complex.jl

# This file is a part of Julia. License is MIT: https://julialang.org/license

"""
    Complex{T<:Real} <: Number

Complex number type with real and imaginary part of type `T`.

`Complex32`, `Complex64` and `Complex128` are aliases for
`Complex{Float16}`, `Complex{Float32}` and `Complex{Float64}` respectively.
"""
struct Complex{T<:Real} <: Number
    re::T
    im::T
end
Complex(x::Real, y::Real) = Complex(promote(x,y)...)
Complex(x::Real) = Complex(x, zero(x))

"""
    im

The imaginary unit.

# Examples
```jldoctest
julia> im * im
-1 + 0im
```
"""
const im = Complex(false, true)

const Complex128 = Complex{Float64}
const Complex64  = Complex{Float32}
const Complex32  = Complex{Float16}

convert(::Type{Complex{T}}, x::Real) where {T<:Real} = Complex{T}(x,0)
convert(::Type{Complex{T}}, z::Complex) where {T<:Real} = Complex{T}(real(z),imag(z))
convert(::Type{T}, z::Complex) where {T<:Real} =
    isreal(z) ? convert(T,real(z)) : throw(InexactError(:convert, T, z))

convert(::Type{Complex}, z::Complex) = z
convert(::Type{Complex}, x::Real) = Complex(x)

promote_rule(::Type{Complex{T}}, ::Type{S}) where {T<:Real,S<:Real} =
    Complex{promote_type(T,S)}
promote_rule(::Type{Complex{T}}, ::Type{Complex{S}}) where {T<:Real,S<:Real} =
    Complex{promote_type(T,S)}

widen(::Type{Complex{T}}) where {T} = Complex{widen(T)}

float(::Type{Complex{T}}) where {T<:AbstractFloat} = Complex{T}
float(::Type{Complex{T}}) where {T} = Complex{float(T)}

"""
    real(z)

Return the real part of the complex number `z`.

# Examples
```jldoctest
julia> real(1 + 3im)
1
```
"""
real(z::Complex) = z.re

"""
    imag(z)

Return the imaginary part of the complex number `z`.

# Examples
```jldoctest
julia> imag(1 + 3im)
3
```
"""
imag(z::Complex) = z.im
real(x::Real) = x
imag(x::Real) = zero(x)

"""
    reim(z)

Return both the real and imaginary parts of the complex number `z`.

# Examples
```jldoctest
julia> reim(1 + 3im)
(1, 3)
```
"""
reim(z) = (real(z), imag(z))

"""
    real(T::Type)

Returns the type that represents the real part of a value of type `T`.
e.g: for `T == Complex{R}`, returns `R`.
Equivalent to `typeof(real(zero(T)))`.

```jldoctest
julia> real(Complex{Int})
Int64

julia> real(Float64)
Float64
```
"""
real(T::Type) = typeof(real(zero(T)))
real(::Type{T}) where {T<:Real} = T
real(::Type{Complex{T}}) where {T<:Real} = T

"""
    isreal(x) -> Bool

Test whether `x` or all its elements are numerically equal to some real number.

# Examples
```jldoctest
julia> isreal(5.)
true

julia> isreal([4.; complex(0,1)])
false
```
"""
isreal(x::Real) = true
isreal(z::Complex) = iszero(imag(z))
isinteger(z::Complex) = isreal(z) & isinteger(real(z))
isfinite(z::Complex) = isfinite(real(z)) & isfinite(imag(z))
isnan(z::Complex) = isnan(real(z)) | isnan(imag(z))
isinf(z::Complex) = isinf(real(z)) | isinf(imag(z))
iszero(z::Complex) = iszero(real(z)) & iszero(imag(z))
isone(z::Complex) = isone(real(z)) & iszero(imag(z))

"""
    complex(r, [i])

Convert real numbers or arrays to complex. `i` defaults to zero.

# Examples
```jldoctest
julia> complex(7)
7 + 0im

julia> complex([1, 2, 3])
3-element Array{Complex{Int64},1}:
 1 + 0im
 2 + 0im
 3 + 0im
```
"""
complex(z::Complex) = z
complex(x::Real) = Complex(x)
complex(x::Real, y::Real) = Complex(x, y)

"""
    complex(T::Type)

Returns an appropriate type which can represent a value of type `T` as a complex number.
Equivalent to `typeof(complex(zero(T)))`.

