complex.jl

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

@test reim(2 + 3im) == (2, 3)

for T in (Int64, Float64)
    @test real(T) == T
    @test real(Complex{T}) == T
    @test complex(T) == Complex{T}
    @test complex(Complex{T}) == Complex{T}
end

#showcompact
@test sprint(showcompact, complex(1, 0)) == "1+0im"
@test sprint(show, complex(true, true)) == "Complex(true,true)"

@testset "arithmetic" begin
    @testset for T in (Float16, Float32, Float64, BigFloat)
        t = true
        f = false

        @testset "add and subtract" begin
            @test isequal(T(+0.0) + im, Complex(T(+0.0), T(+1.0)))
            @test isequal(T(-0.0) + im, Complex(T(-0.0), T(+1.0)))
            @test isequal(T(+0.0) - im, Complex(T(+0.0), T(-1.0)))
            @test isequal(T(-0.0) - im, Complex(T(-0.0), T(-1.0)))
            @test isequal(T(+1.0) + im, Complex(T(+1.0), T(+1.0)))
            @test isequal(T(-1.0) + im, Complex(T(-1.0), T(+1.0)))
            @test isequal(T(+1.0) - im, Complex(T(+1.0), T(-1.0)))
            @test isequal(T(-1.0) - im, Complex(T(-1.0), T(-1.0)))
            @test isequal(im + T(+0.0), Complex(T(+0.0), T(+1.0)))
            @test isequal(im + T(-0.0), Complex(T(-0.0), T(+1.0)))
            @test isequal(im - T(+0.0), Complex(T(+0.0), T(+1.0)))
            @test isequal(im - T(-0.0), Complex(T(+0.0), T(+1.0)))
            @test isequal(im + T(+1.0), Complex(T(+1.0), T(+1.0)))
            @test isequal(im + T(-1.0), Complex(T(-1.0), T(+1.0)))
            @test isequal(im - T(+1.0), Complex(T(-1.0), T(+1.0)))
            @test isequal(im - T(-1.0), Complex(T(+1.0), T(+1.0)))
            @test isequal(T(f) + im, Complex(T(+0.0), T(+1.0)))
            @test isequal(T(t) + im, Complex(T(+1.0), T(+1.0)))
            @test isequal(T(f) - im, Complex(T(+0.0), T(-1.0)))
            @test isequal(T(t) - im, Complex(T(+1.0), T(-1.0)))
            @test isequal(im + T(f), Complex(T(+0.0), T(+1.0)))
            @test isequal(im + T(t), Complex(T(+1.0), T(+1.0)))
            @test isequal(im - T(f), Complex(T(+0.0), T(+1.0)))
            @test isequal(im - T(t), Complex(T(-1.0), T(+1.0)))
        end

        @testset "multiply" begin
            @test isequal(T(+0.0) * im, Complex(T(+0.0), T(+0.0)))
            @test isequal(T(-0.0) * im, Complex(T(-0.0), T(-0.0)))
            @test isequal(T(+1.0) * im, Complex(T(+0.0), T(+1.0)))
            @test isequal(T(-1.0) * im, Complex(T(-0.0), T(-1.0)))
            @test isequal(im * T(+0.0), Complex(T(+0.0), T(+0.0)))
            @test isequal(im * T(-0.0), Complex(T(-0.0), T(-0.0)))
            @test isequal(im * T(+1.0), Complex(T(+0.0), T(+1.0)))
            @test isequal(im * T(-1.0), Complex(T(-0.0), T(-1.0)))
        end

        @testset "divide" begin
            @test isequal(T(+0.0) / im, Complex(T(+0.0), T(-0.0)))
            @test isequal(T(-0.0) / im, Complex(T(-0.0), T(+0.0)))
            @test isequal(T(+1.0) / im, Complex(T(+0.0), T(-1.0)))
            @test isequal(T(-1.0) / im, Complex(T(-0.0), T(+1.0)))
        end
    end
    @test isequal(true + complex(true,false), complex(true,false) + complex(true,false))
    @test isequal(complex(true,false) + true, complex(true,false) + complex(true,false))
    @test isequal(true - complex(true,false), complex(true,false) - complex(true,false))
    @test isequal(complex(true,false) - true, complex(true,false) - complex(true,false))
    @test isequal(true * complex(true,false), complex(true,false) * complex(true,false))
    @test isequal(complex(true,false) * true, complex(true,false) * complex(true,false))
end

