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Complex.jl
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Complex.jl
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###############################################################################
#
# acb.jl : Arb complex numbers
#
# Copyright (C) 2015 Tommy Hofmann
# Copyright (C) 2015 Fredrik Johansson
#
###############################################################################
import Base: real, imag, abs, conj, angle, log, log1p, sin, cos,
tan, cot, sinpi, cospi, sinh, cosh, tanh, coth, atan, expm1
import Base: cispi
export one, onei, real, imag, conj, abs, inv, angle, isreal, polygamma, erf,
erfi, erfc, bessel_j, bessel_k, bessel_i, bessel_y, airy_ai, airy_bi,
airy_ai_prime, airy_bi_prime
export rsqrt, log, log1p, cispi, sin, cos, tan, cot, sinpi, cospi, tanpi,
cotpi, sincos, sincospi, sinh, cosh, tanh, coth, sinhcosh, atan,
log_sinpi, gamma, rgamma, lgamma, gamma_regularized, gamma_lower,
gamma_lower_regularized, rising_factorial, rising_factorial2, polylog,
barnes_g, log_barnes_g, agm, exp_integral_ei, sin_integral,
cos_integral, sinh_integral, cosh_integral, log_integral,
log_integral_offset, exp_integral_e, gamma, hypergeometric_1f1,
hypergeometric_1f1_regularized, hypergeometric_u, hypergeometric_2f1,
jacobi_theta, modular_delta, dedekind_eta, eisenstein_g, j_invariant,
modular_lambda, modular_weber_f, modular_weber_f1, modular_weber_f2,
weierstrass_p, weierstrass_p_prime, elliptic_k, elliptic_e,
canonical_unit, root_of_unity, hilbert_class_polynomial
###############################################################################
#
# Basic manipulation
#
###############################################################################
elem_type(::Type{ComplexField}) = ComplexFieldElem
parent_type(::Type{ComplexFieldElem}) = ComplexField
base_ring(R::ComplexField) = Union{}
base_ring(a::ComplexFieldElem) = Union{}
parent(x::ComplexFieldElem) = ComplexField()
is_domain_type(::Type{ComplexFieldElem}) = true
is_exact_type(::Type{ComplexFieldElem}) = false
function zero(r::ComplexField)
z = ComplexFieldElem()
return z
end
function one(r::ComplexField)
z = ComplexFieldElem()
ccall((:acb_one, libarb), Nothing, (Ref{ComplexFieldElem}, ), z)
return z
end
@doc raw"""
onei(r::ComplexField)
Return exact one times $i$ in the given Arb complex field.
"""
function onei(r::ComplexField)
z = ComplexFieldElem()
ccall((:acb_onei, libarb), Nothing, (Ref{ComplexFieldElem}, ), z)
return z
end
@doc raw"""
accuracy_bits(x::ComplexFieldElem)
Return the relative accuracy of $x$ measured in bits, capped between
`typemax(Int)` and `-typemax(Int)`.
"""
function accuracy_bits(x::ComplexFieldElem)
# bug in acb.h: rel_accuracy_bits is not in the library
return -ccall((:acb_rel_error_bits, libarb), Int, (Ref{ComplexFieldElem},), x)
end
function deepcopy_internal(a::ComplexFieldElem, dict::IdDict)
b = ComplexFieldElem()
ccall((:acb_set, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}), b, a)
return b
end
function canonical_unit(x::ComplexFieldElem)
return x
end
# TODO: implement hash
function check_parent(a::ComplexFieldElem, b::ComplexFieldElem)
return true
end
characteristic(::ComplexField) = 0
################################################################################
#
# Conversions
#
################################################################################
function convert(::Type{ComplexF64}, x::ComplexFieldElem)
GC.@preserve x begin
re = ccall((:acb_real_ptr, libarb), Ptr{arb_struct}, (Ref{ComplexFieldElem}, ), x)
im = ccall((:acb_imag_ptr, libarb), Ptr{arb_struct}, (Ref{ComplexFieldElem}, ), x)
t = ccall((:arb_mid_ptr, libarb), Ptr{arf_struct}, (Ptr{RealFieldElem}, ), re)
u = ccall((:arb_mid_ptr, libarb), Ptr{arf_struct}, (Ptr{RealFieldElem}, ), im)
# 4 == round to nearest
v = ccall((:arf_get_d, libarb), Float64, (Ptr{arf_struct}, Int), t, 4)
w = ccall((:arf_get_d, libarb), Float64, (Ptr{arf_struct}, Int), u, 4)
end
return complex(v, w)
end
################################################################################
#
# Real and imaginary part
#
################################################################################
function real(x::ComplexFieldElem)
z = RealFieldElem()
ccall((:acb_get_real, libarb), Nothing, (Ref{RealFieldElem}, Ref{ComplexFieldElem}), z, x)
return z
end
function imag(x::ComplexFieldElem)
z = RealFieldElem()
ccall((:acb_get_imag, libarb), Nothing, (Ref{RealFieldElem}, Ref{ComplexFieldElem}), z, x)
return z
end
################################################################################
#
# String I/O
#
################################################################################
function expressify(z::ComplexFieldElem; context = nothing)
x = real(z)
y = imag(z)
if iszero(y) # is exact zero!