# Examples
```jldoctest
julia> complex(Complex{Int})
Complex{Int64}

julia> complex(Int)
Complex{Int64}
```
"""
complex(::Type{T}) where {T<:Real} = Complex{T}
complex(::Type{Complex{T}}) where {T<:Real} = Complex{T}

flipsign(x::Complex, y::Real) = ifelse(signbit(y), -x, x)

function show(io::IO, z::Complex)
    r, i = reim(z)
    compact = get(io, :compact, false)
    show(io, r)
    if signbit(i) && !isnan(i)
        i = -i
        print(io, compact ? "-" : " - ")
    else
        print(io, compact ? "+" : " + ")
    end
    show(io, i)
    if !(isa(i,Integer) && !isa(i,Bool) || isa(i,AbstractFloat) && isfinite(i))
        print(io, "*")
    end
    print(io, "im")
end
show(io::IO, z::Complex{Bool}) =
    print(io, z == im ? "im" : "Complex($(z.re),$(z.im))")

function show_unquoted(io::IO, z::Complex, ::Int, prec::Int)
    if operator_precedence(:+) <= prec
        print(io, "(")
        show(io, z)
        print(io, ")")
    else
        show(io, z)
    end
end

function read(s::IO, ::Type{Complex{T}}) where T<:Real
    r = read(s,T)
    i = read(s,T)
    Complex{T}(r,i)
end
function write(s::IO, z::Complex)
    write(s,real(z),imag(z))
end

## byte order swaps: real and imaginary part are swapped individually
bswap(z::Complex) = Complex(bswap(real(z)), bswap(imag(z)))

## equality and hashing of complex numbers ##

==(z::Complex, w::Complex) = (real(z) == real(w)) & (imag(z) == imag(w))
==(z::Complex, x::Real) = isreal(z) && real(z) == x
==(x::Real, z::Complex) = isreal(z) && real(z) == x

isequal(z::Complex, w::Complex) = isequal(real(z),real(w)) & isequal(imag(z),imag(w))

in(x::Complex, r::AbstractRange{<:Real}) = isreal(x) && real(x) in r

if UInt === UInt64
    const h_imag = 0x32a7a07f3e7cd1f9
else
    const h_imag = 0x3e7cd1f9
end
const hash_0_imag = hash(0, h_imag)

function hash(z::Complex, h::UInt)
    # TODO: with default argument specialization, this would be better:
    # hash(real(z), h ⊻ hash(imag(z), h ⊻ h_imag) ⊻ hash(0, h ⊻ h_imag))
    hash(real(z), h ⊻ hash(imag(z), h_imag) ⊻ hash_0_imag)
end

## generic functions of complex numbers ##

"""
    conj(z)

Compute the complex conjugate of a complex number `z`.

# Examples
```jldoctest
julia> conj(1 + 3im)
1 - 3im
```
"""
conj(z::Complex) = Complex(real(z),-imag(z))
abs(z::Complex)  = hypot(real(z), imag(z))
abs2(z::Complex) = real(z)*real(z) + imag(z)*imag(z)
inv(z::Complex)  = conj(z)/abs2(z)
inv(z::Complex{<:Integer}) = inv(float(z))

-(z::Complex) = Complex(-real(z), -imag(z))
+(z::Complex, w::Complex) = Complex(real(z) + real(w), imag(z) + imag(w))
-(z::Complex, w::Complex) = Complex(real(z) - real(w), imag(z) - imag(w))
*(z::Complex, w::Complex) = Complex(real(z) * real(w) - imag(z) * imag(w),
                                    real(z) * imag(w) + imag(z) * real(w))

muladd(z::Complex, w::Complex, x::Complex) =
    Complex(muladd(real(z), real(w), real(x)) - imag(z)*imag(w), # TODO: use mulsub given #15985
            muladd(real(z), imag(w), muladd(imag(z), real(w), imag(x))))