@testset "basic math functions" begin
    # We compare to BigFloat instead of hard-coding
    # values, assuming that BigFloat has an independent and independently
    # tested implementation.
    @testset for T in (Float32, Float64)
        x = Complex{T}(1//3 + 1//4*im)
        y = Complex{T}(1//2 + 1//5*im)
        yi = 4
        @testset "Random values" begin
            @test x^y ≈ big(x)^big(y)
            @test x^yi ≈ big(x)^yi
            @test x^true ≈ big(x)^true
            @test x^false ≈ big(x)^false
            @test x^1 ≈ big(x)^1
            @test abs(x) ≈ abs(big(x))
            @test abs2(x) ≈ abs2(big(x))
            @test acos(x) ≈ acos(big(x))
            @test acosh(1+x) ≈ acosh(1+big(x))
            @test angle(x) ≈ angle(big(x))
            @test asin(x) ≈ asin(big(x))
            @test asinh(x) ≈ asinh(big(x))
            @test atan(x) ≈ atan(big(x))
            @test atanh(x) ≈ atanh(big(x))
            @test cis(real(x)) ≈ cis(real(big(x)))
            @test cis(x) ≈ cis(big(x))
            @test cos(x) ≈ cos(big(x))
            @test cosh(x) ≈ cosh(big(x))
            @test exp(x) ≈ exp(big(x))
            @test exp10(x) ≈ exp10(big(x))
            @test exp2(x) ≈ exp2(big(x))
            @test expm1(x) ≈ expm1(big(x)) atol=eps(T)
            @test log(x) ≈ log(big(x))
            @test log10(x) ≈ log10(big(x))
            @test log1p(x) ≈ log1p(big(x))
            @test log2(x) ≈ log2(big(x))
            @test sin(x) ≈ sin(big(x))
            @test sinh(x) ≈ sinh(big(x))
            @test sqrt(x) ≈ sqrt(big(x))
            @test tan(x) ≈ tan(big(x))
            @test tanh(x) ≈ tanh(big(x))
        end
        @testset "Inverses" begin
            @test acos(cos(x)) ≈ x
            @test acosh(cosh(x)) ≈ x
            @test asin(sin(x)) ≈ x
            @test asinh(sinh(x)) ≈ x
            @test atan(tan(x)) ≈ x
            @test atanh(tanh(x)) ≈ x
            @test cos(acos(x)) ≈ x
            @test cosh(acosh(1+x)) ≈ 1+x
            @test exp(log(x)) ≈ x
            @test exp10(log10(x)) ≈ x
            @test exp2(log2(x)) ≈ x
            @test expm1(log1p(x)) ≈ x
            @test log(exp(x)) ≈ x
            @test log10(exp10(x)) ≈ x
            @test log1p(expm1(x)) ≈ x
            @test log2(exp2(x)) ≈ x
            @test sin(asin(x)) ≈ x
            @test sinh(asinh(x)) ≈ x
            @test sqrt(x)^2 ≈ x
            @test sqrt(x^2) ≈ x
            @test tan(atan(x)) ≈ x
            @test tanh(atanh(x)) ≈ x
        end
        @testset "Relations between functions" begin
            @test cosh(x) ≈ (exp(x)+exp(-x))/2
            @test cosh(x)^2-sinh(x)^2 ≈ 1
            @test sin(x)^2+cos(x)^2 ≈ 1
            @test sinh(x) ≈ (exp(x)-exp(-x))/2
            @test tan(x) ≈ sin(x)/cos(x)
            @test tanh(x) ≈ sinh(x)/cosh(x)
        end
    end
end

@testset "isinf" begin
    @test iszero(real(complex(0.0,1.0))) # isimag deprecated
    @test !iszero(real(complex(1.0,1.0))) # isimag deprecated
    @test isinf(complex(Inf,0))
    @test isinf(complex(-Inf,0))
    @test isinf(complex(0,Inf))
    @test isinf(complex(0,-Inf))
    @test !isinf(complex(0,0))
end

@testset "flipsign" begin
    @test isequal(complex( 0.0, 0.0 ), flipsign(complex( 0.0, 0.0 ), 1))
    @test isequal(complex( -0.0, -0.0 ), flipsign(complex( 0.0, 0.0 ), -1))
    @test isequal(complex( Inf, 0.0 ), flipsign(complex( Inf, 0.0 ), 1))
    @test isequal(complex( -Inf, -0.0 ), flipsign(complex( Inf, 0.0 ), -1))
    @test isequal(complex( 0.0, NaN ), flipsign(complex( 0.0, NaN ), 1.0))
    @test isequal(complex( -0.0, NaN ), flipsign(complex( 0.0, NaN ), -1.0))