return expressify(x, context = context)
else
y = Expr(:call, :*, expressify(y, context = context), :im)
if iszero(x)
return y
else
x = expressify(x, context = context)
return Expr(:call, :+, x, y)
end
end
end
function Base.show(io::IO, ::MIME"text/plain", z::ComplexFieldElem)
print(io, AbstractAlgebra.obj_to_string(z, context = io))
end
function Base.show(io::IO, z::ComplexFieldElem)
print(io, AbstractAlgebra.obj_to_string(z, context = io))
end
function show(io::IO, x::ComplexField)
if get(io, :supercompact, false)
print(io, LowercaseOff(), "CC")
else
print(io, "Complex field")
end
end
################################################################################
#
# Unary operations
#
################################################################################
function -(x::ComplexFieldElem)
z = ComplexFieldElem()
ccall((:acb_neg, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}), z, x)
return z
end
################################################################################
#
# Binary operations
#
################################################################################
# acb - acb
for (s,f) in ((:+,"acb_add"), (:*,"acb_mul"), (://, "acb_div"), (:-,"acb_sub"), (:^,"acb_pow"))
@eval begin
function ($s)(x::ComplexFieldElem, y::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall(($f, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int),
z, x, y, prec)
return z
end
end
end
for (f,s) in ((:+, "add"), (:-, "sub"), (:*, "mul"), (://, "div"), (:^, "pow"))
@eval begin
function ($f)(x::ComplexFieldElem, y::UInt, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall(($("acb_"*s*"_ui"), libarb), Nothing,
(Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, UInt, Int),
z, x, y, prec)
return z
end
function ($f)(x::ComplexFieldElem, y::Int, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall(($("acb_"*s*"_si"), libarb), Nothing,
(Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int, Int), z, x, y, prec)
return z
end
function ($f)(x::ComplexFieldElem, y::ZZRingElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall(($("acb_"*s*"_fmpz"), libarb), Nothing,
(Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Ref{ZZRingElem}, Int),
z, x, y, prec)
return z
end
function ($f)(x::ComplexFieldElem, y::RealFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall(($("acb_"*s*"_arb"), libarb), Nothing,
(Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Ref{RealFieldElem}, Int),
z, x, y, prec)
return z
end
end
end
+(x::UInt,y::ComplexFieldElem) = +(y,x)
+(x::Int,y::ComplexFieldElem) = +(y,x)
+(x::ZZRingElem,y::ComplexFieldElem) = +(y,x)
+(x::RealFieldElem,y::ComplexFieldElem) = +(y,x)
*(x::UInt,y::ComplexFieldElem) = *(y,x)
*(x::Int,y::ComplexFieldElem) = *(y,x)
*(x::ZZRingElem,y::ComplexFieldElem) = *(y,x)
*(x::RealFieldElem,y::ComplexFieldElem) = *(y,x)
//(x::UInt,y::ComplexFieldElem) = (x == 1) ? inv(y) : parent(y)(x) // y
//(x::Int,y::ComplexFieldElem) = (x == 1) ? inv(y) : parent(y)(x) // y
//(x::ZZRingElem,y::ComplexFieldElem) = isone(x) ? inv(y) : parent(y)(x) // y
//(x::RealFieldElem,y::ComplexFieldElem) = isone(x) ? inv(y) : parent(y)(x) // y
^(x::UInt,y::ComplexFieldElem) = parent(y)(x) ^ y
^(x::Int,y::ComplexFieldElem) = parent(y)(x) ^ y