# handle Bool and Complex{Bool}
# avoid type signature ambiguity warnings
+(x::Bool, z::Complex{Bool}) = Complex(x + real(z), imag(z))
+(z::Complex{Bool}, x::Bool) = Complex(real(z) + x, imag(z))
-(x::Bool, z::Complex{Bool}) = Complex(x - real(z), - imag(z))
-(z::Complex{Bool}, x::Bool) = Complex(real(z) - x, imag(z))
*(x::Bool, z::Complex{Bool}) = Complex(x * real(z), x * imag(z))
*(z::Complex{Bool}, x::Bool) = Complex(real(z) * x, imag(z) * x)

+(x::Bool, z::Complex) = Complex(x + real(z), imag(z))
+(z::Complex, x::Bool) = Complex(real(z) + x, imag(z))
-(x::Bool, z::Complex) = Complex(x - real(z), - imag(z))
-(z::Complex, x::Bool) = Complex(real(z) - x, imag(z))
*(x::Bool, z::Complex) = Complex(x * real(z), x * imag(z))
*(z::Complex, x::Bool) = Complex(real(z) * x, imag(z) * x)

+(x::Real, z::Complex{Bool}) = Complex(x + real(z), imag(z))
+(z::Complex{Bool}, x::Real) = Complex(real(z) + x, imag(z))
function -(x::Real, z::Complex{Bool})
    # we don't want the default type for -(Bool)
    re = x-real(z)
    Complex(re, - oftype(re, imag(z)))
end
-(z::Complex{Bool}, x::Real) = Complex(real(z) - x, imag(z))
*(x::Real, z::Complex{Bool}) = Complex(x * real(z), x * imag(z))
*(z::Complex{Bool}, x::Real) = Complex(real(z) * x, imag(z) * x)

# adding or multiplying real & complex is common
+(x::Real, z::Complex) = Complex(x + real(z), imag(z))
+(z::Complex, x::Real) = Complex(x + real(z), imag(z))
function -(x::Real, z::Complex)
    # we don't want the default type for -(Bool)
    re = x - real(z)
    Complex(re, - oftype(re, imag(z)))
end
-(z::Complex, x::Real) = Complex(real(z) - x, imag(z))
*(x::Real, z::Complex) = Complex(x * real(z), x * imag(z))
*(z::Complex, x::Real) = Complex(x * real(z), x * imag(z))

muladd(x::Real, z::Complex, y::Number) = muladd(z, x, y)
muladd(z::Complex, x::Real, y::Real) = Complex(muladd(real(z),x,y), imag(z)*x)
muladd(z::Complex, x::Real, w::Complex) =
    Complex(muladd(real(z),x,real(w)), muladd(imag(z),x,imag(w)))
muladd(x::Real, y::Real, z::Complex) = Complex(muladd(x,y,real(z)), imag(z))
muladd(z::Complex, w::Complex, x::Real) =
    Complex(muladd(real(z), real(w), x) - imag(z)*imag(w), # TODO: use mulsub given #15985
            muladd(real(z), imag(w), imag(z) * real(w)))

/(a::R, z::S) where {R<:Real,S<:Complex} = (T = promote_type(R,S); a*inv(T(z)))
/(z::Complex, x::Real) = Complex(real(z)/x, imag(z)/x)

function /(a::Complex{T}, b::Complex{T}) where T<:Real
    are = real(a); aim = imag(a); bre = real(b); bim = imag(b)
    if abs(bre) <= abs(bim)
        if isinf(bre) && isinf(bim)
            r = sign(bre)/sign(bim)
        else
            r = bre / bim
        end
        den = bim + r*bre
        Complex((are*r + aim)/den, (aim*r - are)/den)
    else
        if isinf(bre) && isinf(bim)
            r = sign(bim)/sign(bre)
        else
            r = bim / bre
        end
        den = bre + r*bim
        Complex((are + aim*r)/den, (aim - are*r)/den)
    end
end

inv(z::Complex{<:Union{Float16,Float32}}) =
    oftype(z, conj(widen(z))/abs2(widen(z)))

/(z::Complex{T}, w::Complex{T}) where {T<:Union{Float16,Float32}} =
    oftype(z, widen(z)*inv(widen(w)))