    @test isequal(complex( 5.0, 4.0 ), flipsign(complex(-5.0, -4.0), -1))
    @test isequal(complex( 0.5, -0.5 ), flipsign(complex(-0.5, 0.5), -2))
end

@testset "sqrt" begin
    # tests special values from csqrt man page
    # as well as conj(sqrt(z)) = sqrt(conj(z))
    @test isequal(sqrt(complex( 0.0, 0.0)), complex( 0.0, 0.0))
    @test isequal(sqrt(complex( 0.0,-0.0)), complex( 0.0,-0.0))
    @test isequal(sqrt(complex( 0.0, Inf)), complex( Inf, Inf))
    @test isequal(sqrt(complex( 0.0,-Inf)), complex( Inf,-Inf))
    @test isequal(sqrt(complex( 0.0, NaN)), complex( NaN, NaN))
    @test isequal(sqrt(complex(-0.0, 0.0)), complex( 0.0, 0.0))
    @test isequal(sqrt(complex(-0.0,-0.0)), complex( 0.0,-0.0))

    @test isequal(sqrt(complex( 5.0, 0.0)), complex(sqrt(5.0), 0.0))
    @test isequal(sqrt(complex( 5.0,-0.0)), complex(sqrt(5.0),-0.0))
    @test isequal(sqrt(complex(-5.0, 0.0)), complex( 0.0, sqrt(5.0)))
    @test isequal(sqrt(complex(-5.0,-0.0)), complex( 0.0,-sqrt(5.0)))

    @test isequal(sqrt(complex( Inf, 0.0)), complex( Inf, 0.0))
    @test isequal(sqrt(complex( Inf,-0.0)), complex( Inf,-0.0))
    @test isequal(sqrt(complex( Inf, 5.0)), complex( Inf, 0.0))
    @test isequal(sqrt(complex( Inf,-5.0)), complex( Inf,-0.0))
    @test isequal(sqrt(complex( Inf, Inf)), complex( Inf, Inf))
    @test isequal(sqrt(complex( Inf,-Inf)), complex( Inf,-Inf))
    @test isequal(sqrt(complex( Inf, NaN)), complex( Inf, NaN))

    @test isequal(sqrt(complex(-Inf, 0.0)), complex( 0.0, Inf))
    @test isequal(sqrt(complex(-Inf,-0.0)), complex( 0.0,-Inf))
    @test isequal(sqrt(complex(-Inf, 5.0)), complex( 0.0, Inf))
    @test isequal(sqrt(complex(-Inf,-5.0)), complex( 0.0,-Inf))
    @test isequal(sqrt(complex(-Inf, Inf)), complex( Inf, Inf))
    @test isequal(sqrt(complex(-Inf,-Inf)), complex( Inf,-Inf))
    @test isequal(sqrt(complex(-Inf, NaN)), complex( NaN, Inf))

    @test isequal(sqrt(complex( NaN, 0.0)), complex( NaN, NaN))
    @test isequal(sqrt(complex( NaN, 0.0)), complex( NaN, NaN))
    @test isequal(sqrt(complex( NaN, Inf)), complex( Inf, Inf))
    @test isequal(sqrt(complex( NaN,-Inf)), complex( Inf,-Inf))
end

@testset "log(conj(z)) == conj(log(z))" begin
    @test isequal(log(complex( 0.0, 0.0)), complex(-Inf, 0.0))
    @test isequal(log(complex( 0.0,-0.0)), complex(-Inf,-0.0))
    @test isequal(log(complex( 0.0, 1.0)), complex( 0.0, pi/2))
    @test isequal(log(complex( 0.0,-1.0)), complex( 0.0,-pi/2))
    @test isequal(log(complex( 0.0, Inf)), complex( Inf, pi/2))
    @test isequal(log(complex( 0.0,-Inf)), complex( Inf,-pi/2))
    @test isequal(log(complex( 0.0, NaN)), complex( NaN, NaN))
    @test isequal(log(complex(-0.0, 0.0)), complex(-Inf, pi))
    @test isequal(log(complex(-0.0,-0.0)), complex(-Inf,-pi))

