt(z, analytic):
...     if (analytic and not z.real() > Integer(0)
...                  and z.imag().contains_zero()):
...         return CBF(NaN)
...     else:
...         return z.sqrt()
>>> CBF.integral(my_sqrt, -Integer(1) + CBF(i), -Integer(1) - CBF(i))                       # needs sage.symbolic
[+/- ...e-14] + [-0.4752076627926 +/- 5...e-14]*I


Note, though, that proper handling of the analytic flag is required even when the path does not touch the branch cut:

sage: correct = CBF.integral(my_sqrt, 1, 2); correct
[1.21895141649746 +/- ...e-15]
sage: RBF(integral(sqrt(x), x, 1, 2))       # long time                     # needs sage.symbolic
[1.21895141649746 +/- ...e-15]
sage: wrong = CBF.integral(lambda z, _: z.sqrt(), 1, 2) # WRONG!
sage: correct - wrong
[-5.640636259e-5 +/- ...e-15]

>>> from sage.all import *
>>> correct = CBF.integral(my_sqrt, Integer(1), Integer(2)); correct
[1.21895141649746 +/- ...e-15]
>>> RBF(integral(sqrt(x), x, Integer(1), Integer(2)))       # long time                     # needs sage.symbolic
[1.21895141649746 +/- ...e-15]
>>> wrong = CBF.integral(lambda z, _: z.sqrt(), Integer(1), Integer(2)) # WRONG!
>>> correct - wrong
[-5.640636259e-5 +/- ...e-15]


We can integrate the real absolute value function by defining a piecewise holomorphic extension:

sage: def real_abs(z, analytic):
....:     if z.real().contains_zero():
....:         if analytic:
....:             return z.parent()(NaN)
....:         else:
....:             return z.union(-z)
....:     elif z.real() > 0:
....:         return z
....:     else:
....:         return -z
sage: CBF.integral(real_abs, -1, 1)
[1.00000000000...]
sage: CBF.integral(lambda z, analytic: real_abs(z.sin(), analytic), 0, 2*CBF.pi())
[4.00000000000...]

>>> from sage.all import *
>>> def real_abs(z, analytic):
...     if z.real().contains_zero():
...         if analytic:
...             return z.parent()(NaN)
...         else:
...             return z.union(-z)
...     elif z.real() > Integer(0):
...         return z
...     else:
...         return -z
>>> CBF.integral(real_abs, -Integer(1), Integer(1))
[1.00000000000...]
>>> CBF.integral(lambda z, analytic: real_abs(z.sin(), analytic), Integer(0), Integer(2)*CBF.pi())
[4.00000000000...]


Some methods of complex balls natively support the analytic flag:

sage: CBF.integral(lambda z, analytic: z.log(analytic=analytic),
....:              -1-CBF(i), -1+CBF(i))
[+/- ...e-14] + [0.26394350735484 +/- ...e-15]*I
sage: from sage.rings.complex_arb import ComplexBall
sage: CBF.integral(ComplexBall.sqrt, -1+CBF(i), -1-CBF(i))
[+/- ...e-14] + [-0.4752076627926 +/- 5...e-14]*I

>>> from sage.all import *
>>> CBF.integral(lambda z, analytic: z.log(analytic=analytic),
...              -Integer(1)-CBF(i), -Integer(1)+CBF(i))
[+/- ...e-14] + [0.26394350735484 +/- ...e-15]*I
>>> from sage.rings.complex_arb import ComplexBall
>>> CBF.integral(ComplexBall.sqrt, -Integer(1)+CBF(i), -Integer(1)-CBF(i))
[+/- ...e-14] + [-0.4752076627926 +/- 5...e-14]*I


Here the integrand has a pole on or very close to the integration path, but there is no need to explicitly handle the analytic flag since the integrand is unbounded:

sage: CBF.integral(lambda x, _: 1/x, -1, 1)
nan + nan*I
sage: CBF.integral(lambda x, _: 1/x, 10^-1000, 1)
nan + nan*I
sage: CBF.integral(lambda x, _: 1/x, 10^-1000, 1, abs_tol=1e-10)
[2302.5850930 +/- ...e-8]

>>> from sage.all import *
>>> CBF.integral(lambda x, _: Integer(1)/x, -Integer(1), Integer(1))
nan + nan*I
>>> CBF.integral(lambda x, _: Integer(1)/x, Integer(10)**-Integer(1000), Integer(1))
nan + nan*I
>>> CBF.integral(lambda x, _: Integer(1)/x, Integer(10)**-Integer(1000), Integer(1), abs_tol=RealNumber('1e-10'))
[2302.5850930 +/- ...e-8]