# robust complex division for double precision
# the first step is to scale variables if appropriate ,then do calculations
# in a way that avoids over/underflow (subfuncs 1 and 2), then undo the scaling.
# scaling variable s and other techniques
# based on arxiv.1210.4539
#             a + i*b
#  p + i*q = ---------
#             c + i*d
function /(z::Complex128, w::Complex128)
    a, b = reim(z); c, d = reim(w)
    half = 0.5
    two = 2.0
    ab = max(abs(a), abs(b))
    cd = max(abs(c), abs(d))
    ov = realmax(a)
    un = realmin(a)
    ϵ = eps(Float64)
    bs = two/(ϵ*ϵ)
    s = 1.0
    ab >= half*ov  && (a=half*a; b=half*b; s=two*s ) # scale down a,b
    cd >= half*ov  && (c=half*c; d=half*d; s=s*half) # scale down c,d
    ab <= un*two/ϵ && (a=a*bs; b=b*bs; s=s/bs      ) # scale up a,b
    cd <= un*two/ϵ && (c=c*bs; d=d*bs; s=s*bs      ) # scale up c,d
    abs(d)<=abs(c) ? ((p,q)=robust_cdiv1(a,b,c,d)  ) : ((p,q)=robust_cdiv1(b,a,d,c); q=-q)
    return Complex128(p*s,q*s) # undo scaling
end
function robust_cdiv1(a::Float64, b::Float64, c::Float64, d::Float64)
    r = d/c
    t = 1.0/(c+d*r)
    p = robust_cdiv2(a,b,c,d,r,t)
    q = robust_cdiv2(b,-a,c,d,r,t)
    return p,q
end
function robust_cdiv2(a::Float64, b::Float64, c::Float64, d::Float64, r::Float64, t::Float64)
    if r != 0
        br = b*r
        return (br != 0 ? (a+br)*t : a*t + (b*t)*r)
    else
        return (a + d*(b/c)) * t
    end
end

function inv(w::Complex128)
    c, d = reim(w)
    half = 0.5
    two = 2.0
    cd = max(abs(c), abs(d))
    ov = realmax(c)
    un = realmin(c)
    ϵ = eps(Float64)
    bs = two/(ϵ*ϵ)
    s = 1.0
    cd >= half*ov  && (c=half*c; d=half*d; s=s*half) # scale down c,d
    cd <= un*two/ϵ && (c=c*bs; d=d*bs; s=s*bs      ) # scale up c,d
    if abs(d)<=abs(c)
        r = d/c
        t = 1.0/(c+d*r)
        p = t
        q = -r * t
    else
        c, d = d, c
        r = d/c
        t = 1.0/(c+d*r)
        p = r * t
        q = -t
    end
    return Complex128(p*s,q*s) # undo scaling
end

function ssqs(x::T, y::T) where T<:AbstractFloat
    k::Int = 0
    ρ = x*x + y*y
    if !isfinite(ρ) && (isinf(x) || isinf(y))
        ρ = convert(T, Inf)
    elseif isinf(ρ) || (ρ==0 && (x!=0 || y!=0)) || ρ<nextfloat(zero(T))/(2*eps(T)^2)
        m::T = max(abs(x), abs(y))
        k = m==0 ? m : exponent(m)
        xk, yk = ldexp(x,-k), ldexp(y,-k)
        ρ = xk*xk + yk*yk
    end
    ρ, k
end

function sqrt(z::Complex{<:AbstractFloat})
    x, y = reim(z)
    if x==y==0
        return Complex(zero(x),y)
    end
    ρ, k::Int = ssqs(x, y)
    if isfinite(x) ρ=ldexp(abs(x),-k)+sqrt(ρ) end
    if isodd(k)
        k = div(k-1,2)
    else
        k = div(k,2)-1
        ρ += ρ
    end
    ρ = ldexp(sqrt(ρ),k) #sqrt((abs(z)+abs(x))/2) without over/underflow
    ξ = ρ
    η = y
    if ρ != 0
        if isfinite(η) η=(η/ρ)/2 end
        if x<0
            ξ = abs(η)
            η = copysign(ρ,y)
        end
    end
    Complex(ξ,η)
end
sqrt(z::Complex) = sqrt(float(z))

# function sqrt(z::Complex)
#     rz = float(real(z))
#     iz = float(imag(z))
#     r = sqrt((hypot(rz,iz)+abs(rz))/2)
#     if r == 0
#         return Complex(zero(iz), iz)
#     end
#     if rz >= 0
#         return Complex(r, iz/r/2)
#     end
#     return Complex(abs(iz)/r/2, copysign(r,iz))
# end

# compute exp(im*theta)
function cis(theta::Real)
    s, c = sincos(theta)
    Complex(c, s)
end

"""
    cis(z)

Return ``\\exp(iz)``.