    @test isequal(log(complex( 5.0, 0.0)),complex(log(5.0), 0.0))
    @test isequal(log(complex( 5.0,-0.0)),complex(log(5.0),-0.0))

    @test isequal(log(complex( Inf, 5.0)), complex( Inf, 0.0))
    @test isequal(log(complex( Inf,-5.0)), complex( Inf,-0.0))
    @test isequal(log(complex( Inf, Inf)), complex( Inf, pi/4))
    @test isequal(log(complex( Inf,-Inf)), complex( Inf,-pi/4))
    @test isequal(log(complex( Inf, NaN)), complex( Inf, NaN))

    @test isequal(log(complex(-Inf, 5.0)), complex( Inf, pi))
    @test isequal(log(complex(-Inf,-5.0)), complex( Inf,-pi))
    @test isequal(log(complex(-Inf, Inf)), complex( Inf, 3*pi/4))
    @test isequal(log(complex(-Inf,-Inf)), complex( Inf,-3*pi/4))
    @test isequal(log(complex(-Inf, NaN)), complex( Inf, NaN))

    @test isequal(log(complex( NaN, 0.0)), complex( NaN, NaN))
    @test isequal(log(complex( NaN, Inf)), complex( Inf, NaN))
    @test isequal(log(complex( NaN,-Inf)), complex( Inf, NaN))
    @test isequal(log(complex( NaN, NaN)), complex( NaN, NaN))
end

@testset "exp(conj(z)) == conj(exp(z))" begin
    @test isequal(exp(complex( 0.0, 0.0)), complex(1.0, 0.0))
    @test isequal(exp(complex( 0.0,-0.0)), complex(1.0,-0.0))
    @test isequal(exp(complex( 0.0, Inf)), complex(NaN, NaN))
    @test isequal(exp(complex( 0.0,-Inf)), complex(NaN, NaN))
    @test isequal(exp(complex( 0.0, NaN)), complex(NaN, NaN))

    @test isequal(exp(complex(-0.0, 0.0)), complex(1.0, 0.0))
    @test isequal(exp(complex(-0.0,-0.0)), complex(1.0,-0.0))

    @test isequal(exp(complex( 5.0, Inf)), complex(NaN, NaN))

    @test isequal(exp(complex( Inf, 0.0)), complex(Inf, 0.0))
    @test isequal(exp(complex( Inf,-0.0)), complex(Inf,-0.0))
    @test isequal(exp(complex( Inf, 5.0)), complex(cos(5.0)*Inf,sin(5.0)* Inf))
    @test isequal(exp(complex( Inf,-5.0)), complex(cos(5.0)*Inf,sin(5.0)*-Inf))
    @test isequal(exp(complex( Inf, NaN)), complex(-Inf, NaN))
    @test isequal(exp(complex( Inf, Inf)), complex(-Inf, NaN))
    @test isequal(exp(complex( Inf,-Inf)), complex(-Inf, NaN))

    @test isequal(exp(complex(-Inf, 0.0)), complex(0.0, 0.0))
    @test isequal(exp(complex(-Inf,-0.0)), complex(0.0,-0.0))
    @test isequal(exp(complex(-Inf, 5.0)), complex(cos(5.0)*0.0,sin(5.0)* 0.0))
    @test isequal(exp(complex(-Inf,-5.0)), complex(cos(5.0)*0.0,sin(5.0)*-0.0))
    @test isequal(exp(complex(-Inf, Inf)), complex(-0.0, 0.0))
    @test isequal(exp(complex(-Inf,-Inf)), complex(-0.0,-0.0))
    @test isequal(exp(complex(-Inf, NaN)), complex(-0.0, 0.0))

    @test isequal(exp(complex( NaN, 0.0)), complex( NaN, 0.0))
    @test isequal(exp(complex( NaN,-0.0)), complex( NaN,-0.0))
    @test isequal(exp(complex( NaN, 5.0)), complex( NaN, NaN))
    @test isequal(exp(complex( NaN, NaN)), complex( NaN, NaN))
end

@testset "expm1(conj(z)) == conj(expm1(z))" begin
    @test isequal(expm1(complex( 0.0, 0.0)), complex(0.0, 0.0))
    @test isequal(expm1(complex( 0.0,-0.0)), complex(0.0,-0.0))
    @test isequal(expm1(complex( 0.0, Inf)), complex(NaN, NaN))
    @test isequal(expm1(complex( 0.0,-Inf)), complex(NaN, NaN))
    @test isequal(expm1(complex( 0.0, NaN)), complex(NaN, NaN))