Tolerances:

sage: CBF.integral(lambda x, _: x.exp(), -1020, -1010)
[+/- ...e-438]
sage: CBF.integral(lambda x, _: x.exp(), -1020, -1010, abs_tol=1e-450)
[2.304377150950e-439 +/- ...e-452]
sage: CBF.integral(lambda x, _: x.exp(), -1020, -1010, abs_tol=0)
[2.304377150950e-439 +/- 7...e-452]
sage: CBF.integral(lambda x, _: x.exp(), -1020, -1010, rel_tol=1e-2, abs_tol=0)
[2.3044e-439 +/- ...e-444]

sage: epsi = CBF(1e-10)
sage: CBF.integral(lambda x, _: x*(1/x).sin(), epsi, 1)
[0.38 +/- ...e-3]
sage: CBF.integral(lambda x, _: x*(1/x).sin(), epsi, 1, use_heap=True)
[0.37853002 +/- ...e-9]

>>> from sage.all import *
>>> CBF.integral(lambda x, _: x.exp(), -Integer(1020), -Integer(1010))
[+/- ...e-438]
>>> CBF.integral(lambda x, _: x.exp(), -Integer(1020), -Integer(1010), abs_tol=RealNumber('1e-450'))
[2.304377150950e-439 +/- ...e-452]
>>> CBF.integral(lambda x, _: x.exp(), -Integer(1020), -Integer(1010), abs_tol=Integer(0))
[2.304377150950e-439 +/- 7...e-452]
>>> CBF.integral(lambda x, _: x.exp(), -Integer(1020), -Integer(1010), rel_tol=RealNumber('1e-2'), abs_tol=Integer(0))
[2.3044e-439 +/- ...e-444]

>>> epsi = CBF(RealNumber('1e-10'))
>>> CBF.integral(lambda x, _: x*(Integer(1)/x).sin(), epsi, Integer(1))
[0.38 +/- ...e-3]
>>> CBF.integral(lambda x, _: x*(Integer(1)/x).sin(), epsi, Integer(1), use_heap=True)
[0.37853002 +/- ...e-9]


ALGORITHM:

Uses the acb_calc module of the FLINT library.

is_exact()[source]#

Complex ball fields are not exact.

EXAMPLES:

sage: ComplexBallField().is_exact()
False

>>> from sage.all import *
>>> ComplexBallField().is_exact()
False

ngens()[source]#

Return 1 as the only generator is the imaginary unit.

EXAMPLES:

sage: CBF.ngens()
1

>>> from sage.all import *
>>> CBF.ngens()
1

pi()[source]#

Return a ball enclosing $$\pi$$.

EXAMPLES:

sage: CBF.pi()
[3.141592653589793 +/- ...e-16]
sage: ComplexBallField(128).pi()
[3.1415926535897932384626433832795028842 +/- ...e-38]

sage: CBF.pi().parent()
Complex ball field with 53 bits of precision

>>> from sage.all import *
>>> CBF.pi()
[3.141592653589793 +/- ...e-16]
>>> ComplexBallField(Integer(128)).pi()
[3.1415926535897932384626433832795028842 +/- ...e-38]

>>> CBF.pi().parent()
Complex ball field with 53 bits of precision

prec()[source]#

Return the bit precision used for operations on elements of this field.

EXAMPLES:

sage: ComplexBallField().precision()
53

>>> from sage.all import *
>>> ComplexBallField().precision()
53

precision()[source]#

Return the bit precision used for operations on elements of this field.

EXAMPLES:

sage: ComplexBallField().precision()
53

>>> from sage.all import *
>>> ComplexBallField().precision()
53

some_elements()[source]#

Complex ball fields contain elements with exact, inexact, infinite, or undefined real and imaginary parts.

EXAMPLES:

sage: CBF.some_elements()
[1.000000000000000,
-0.5000000000000000*I,
1.000000000000000 + [0.3333333333333333 +/- ...e-17]*I,
[-0.3333333333333333 +/- ...e-17] + 0.2500000000000000*I,
[-2.175556475109056e+181961467118333366510562 +/- ...e+181961467118333366510545],
[+/- inf],
[0.3333333333333333 +/- ...e-17] + [+/- inf]*I,
[+/- inf] + [+/- inf]*I,
nan,
nan + nan*I,
[+/- inf] + nan*I]

>>> from sage.all import *
>>> CBF.some_elements()
[1.000000000000000,
-0.5000000000000000*I,
1.000000000000000 + [0.3333333333333333 +/- ...e-17]*I,
[-0.3333333333333333 +/- ...e-17] + 0.2500000000000000*I,
[-2.175556475109056e+181961467118333366510562 +/- ...e+181961467118333366510545],
[+/- inf],
[0.3333333333333333 +/- ...e-17] + [+/- inf]*I,
[+/- inf] + [+/- inf]*I,
nan,
nan + nan*I,
[+/- inf] + nan*I]

class sage.rings.complex_arb.IntegrationContext[source]#

Bases: object

Used to wrap the integrand and hold some context information during numerical integration.