# Examples
```jldoctest
julia> cis(π) ≈ -1
true
```
"""
function cis(z::Complex)
    v = exp(-imag(z))
    s, c = sincos(real(z))
    Complex(v * c, v * s)
end

"""
    angle(z)

Compute the phase angle in radians of a complex number `z`.

# Examples
```jldoctest
julia> rad2deg(angle(1 + im))
45.0

julia> rad2deg(angle(1 - im))
-45.0

julia> rad2deg(angle(-1 - im))
-135.0
```
"""
angle(z::Complex) = atan2(imag(z), real(z))

function log(z::Complex{T}) where T<:AbstractFloat
    T1::T  = 1.25
    T2::T  = 3
    ln2::T = log(convert(T,2))  #0.6931471805599453
    x, y = reim(z)
    ρ, k = ssqs(x,y)
    ax = abs(x)
    ay = abs(y)
    if ax < ay
        θ, β = ax, ay
    else
        θ, β = ay, ax
    end
    if k==0 && (0.5 < β*β) && (β <= T1 || ρ < T2)
        ρρ = log1p((β-1)*(β+1)+θ*θ)/2
    else
        ρρ = log(ρ)/2 + k*ln2
    end
    Complex(ρρ, angle(z))
end
log(z::Complex) = log(float(z))

# function log(z::Complex)
#     ar = abs(real(z))
#     ai = abs(imag(z))
#     if ar < ai
#         r = ar/ai
#         re = log(ai) + log1p(r*r)/2
#     else
#         if ar == 0
#             re = isnan(ai) ? ai : -inv(ar)
#         elseif isinf(ai)
#             re = oftype(ar,Inf)
#         else
#             r = ai/ar
#             re = log(ar) + log1p(r*r)/2
#         end
#     end
#     Complex(re, angle(z))
# end

function log10(z::Complex)
    a = log(z)
    a/log(oftype(real(a),10))
end
function log2(z::Complex)
    a = log(z)
    a/log(oftype(real(a),2))
end

function exp(z::Complex)
    zr, zi = reim(z)
    if isnan(zr)
        Complex(zr, zi==0 ? zi : zr)
    elseif !isfinite(zi)
        if zr == Inf
            Complex(-zr, oftype(zr,NaN))
        elseif zr == -Inf
            Complex(-zero(zr), copysign(zero(zi), zi))
        else
            Complex(oftype(zr,NaN), oftype(zi,NaN))
        end
    else
        er = exp(zr)
        if iszero(zi)
            Complex(er, zi)
        else
            s, c = sincos(zi)
            Complex(er * c, er * s)
        end
    end
end

function expm1(z::Complex{T}) where T<:Real
    Tf = float(T)
    zr,zi = reim(z)
    if isnan(zr)
        Complex(zr, zi==0 ? zi : zr)
    elseif !isfinite(zi)
        if zr == Inf
            Complex(-zr, oftype(zr,NaN))
        elseif zr == -Inf
            Complex(-one(zr), copysign(zero(zi), zi))
        else
            Complex(oftype(zr,NaN), oftype(zi,NaN))
        end
    else
        erm1 = expm1(zr)
        if zi == 0
            Complex(erm1, zi)
        else
            er = erm1+one(erm1)
            if isfinite(er)
                wr = erm1 - 2 * er * (sin(convert(Tf, 0.5) * zi))^2
                return Complex(wr, er * sin(zi))
            else
                s, c = sincos(zi)
                return Complex(er * c, er * s)
            end
        end
    end
end

function log1p(z::Complex{T}) where T
    zr,zi = reim(z)
    if isfinite(zr)
        isinf(zi) && return log(z)
        # This is based on a well-known trick for log1p of real z,
        # allegedly due to Kahan, only modified to handle real(u) <= 0
        # differently to avoid inaccuracy near z==-2 and for correct branch cut
        u = one(float(T)) + z
        u == 1 ? convert(typeof(u), z) : real(u) <= 0 ? log(u) : log(u)*z/(u-1)
    elseif isnan(zr)
        Complex(zr, zr)
    elseif isfinite(zi)
        Complex(T(Inf), copysign(zr > 0 ? zero(T) : convert(T, pi), zi))
    else
        Complex(T(Inf), T(NaN))
    end
end