    @test isequal(expm1(complex(-0.0, 0.0)), complex(-0.0, 0.0))
    @test isequal(expm1(complex(-0.0,-0.0)), complex(-0.0,-0.0))

    @test isequal(expm1(complex( 5.0, Inf)), complex(NaN, NaN))

    @test isequal(expm1(complex( Inf, 0.0)), complex(Inf, 0.0))
    @test isequal(expm1(complex( Inf,-0.0)), complex(Inf,-0.0))
    @test isequal(expm1(complex( Inf, 5.0)), complex(cos(5.0)*Inf,sin(5.0)* Inf))
    @test isequal(expm1(complex( Inf,-5.0)), complex(cos(5.0)*Inf,sin(5.0)*-Inf))
    @test isequal(expm1(complex( Inf, NaN)), complex(-Inf, NaN))
    @test isequal(expm1(complex( Inf, Inf)), complex(-Inf, NaN))
    @test isequal(expm1(complex( Inf,-Inf)), complex(-Inf, NaN))

    @test isequal(expm1(complex(-Inf, 0.0)), complex(-1.0, 0.0))
    @test isequal(expm1(complex(-Inf,-0.0)), complex(-1.0,-0.0))
    @test isequal(expm1(complex(-Inf, 5.0)), complex(-1.0,sin(5.0)* 0.0))
    @test isequal(expm1(complex(-Inf,-5.0)), complex(-1.0,sin(5.0)*-0.0))
    @test isequal(expm1(complex(-Inf, Inf)), complex(-1.0, 0.0))
    @test isequal(expm1(complex(-Inf,-Inf)), complex(-1.0,-0.0))
    @test isequal(expm1(complex(-Inf, NaN)), complex(-1.0, 0.0))

    @test isequal(expm1(complex( NaN, 0.0)), complex( NaN, 0.0))
    @test isequal(expm1(complex( NaN,-0.0)), complex( NaN,-0.0))
    @test isequal(expm1(complex( NaN, 5.0)), complex( NaN, NaN))
    @test isequal(expm1(complex( NaN, NaN)), complex( NaN, NaN))

    @test isequal(expm1(complex( 1e-20, 0.0)), complex(expm1( 1e-20), 0.0))
    @test isequal(expm1(complex(-1e-20, 0.0)), complex(expm1(-1e-20), 0.0))

    @test expm1(complex( 1e-20, 1e-10)) ≈ complex( 5e-21, 1e-10)
    @test expm1(complex( 1e-20,-1e-10)) ≈ complex( 5e-21,-1e-10)
    @test expm1(complex(-1e-20, 1e-10)) ≈ complex(-1.5e-20, 1e-10)
    @test expm1(complex(-1e-20,-1e-10)) ≈ complex(-1.5e-20,-1e-10)

    @test expm1(complex( 10.0, 10.0)) ≈ exp(complex( 10.0, 10.0))-1
    @test expm1(complex( 10.0,-10.0)) ≈ exp(complex( 10.0,-10.0))-1
    @test expm1(complex(-10.0, 10.0)) ≈ exp(complex(-10.0, 10.0))-1
    @test expm1(complex(-10.0,-10.0)) ≈ exp(complex(-10.0,-10.0))-1
end

import Base.Math.@horner
@testset "log1p" begin
    @test isequal(log1p(complex(Inf, 3)), complex(Inf, +0.))
    @test isequal(log1p(complex(Inf, -3)), complex(Inf, -0.))
    @test isequal(log1p(complex(-Inf, 3)), complex(Inf, +pi))
    @test isequal(log1p(complex(-Inf, -3)), complex(Inf, -pi))
    @test isequal(log1p(complex(Inf, NaN)), complex(Inf, NaN))
    @test isequal(log1p(complex(NaN, 0)), complex(NaN, NaN))
    @test isequal(log1p(complex(0, NaN)), complex(NaN, NaN))
    @test isequal(log1p(complex(-1, +0.)), complex(-Inf, +0.))
    @test isequal(log1p(complex(-1, -0.)), complex(-Inf, -0.))
    @test isequal(log1p(complex(-2, 1e-10)), log(1 + complex(-2, 1e-10)))
    @test isequal(log1p(complex(1, Inf)), complex(Inf, pi/2))
    @test isequal(log1p(complex(1, -Inf)), complex(Inf, -pi/2))

    for z in (1e-10+1e-9im, 1e-10-1e-9im, -1e-10+1e-9im, -1e-10-1e-9im)
        @test log1p(z) ≈ @horner(z, 0, 1, -0.5, 1/3, -0.25, 0.2)
    end
    for z in (15+4im, 0.2+3im, 0.08-0.9im)
        @test log1p(z) ≈ log(1+z)
    end
end