^(x::ZZRingElem,y::ComplexFieldElem) = parent(y)(x) ^ y
^(x::RealFieldElem,y::ComplexFieldElem) = parent(y)(x) ^ y
^(x::Integer, y::ComplexFieldElem) = ZZRingElem(x)^y
function -(x::UInt, y::ComplexFieldElem)
z = ComplexFieldElem()
ccall((:acb_sub_ui, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, UInt, Int), z, y, x, precision(Balls))
ccall((:acb_neg, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}), z, z)
return z
end
function -(x::Int, y::ComplexFieldElem)
z = ComplexFieldElem()
ccall((:acb_sub_si, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int, Int), z, y, x, precision(Balls))
ccall((:acb_neg, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}), z, z)
return z
end
function -(x::ZZRingElem, y::ComplexFieldElem)
z = ComplexFieldElem()
ccall((:acb_sub_fmpz, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Ref{ZZRingElem}, Int), z, y, x, precision(Balls))
ccall((:acb_neg, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}), z, z)
return z
end
function -(x::RealFieldElem, y::ComplexFieldElem)
z = ComplexFieldElem()
ccall((:acb_sub_arb, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Ref{RealFieldElem}, Int), z, y, x, precision(Balls))
ccall((:acb_neg, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}), z, z)
return z
end
+(x::ComplexFieldElem, y::Integer) = x + ZZRingElem(y)
-(x::ComplexFieldElem, y::Integer) = x - ZZRingElem(y)
*(x::ComplexFieldElem, y::Integer) = x*ZZRingElem(y)
//(x::ComplexFieldElem, y::Integer) = x//ZZRingElem(y)
+(x::Integer, y::ComplexFieldElem) = ZZRingElem(x) + y
-(x::Integer, y::ComplexFieldElem) = ZZRingElem(x) - y
*(x::Integer, y::ComplexFieldElem) = ZZRingElem(x)*y
//(x::Integer, y::ComplexFieldElem) = ZZRingElem(x)//y
^(x::ComplexFieldElem, y::Integer) = x ^ parent(x)(y)
+(x::ComplexFieldElem, y::QQFieldElem) = x + parent(x)(y)
-(x::ComplexFieldElem, y::QQFieldElem) = x - parent(x)(y)
*(x::ComplexFieldElem, y::QQFieldElem) = x * parent(x)(y)
//(x::ComplexFieldElem, y::QQFieldElem) = x // parent(x)(y)
^(x::ComplexFieldElem, y::QQFieldElem) = x ^ parent(x)(y)
+(x::QQFieldElem, y::ComplexFieldElem) = parent(y)(x) + y
-(x::QQFieldElem, y::ComplexFieldElem) = parent(y)(x) - y
*(x::QQFieldElem, y::ComplexFieldElem) = parent(y)(x) * y
//(x::QQFieldElem, y::ComplexFieldElem) = parent(y)(x) // y
^(x::QQFieldElem, y::ComplexFieldElem) = parent(y)(x) ^ y
divexact(x::ComplexFieldElem, y::ComplexFieldElem; check::Bool=true) = x // y
divexact(x::ZZRingElem, y::ComplexFieldElem; check::Bool=true) = x // y
divexact(x::ComplexFieldElem, y::ZZRingElem; check::Bool=true) = x // y
divexact(x::Int, y::ComplexFieldElem; check::Bool=true) = x // y
divexact(x::ComplexFieldElem, y::Int; check::Bool=true) = x // y
divexact(x::UInt, y::ComplexFieldElem; check::Bool=true) = x // y
divexact(x::ComplexFieldElem, y::UInt; check::Bool=true) = x // y
divexact(x::QQFieldElem, y::ComplexFieldElem; check::Bool=true) = x // y
divexact(x::ComplexFieldElem, y::QQFieldElem; check::Bool=true) = x // y
divexact(x::RealFieldElem, y::ComplexFieldElem; check::Bool=true) = x // y
divexact(x::ComplexFieldElem, y::RealFieldElem; check::Bool=true) = x // y
divexact(x::Float64, y::ComplexFieldElem; check::Bool=true) = x // y
divexact(x::ComplexFieldElem, y::Float64; check::Bool=true) = x // y