function exp2(z::Complex{T}) where T<:AbstractFloat
    er = exp2(real(z))
    theta = imag(z) * log(convert(T, 2))
    s, c = sincos(theta)
    Complex(er * c, er * s)
end
exp2(z::Complex) = exp2(float(z))

function exp10(z::Complex{T}) where T<:AbstractFloat
    er = exp10(real(z))
    theta = imag(z) * log(convert(T, 10))
    s, c = sincos(theta)
    Complex(er * c, er * s)
end
exp10(z::Complex) = exp10(float(z))

# _cpow helper function to avoid method ambiguity with ^(::Complex,::Real)
function _cpow(z::Complex{T}, p::Union{T,Complex{T}})::Complex{T} where T<:AbstractFloat
    if p == 2 #square
        zr, zi = reim(z)
        x = (zr-zi)*(zr+zi)
        y = 2*zr*zi
        if isnan(x)
            if isinf(y)
                x = copysign(zero(T),zr)
            elseif isinf(zi)
                x = convert(T,-Inf)
            elseif isinf(zr)
                x = convert(T,Inf)
            end
        elseif isnan(y) && isinf(x)
            y = copysign(zero(T), y)
        end
        Complex(x,y)
    elseif z!=0
        if p!=0 && isinteger(p)
            rp = real(p)
            if rp < 0
                return power_by_squaring(inv(z), convert(Integer, -rp))
            else
                return power_by_squaring(z, convert(Integer, rp))
            end
        end
        exp(p*log(z))
    elseif p!=0 #0^p
        zero(z) #CHECK SIGNS
    else #0^0
        zer = copysign(zero(T),real(p))*copysign(zero(T),imag(z))
        Complex(one(T), zer)
    end
end
function _cpow(z::Complex{T}, p::Union{T,Complex{T}}) where T<:Real
    if isinteger(p)
        rp = real(p)
        if rp < 0
            return power_by_squaring(inv(float(z)), convert(Integer, -rp))
        else
            return power_by_squaring(float(z), convert(Integer, rp))
        end
    end
    pr, pim = reim(p)
    zr, zi = reim(z)
    r = abs(z)
    rp = r^pr
    theta = atan2(zi, zr)
    ntheta = pr*theta
    if pim != 0 && r != 0
        rp = rp*exp(-pim*theta)
        ntheta = ntheta + pim*log(r)
    end
    sinntheta, cosntheta = sincos(ntheta)
    re, im = rp*cosntheta, rp*sinntheta
    if isinf(rp)
        if isnan(re)
            re = copysign(zero(re), cosntheta)
        end
        if isnan(im)
            im = copysign(zero(im), sinntheta)
        end
    end

    # apply some corrections to force known zeros
    if pim == 0
        if isinteger(pr)
            if zi == 0
                im = copysign(zero(im), im)
            elseif zr == 0
                if isinteger(0.5*pr) # pr is even
                    im = copysign(zero(im), im)
                else
                    re = copysign(zero(re), re)
                end
            end
        else
            dr = pr*2
            if isinteger(dr) && zi == 0
                if zr < 0
                    re = copysign(zero(re), re)
                else
                    im = copysign(zero(im), im)
                end
            end
        end
    end

    Complex(re, im)
end
^(z::Complex{T}, p::Complex{T}) where T<:Real = _cpow(z, p)
^(z::Complex{T}, p::T) where T<:Real = _cpow(z, p)

^(z::Complex, n::Bool) = n ? z : one(z)
^(z::Complex, n::Integer) = z^Complex(n)

^(z::Complex{<:AbstractFloat}, n::Bool) = n ? z : one(z)  # to resolve ambiguity
^(z::Complex{<:Integer}, n::Bool) = n ? z : one(z)        # to resolve ambiguity