@testset "^ (cpow)" begin
    # equivalent to exp(y*log(x))
    #   except for 0^0?
    # conj(x)^conj(y) = conj(x^y)
    @test isequal(complex( 0.0, 0.0)^complex( 0.0, 0.0), complex(1.0, 0.0))
    @test isequal(complex( 0.0, 0.0)^complex( 0.0,-0.0), complex(1.0, 0.0))
    @test isequal(complex( 0.0, 0.0)^complex(-0.0, 0.0), complex(1.0,-0.0))
    @test isequal(complex( 0.0, 0.0)^complex(-0.0,-0.0), complex(1.0,-0.0))

    @test isequal(complex( 0.0,-0.0)^complex( 0.0, 0.0), complex(1.0,-0.0))
    @test isequal(complex( 0.0,-0.0)^complex( 0.0,-0.0), complex(1.0,-0.0))
    @test isequal(complex( 0.0,-0.0)^complex(-0.0, 0.0), complex(1.0, 0.0))
    @test isequal(complex( 0.0,-0.0)^complex(-0.0,-0.0), complex(1.0, 0.0))

    @test isequal(complex(-0.0, 0.0)^complex( 0.0, 0.0), complex(1.0, 0.0))
    @test isequal(complex(-0.0, 0.0)^complex( 0.0,-0.0), complex(1.0, 0.0))
    @test isequal(complex(-0.0, 0.0)^complex(-0.0, 0.0), complex(1.0,-0.0))
    @test isequal(complex(-0.0, 0.0)^complex(-0.0,-0.0), complex(1.0,-0.0))

    @test isequal(complex(-0.0,-0.0)^complex( 0.0, 0.0), complex(1.0,-0.0))
    @test isequal(complex(-0.0,-0.0)^complex( 0.0,-0.0), complex(1.0,-0.0))
    @test isequal(complex(-0.0,-0.0)^complex(-0.0, 0.0), complex(1.0, 0.0))
    @test isequal(complex(-0.0,-0.0)^complex(-0.0,-0.0), complex(1.0, 0.0))

    @test complex(0.0,1.0)^complex(2.0,0) ≈ complex(-1.0, 0.0)
    @test complex(1.0,2.0)^complex(3.0,0) ≈ complex(-11.0, -2.0)

    @test isequal(complex(0.0,0.0)^false, complex(1.0,0.0))
    @test isequal(complex(0.0,0.0)^0, complex(1.0,0.0))
end

@testset "sinh and sin" begin
    # sinh: has properties
    #  sinh(conj(z)) = conj(sinh(z))
    #  sinh(-z) = -sinh(z)

    # sin: defined in terms of sinh
    #  sin(z) = -i*sinh(i*z)
    #  i.e. if sinh(a+ib) = x+iy
    #    then  sin(b-ia) = y-ix
    #  sin(conj(z)) = conj(sin(z))
    #  sin(-z) = -sin(z)

    # @test isequal(sin(complex( 0, 10000)),complex( 0.0, Inf))
    # @test isequal(sin(complex( 0,-10000)),complex( 0.0,-Inf))
    for (x,y) in [(complex( 0.0, 0.0), complex( 0.0, 0.0)),
                  (complex( 0.0, Inf), complex( 0.0, NaN)),
                  (complex( 0.0, NaN), complex( 0.0, NaN)),
                  (complex( 7.2, Inf), complex( NaN, NaN)),
                  (complex( 7.2, NaN), complex( NaN, NaN)),
                  (complex( 7.2, 0.0), complex( sinh(7.2), 0.0)),
                  (complex( 1e5, 0.0), complex( sinh(1e5), 0.0)),
                  (complex( Inf, 0.0), complex( Inf, 0.0)),
                  (complex( Inf, 7.2), Inf*cis(7.2)),
                  (complex( Inf, Inf), complex( Inf, NaN)),
                  (complex( Inf, NaN), complex( Inf, NaN)),
                  (complex( NaN, 0.0), complex( NaN, 0.0)),
                  (complex( NaN, 7.2), complex( NaN, NaN)),
                  (complex( NaN, NaN), complex( NaN, NaN)),
                  ]
        @test isequal(sinh(x), y)
        @test isequal(sinh(conj(x)), conj(y))
        @test isequal(sinh(-x), -y)
        @test isequal(sinh(-conj(x)), -conj(y))

        xx = complex(imag(x),-real(x))
        yy = complex(imag(y),-real(y))