divexact(x::BigFloat, y::ComplexFieldElem; check::Bool=true) = x // y
divexact(x::ComplexFieldElem, y::BigFloat; check::Bool=true) = x // y
divexact(x::Integer, y::ComplexFieldElem; check::Bool=true) = x // y
divexact(x::ComplexFieldElem, y::Integer; check::Bool=true) = x // y
divexact(x::Rational{T}, y::ComplexFieldElem; check::Bool=true) where {T <: Integer} = x // y
divexact(x::ComplexFieldElem, y::Rational{T}; check::Bool=true) where {T <: Integer} = x // y
/(x::ComplexFieldElem, y::ComplexFieldElem) = x // y
/(x::ZZRingElem, y::ComplexFieldElem) = x // y
/(x::ComplexFieldElem, y::ZZRingElem) = x // y
/(x::Int, y::ComplexFieldElem) = x // y
/(x::ComplexFieldElem, y::Int) = x // y
/(x::UInt, y::ComplexFieldElem) = x // y
/(x::ComplexFieldElem, y::UInt) = x // y
/(x::QQFieldElem, y::ComplexFieldElem) = x // y
/(x::ComplexFieldElem, y::QQFieldElem) = x // y
/(x::RealFieldElem, y::ComplexFieldElem) = x // y
/(x::ComplexFieldElem, y::RealFieldElem) = x // y
+(x::Rational{T}, y::ComplexFieldElem) where {T <: Integer} = QQFieldElem(x) + y
+(x::ComplexFieldElem, y::Rational{T}) where {T <: Integer} = x + QQFieldElem(y)
-(x::Rational{T}, y::ComplexFieldElem) where {T <: Integer} = QQFieldElem(x) - y
-(x::ComplexFieldElem, y::Rational{T}) where {T <: Integer} = x - QQFieldElem(y)
*(x::Rational{T}, y::ComplexFieldElem) where {T <: Integer} = QQFieldElem(x) * y
*(x::ComplexFieldElem, y::Rational{T}) where {T <: Integer} = x * QQFieldElem(y)
//(x::Rational{T}, y::ComplexFieldElem) where {T <: Integer} = QQFieldElem(x) // y
//(x::ComplexFieldElem, y::Rational{T}) where {T <: Integer} = x // QQFieldElem(y)
^(x::Rational{T}, y::ComplexFieldElem) where {T <: Integer} = QQFieldElem(x)^y
^(x::ComplexFieldElem, y::Rational{T}) where {T <: Integer} = x ^ QQFieldElem(y)
+(x::Float64, y::ComplexFieldElem) = parent(y)(x) + y
+(x::ComplexFieldElem, y::Float64) = x + parent(x)(y)
-(x::Float64, y::ComplexFieldElem) = parent(y)(x) - y
-(x::ComplexFieldElem, y::Float64) = x - parent(x)(y)
*(x::Float64, y::ComplexFieldElem) = parent(y)(x) * y
*(x::ComplexFieldElem, y::Float64) = x * parent(x)(y)
//(x::Float64, y::ComplexFieldElem) = parent(y)(x) // y
//(x::ComplexFieldElem, y::Float64) = x // parent(x)(y)
^(x::Float64, y::ComplexFieldElem) = parent(y)(x)^y
^(x::ComplexFieldElem, y::Float64) = x ^ parent(x)(y)
+(x::BigFloat, y::ComplexFieldElem) = parent(y)(x) + y
+(x::ComplexFieldElem, y::BigFloat) = x + parent(x)(y)
-(x::BigFloat, y::ComplexFieldElem) = parent(y)(x) - y
-(x::ComplexFieldElem, y::BigFloat) = x - parent(x)(y)
*(x::BigFloat, y::ComplexFieldElem) = parent(y)(x) * y
*(x::ComplexFieldElem, y::BigFloat) = x * parent(x)(y)
//(x::BigFloat, y::ComplexFieldElem) = parent(y)(x) // y
//(x::ComplexFieldElem, y::BigFloat) = x // parent(x)(y)
^(x::BigFloat, y::ComplexFieldElem) = parent(y)(x)^y
^(x::ComplexFieldElem, y::BigFloat) = x ^ parent(x)(y)
################################################################################
#
# Comparison
#
################################################################################
@doc raw"""
isequal(x::ComplexFieldElem, y::ComplexFieldElem)
Return `true` if the boxes $x$ and $y$ are precisely equal, i.e. their real
and imaginary parts have the same midpoints and radii.