^(z::Complex{<:AbstractFloat}, n::Integer) =
    n>=0 ? power_by_squaring(z,n) : power_by_squaring(inv(z),-n)
^(z::Complex{<:Integer}, n::Integer) = power_by_squaring(z,n) # DomainError for n<0

function ^(z::Complex{T}, p::S) where {T<:Real,S<:Real}
    P = promote_type(T,S)
    return Complex{P}(z) ^ P(p)
end

function sin(z::Complex{T}) where T
    F = float(T)
    zr, zi = reim(z)
    if zr == 0
        Complex(F(zr), sinh(zi))
    elseif !isfinite(zr)
        if zi == 0 || isinf(zi)
            Complex(F(NaN), F(zi))
        else
            Complex(F(NaN), F(NaN))
        end
    else
        s, c = sincos(zr)
        Complex(s * cosh(zi), c * sinh(zi))
    end
end


function cos(z::Complex{T}) where T
    F = float(T)
    zr, zi = reim(z)
    if zr == 0
        Complex(cosh(zi), isnan(zi) ? F(zr) : -flipsign(F(zr),zi))
    elseif !isfinite(zr)
        if zi == 0
            Complex(F(NaN), isnan(zr) ? zero(F) : -flipsign(F(zi),zr))
        elseif isinf(zi)
            Complex(F(Inf), F(NaN))
        else
            Complex(F(NaN), F(NaN))
        end
    else
        s, c = sincos(zr)
        Complex(c * cosh(zi), -s * sinh(zi))
    end
end


function tan(z::Complex)
    zr, zi = reim(z)
    w = tanh(Complex(-zi, zr))
    Complex(imag(w), -real(w))
end

function asin(z::Complex)
    zr, zi = reim(z)
    if isinf(zr) && isinf(zi)
        return Complex(copysign(oftype(zr,pi)/4, zr),zi)
    elseif isnan(zi) && isinf(zr)
        return Complex(zi, oftype(zr, Inf))
    end
    ξ = zr == 0       ? zr :
        !isfinite(zr) ? oftype(zr,pi)/2 * sign(zr) :
        atan2(zr, real(sqrt(1-z)*sqrt(1+z)))
    η = asinh(copysign(imag(sqrt(conj(1-z))*sqrt(1+z)), imag(z)))
    Complex(ξ,η)
end

function acos(z::Complex{<:AbstractFloat})
    zr, zi = reim(z)
    if isnan(zr)
        if isinf(zi) return Complex(zr, -zi)
        else         return Complex(zr, zr) end
    elseif isnan(zi)
        if isinf(zr) return Complex(zi, abs(zr))
        elseif zr==0 return Complex(oftype(zr,pi)/2, zi)
        else         return Complex(zi, zi) end
    elseif zr==zi==0
        return Complex(oftype(zr,pi)/2, -zi)
    elseif zr==Inf && zi===0.0
        return Complex(zi, -zr)
    elseif zr==-Inf && zi===-0.0
        return Complex(oftype(zi,pi), -zr)
    end
    ξ = 2*atan2(real(sqrt(1-z)), real(sqrt(1+z)))
    η = asinh(imag(sqrt(conj(1+z))*sqrt(1-z)))
    if isinf(zr) && isinf(zi) ξ -= oftype(η,pi)/4 * sign(zr) end
    Complex(ξ,η)
end
acos(z::Complex) = acos(float(z))

function atan(z::Complex)
    w = atanh(Complex(-imag(z),real(z)))
    Complex(imag(w),-real(w))
end

function sinh(z::Complex)
    zr, zi = reim(z)
    w = sin(Complex(zi, zr))
    Complex(imag(w),real(w))
end

function cosh(z::Complex)
    zr, zi = reim(z)
    cos(Complex(zi,-zr))
end

function tanh(z::Complex{T}) where T<:AbstractFloat
    Ω = prevfloat(typemax(T))
    ξ, η = reim(z)
    if isnan(ξ) && η==0 return Complex(ξ, η) end
    if 4*abs(ξ) > asinh(Ω) #Overflow?
        Complex(copysign(one(T),ξ),
                copysign(zero(T),η*(isfinite(η) ? sin(2*abs(η)) : one(η))))
    else
        t = tan(η)
        β = 1+t*t #sec(η)^2
        s = sinh(ξ)
        ρ = sqrt(1 + s*s) #cosh(ξ)
        if isinf(t)
            Complex(ρ/s,1/t)
        else
            Complex(β*ρ*s,t)/(1+β*s*s)
        end
    end
end
tanh(z::Complex) = tanh(float(z))