"""
function isequal(x::ComplexFieldElem, y::ComplexFieldElem)
r = ccall((:acb_equal, libarb), Cint, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}), x, y)
return Bool(r)
end
function ==(x::ComplexFieldElem, y::ComplexFieldElem)
r = ccall((:acb_eq, libarb), Cint, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}), x, y)
return Bool(r)
end
function !=(x::ComplexFieldElem, y::ComplexFieldElem)
r = ccall((:acb_ne, libarb), Cint, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}), x, y)
return Bool(r)
end
==(x::ComplexFieldElem,y::Int) = (x == parent(x)(y))
==(x::Int,y::ComplexFieldElem) = (y == parent(y)(x))
==(x::ComplexFieldElem,y::RealFieldElem) = (x == parent(x)(y))
==(x::RealFieldElem,y::ComplexFieldElem) = (y == parent(y)(x))
==(x::ComplexFieldElem,y::ZZRingElem) = (x == parent(x)(y))
==(x::ZZRingElem,y::ComplexFieldElem) = (y == parent(y)(x))
==(x::ComplexFieldElem,y::Integer) = x == ZZRingElem(y)
==(x::Integer,y::ComplexFieldElem) = ZZRingElem(x) == y
==(x::ComplexFieldElem,y::Float64) = (x == parent(x)(y))
==(x::Float64,y::ComplexFieldElem) = (y == parent(y)(x))
!=(x::ComplexFieldElem,y::Int) = (x != parent(x)(y))
!=(x::Int,y::ComplexFieldElem) = (y != parent(y)(x))
!=(x::ComplexFieldElem,y::RealFieldElem) = (x != parent(x)(y))
!=(x::RealFieldElem,y::ComplexFieldElem) = (y != parent(y)(x))
!=(x::ComplexFieldElem,y::ZZRingElem) = (x != parent(x)(y))
!=(x::ZZRingElem,y::ComplexFieldElem) = (y != parent(y)(x))
!=(x::ComplexFieldElem,y::Float64) = (x != parent(x)(y))
!=(x::Float64,y::ComplexFieldElem) = (y != parent(y)(x))
################################################################################
#
# Containment
#
################################################################################
@doc raw"""
overlaps(x::ComplexFieldElem, y::ComplexFieldElem)
Returns `true` if any part of the box $x$ overlaps any part of the box $y$,
otherwise return `false`.
"""
function overlaps(x::ComplexFieldElem, y::ComplexFieldElem)
r = ccall((:acb_overlaps, libarb), Cint, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}), x, y)
return Bool(r)
end
@doc raw"""
contains(x::ComplexFieldElem, y::ComplexFieldElem)
Returns `true` if the box $x$ contains the box $y$, otherwise return
`false`.
"""
function contains(x::ComplexFieldElem, y::ComplexFieldElem)
r = ccall((:acb_contains, libarb), Cint, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}), x, y)
return Bool(r)
end
@doc raw"""
contains(x::ComplexFieldElem, y::QQFieldElem)
Returns `true` if the box $x$ contains the given rational value, otherwise
return `false`.
"""
function contains(x::ComplexFieldElem, y::QQFieldElem)
r = ccall((:acb_contains_fmpq, libarb), Cint, (Ref{ComplexFieldElem}, Ref{QQFieldElem}), x, y)
return Bool(r)
end
@doc raw"""
contains(x::ComplexFieldElem, y::ZZRingElem)
Returns `true` if the box $x$ contains the given integer value, otherwise
return `false`.
"""
function contains(x::ComplexFieldElem, y::ZZRingElem)
r = ccall((:acb_contains_fmpz, libarb), Cint, (Ref{ComplexFieldElem}, Ref{ZZRingElem}), x, y)
return Bool(r)
end
function contains(x::ComplexFieldElem, y::Int)
v = ZZRingElem(y)
r = ccall((:acb_contains_fmpz, libarb), Cint, (Ref{ComplexFieldElem}, Ref{ZZRingElem}), x, v)
return Bool(r)
end
@doc raw"""
contains(x::ComplexFieldElem, y::Integer)
Returns `true` if the box $x$ contains the given integer value, otherwise
return `false`.
"""
contains(x::ComplexFieldElem, y::Integer) = contains(x, ZZRingElem(y))
@doc raw"""
contains(x::ComplexFieldElem, y::Rational{T}) where {T <: Integer}
Returns `true` if the box $x$ contains the given rational value, otherwise
return `false`.
"""
contains(x::ComplexFieldElem, y::Rational{T}) where {T <: Integer} = contains(x, ZZRingElem(y))
@doc raw"""
contains_zero(x::ComplexFieldElem)
Returns `true` if the box $x$ contains zero, otherwise return `false`.
"""
function contains_zero(x::ComplexFieldElem)
return Bool(ccall((:acb_contains_zero, libarb), Cint, (Ref{ComplexFieldElem},), x))
end
################################################################################
#
# Predicates
#
################################################################################
function is_unit(x::ComplexFieldElem)
!iszero(x)
end
@doc raw"""
iszero(x::ComplexFieldElem)
Return `true` if $x$ is certainly zero, otherwise return `false`.
"""
function iszero(x::ComplexFieldElem)
return Bool(ccall((:acb_is_zero, libarb), Cint, (Ref{ComplexFieldElem},), x))
end
@doc raw"""
isone(x::ComplexFieldElem)
Return `true` if $x$ is certainly one, otherwise return `false`.
"""
function isone(x::ComplexFieldElem)
return Bool(ccall((:acb_is_one, libarb), Cint, (Ref{ComplexFieldElem},), x))
end
@doc raw"""
isfinite(x::ComplexFieldElem)
Return `true` if $x$ is finite, i.e. its real and imaginary parts have finite
midpoint and radius, otherwise return `false`.
"""
function isfinite(x::ComplexFieldElem)
return Bool(ccall((:acb_is_finite, libarb), Cint, (Ref{ComplexFieldElem},), x))
end
@doc raw"""
is_exact(x::ComplexFieldElem)
Return `true` if $x$ is exact, i.e. has its real and imaginary parts have
zero radius, otherwise return `false`.
"""
function is_exact(x::ComplexFieldElem)
return Bool(ccall((:acb_is_exact, libarb), Cint, (Ref{ComplexFieldElem},), x))
end
@doc raw"""
isinteger(x::ComplexFieldElem)
Return `true` if $x$ is an exact integer, otherwise return `false`.
"""
function isinteger(x::ComplexFieldElem)
return Bool(ccall((:acb_is_int, libarb), Cint, (Ref{ComplexFieldElem},), x))
end
function isreal(x::ComplexFieldElem)
return Bool(ccall((:acb_is_real, libarb), Cint, (Ref{ComplexFieldElem},), x))
end
is_negative(x::ComplexFieldElem) = isreal(x) && is_negative(real(x))
################################################################################
#
# Absolute value
#
################################################################################
function abs(x::ComplexFieldElem, prec::Int = precision(Balls))
z = RealFieldElem()
ccall((:acb_abs, libarb), Nothing,
(Ref{RealFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
################################################################################
#
# Inversion
#
################################################################################
function inv(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_inv, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
################################################################################
#
# Shifting
#
################################################################################
function ldexp(x::ComplexFieldElem, y::Int)
z = ComplexFieldElem()
ccall((:acb_mul_2exp_si, libarb), Nothing,
(Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, y)
return z
end
function ldexp(x::ComplexFieldElem, y::ZZRingElem)
z = ComplexFieldElem()
ccall((:acb_mul_2exp_fmpz, libarb), Nothing,
(Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Ref{ZZRingElem}), z, x, y)
return z
end
################################################################################
#
# Miscellaneous
#
################################################################################
@doc raw"""
trim(x::ComplexFieldElem)
Return an `acb` box containing $x$ but which may be more economical,
by rounding off insignificant bits from midpoints.
"""
function trim(x::ComplexFieldElem)
z = ComplexFieldElem()
ccall((:acb_trim, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}), z, x)
return z
end
@doc raw"""
unique_integer(x::ComplexFieldElem)
Return a pair where the first value is a boolean and the second is an `ZZRingElem`
integer. The boolean indicates whether the box $x$ contains a unique
integer. If this is the case, the second return value is set to this unique
integer.
"""
function unique_integer(x::ComplexFieldElem)
z = ZZRingElem()
unique = ccall((:acb_get_unique_fmpz, libarb), Int,
(Ref{ZZRingElem}, Ref{ComplexFieldElem}), z, x)
return (unique != 0, z)
end
function conj(x::ComplexFieldElem)
z = ComplexFieldElem()
ccall((:acb_conj, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}), z, x)
return z
end
function angle(x::ComplexFieldElem, prec::Int = precision(Balls))
z = RealFieldElem()
ccall((:acb_arg, libarb), Nothing,
(Ref{RealFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
z.parent = RealField(precision(Balls))
return z
end
################################################################################
#
# Constants
#
################################################################################
@doc raw"""
const_pi(r::ComplexField)
Return $\pi = 3.14159\ldots$ as an element of $r$.