function asinh(z::Complex)
    w = asin(Complex(-imag(z),real(z)))
    Complex(imag(w),-real(w))
end

function acosh(z::Complex)
    zr, zi = reim(z)
    if isnan(zr) || isnan(zi)
        if isinf(zr) || isinf(zi)
            return Complex(oftype(zr, Inf), oftype(zi, NaN))
        else
            return Complex(oftype(zr, NaN), oftype(zi, NaN))
        end
    elseif zr==-Inf && zi===-0.0 #Edge case is wrong - WHY?
        return Complex(oftype(zr,Inf), oftype(zi, -pi))
    end
    ξ = asinh(real(sqrt(conj(z-1))*sqrt(z+1)))
    η = 2atan2(imag(sqrt(z-1)),real(sqrt(z+1)))
    if isinf(zr) && isinf(zi)
        η -= oftype(η,pi)/4 * sign(zi) * sign(zr)
    end
    Complex(ξ, η)
end

function atanh(z::Complex{T}) where T<:AbstractFloat
    Ω = prevfloat(typemax(T))
    θ = sqrt(Ω)/4
    ρ = 1/θ
    x, y = reim(z)
    ax = abs(x)
    ay = abs(y)
    if ax > θ || ay > θ #Prevent overflow
        if isnan(y)
            if isinf(x)
                return Complex(copysign(zero(x),x), y)
            else
                return Complex(real(1/z), y)
            end
        end
        if isinf(y)
            return Complex(copysign(zero(x),x), copysign(oftype(y,pi)/2, y))
        end
        return Complex(real(1/z), copysign(oftype(y,pi)/2, y))
    elseif ax==1
        if y == 0
            ξ = copysign(oftype(x,Inf),x)
            η = zero(y)
        else
            ym = ay+ρ
            ξ = log(sqrt(sqrt(4+y*y))/sqrt(ym))
            η = copysign(oftype(y,pi)/2 + atan(ym/2), y)/2
        end
    else #Normal case
        ysq = (ay+ρ)^2
        if x == 0
            ξ = x
        else
            ξ = log1p(4x/((1-x)^2 + ysq))/4
        end
        η = angle(Complex((1-x)*(1+x)-ysq, 2y))/2
    end
    Complex(ξ, η)
end
atanh(z::Complex) = atanh(float(z))

function lexcmp(a::Complex, b::Complex)
    c = cmp(real(a), real(b))
    c == 0 || return c
    cmp(imag(a), imag(b))
end

#Rounding complex numbers
#Requires two different RoundingModes for the real and imaginary components
"""
    round(z, RoundingModeReal, RoundingModeImaginary)

Returns the nearest integral value of the same type as the complex-valued `z` to `z`,
breaking ties using the specified [`RoundingMode`](@ref)s. The first
[`RoundingMode`](@ref) is used for rounding the real components while the
second is used for rounding the imaginary components.

# Example
```jldoctest
julia> round(3.14 + 4.5im)
3.0 + 4.0im
```
"""
function round(z::Complex{<:AbstractFloat}, ::RoundingMode{MR}, ::RoundingMode{MI}) where {MR,MI}
    Complex(round(real(z), RoundingMode{MR}()),
            round(imag(z), RoundingMode{MI}()))
end
round(z::Complex) = Complex(round(real(z)), round(imag(z)))

function round(z::Complex, digits::Integer, base::Integer=10)
    Complex(round(real(z), digits, base),
            round(imag(z), digits, base))
end

float(z::Complex{<:AbstractFloat}) = z
float(z::Complex) = Complex(float(real(z)), float(imag(z)))

big(::Type{Complex{T}}) where {T<:Real} = Complex{big(T)}
big(z::Complex{T}) where {T<:Real} = Complex{big(T)}(z)

## Array operations on complex numbers ##

complex(A::AbstractArray{<:Complex}) = A

function complex(A::AbstractArray{T}) where T
    if !isconcrete(T)
        error("`complex` not defined on abstractly-typed arrays; please convert to a more specific type")
    end
    convert(AbstractArray{typeof(complex(zero(T)))}, A)
end

## promotion to complex ##

_default_type(T::Type{Complex}) = Complex{Int}