"""
function const_pi(r::ComplexField, prec::Int = precision(Balls))
z = r()
ccall((:acb_const_pi, libarb), Nothing, (Ref{ComplexFieldElem}, Int), z, prec)
return z
end
################################################################################
#
# Complex valued functions
#
################################################################################
# complex - complex functions
function Base.sqrt(x::ComplexFieldElem, prec::Int = precision(Balls); check::Bool=true)
z = ComplexFieldElem()
ccall((:acb_sqrt, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
rsqrt(x::ComplexFieldElem)
Return the reciprocal of the square root of $x$, i.e. $1/\sqrt{x}$.
"""
function rsqrt(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_rsqrt, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function log(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_log, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function log1p(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_log1p, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function Base.exp(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_exp, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function Base.expm1(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_expm1, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
cispi(x::ComplexFieldElem)
Return the exponential of $\pi i x$.
"""
function cispi(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_exp_pi_i, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
root_of_unity(C::ComplexField, k::Int)
Return $\exp(2\pi i/k)$.
"""
function root_of_unity(C::ComplexField, k::Int, prec::Int = precision(Balls))
k <= 0 && throw(ArgumentError("Order must be positive ($k)"))
z = C()
ccall((:acb_unit_root, libarb), Nothing, (Ref{ComplexFieldElem}, UInt, Int), z, k, prec)
return z
end
function sin(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_sin, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function cos(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_cos, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function tan(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_tan, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function cot(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_cot, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function sinpi(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_sin_pi, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function cospi(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_cos_pi, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function tanpi(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_tan_pi, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function cotpi(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_cot_pi, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function sinh(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_sinh, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function cosh(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_cosh, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function tanh(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_tanh, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function coth(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_coth, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
function atan(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_atan, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
log_sinpi(x::ComplexFieldElem)
Return $\log\sin(\pi x)$, constructed without branch cuts off the real line.
"""
function log_sinpi(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_log_sin_pi, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
gamma(x::ComplexFieldElem)
Return the Gamma function evaluated at $x$.
"""
function gamma(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_gamma, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
rgamma(x::ComplexFieldElem)
Return the reciprocal of the Gamma function evaluated at $x$.
"""
function rgamma(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_rgamma, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
lgamma(x::ComplexFieldElem)
Return the logarithm of the Gamma function evaluated at $x$.
"""
function lgamma(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_lgamma, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
digamma(x::ComplexFieldElem)
Return the logarithmic derivative of the gamma function evaluated at $x$,
i.e. $\psi(x)$.
"""
function digamma(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_digamma, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
zeta(x::ComplexFieldElem)
Return the Riemann zeta function evaluated at $x$.
"""
function zeta(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_zeta, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
barnes_g(x::ComplexFieldElem)
Return the Barnes $G$-function, evaluated at $x$.
"""
function barnes_g(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_barnes_g, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
log_barnes_g(x::ComplexFieldElem)
Return the logarithm of the Barnes $G$-function, evaluated at $x$.
"""
function log_barnes_g(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_log_barnes_g, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
agm(x::ComplexFieldElem)
Return the arithmetic-geometric mean of $1$ and $x$.
"""
function agm(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_agm1, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
erf(x::ComplexFieldElem)
Return the error function evaluated at $x$.
"""
function erf(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_hypgeom_erf, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
erfi(x::ComplexFieldElem)
Return the imaginary error function evaluated at $x$.
"""
function erfi(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_hypgeom_erfi, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
erfc(x::ComplexFieldElem)
Return the complementary error function evaluated at $x$.
"""
function erfc(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_hypgeom_erfc, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
exp_integral_ei(x::ComplexFieldElem)
Return the exponential integral evaluated at $x$.
"""
function exp_integral_ei(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_hypgeom_ei, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end
@doc raw"""
sin_integral(x::ComplexFieldElem)
Return the sine integral evaluated at $x$.
"""
function sin_integral(x::ComplexFieldElem, prec::Int = precision(Balls))
z = ComplexFieldElem()
ccall((:acb_hypgeom_si, libarb), Nothing, (Ref{ComplexFieldElem}, Ref{ComplexFieldElem}, Int), z, x, prec)
return z
end