        @test isequal(sin(xx),yy)
        @test isequal(sin(conj(xx)), conj(yy))
        @test isequal(sin(-xx), -yy)
        @test isequal(sin(-conj(xx)), -conj(yy))

        yyy = sin(pi*xx)
        @test isequal(sinpi(xx), yyy)
        @test isequal(sinpi(conj(xx)),conj(yyy))
        @test isequal(sinpi(-xx),-yyy)
        @test isequal(sinpi(-conj(xx)),-conj(yyy))
    end
end

@testset "cosh and cos" begin
    # cosh: has properties
    #  cosh(conj(z)) = conj(cosh(z))
    #  coshh(-z) = cosh(z)

    # cos
    #  cos(z) = cosh(iz)
    #   i.e cos(b-ia) = cosh(a+ib)
    #   and cos(b+ia) = cosh(a-ib)
    #  cos(conj(z)) = conj(cos(z))
    #  cos(-z) = cos(z)
    for (x,y) in [(complex( 0.0, 0.0), complex( 1.0, 0.0)),
                  (complex( 0.0, Inf), complex( NaN, 0.0)),
                  (complex( 0.0, NaN), complex( NaN, 0.0)),
                  (complex( 7.2, Inf), complex( NaN, NaN)),
                  (complex( 7.2, NaN), complex( NaN, NaN)),
                  (complex( 7.2, 0.0), complex( cosh(7.2), 0.0)),
                  (complex( 1e5, 0.0), complex( Inf, 0.0)),
                  (complex( Inf, 0.0), complex( Inf, 0.0)),
                  (complex( Inf, 7.2), Inf*cis(7.2)),
                  (complex( Inf, Inf), complex( Inf, NaN)),
                  (complex( Inf, NaN), complex( Inf, NaN)),
                  (complex( NaN, 0.0), complex( NaN, 0.0)),
                  (complex( NaN, 7.2), complex( NaN, NaN)),
                  (complex( NaN, NaN), complex( NaN, NaN)),
                  ]
        undef_sign = isequal(x,complex( NaN, 0.0)) || isequal(x,complex( 0.0, NaN))

        @test isequal(cosh(x), y)
        if !undef_sign
            @test isequal(cosh(conj(x)), conj(y))
            @test isequal(cosh(-x), y)
            @test isequal(cosh(-conj(x)), conj(y))
        end

        xx = complex(imag(x),-real(x))
        yy = y
        @test isequal(cos(xx),yy)
        if !undef_sign
            @test isequal(cos(conj(xx)), conj(yy))
            @test isequal(cos(-xx), yy)
            @test isequal(cos(-conj(xx)), conj(yy))
        end

        yyy = cos(pi*xx)
        @test isequal(cospi(xx), yyy)
        if !undef_sign
            @test isequal(cospi(conj(xx)), conj(yyy))
            @test isequal(cospi(-xx), yyy)
            @test isequal(cospi(-conj(xx)), conj(yyy))
        end
    end
end

@testset "tanh(op(z)) == op(tanh(z)) for op in (conj, -)" begin
    @test isequal(tanh(complex( 0, 0)),complex(0.0,0.0)) #integer fallback
    @test isequal(tanh(complex( 0.0, 0.0)),complex(0.0,0.0))
    @test isequal(tanh(complex( 0.0,-0.0)),complex(0.0,-0.0))
    @test_throws DomainError  tanh(complex( 0.0, Inf))
    @test_throws DomainError  tanh(complex( 0.0,-Inf))
    @test isequal(tanh(complex( 0.0, NaN)),complex(NaN,NaN))

    @test isequal(tanh(complex(-0.0, 0.0)),complex(-0.0,0.0))
    @test isequal(tanh(complex(-0.0,-0.0)),complex(-0.0,-0.0))

    @test_throws DomainError  tanh(complex( 5.0, Inf))
    @test isequal(tanh(complex( 5.0, NaN)),complex(NaN,NaN))

    @test isequal(tanh(complex( Inf, 0.0)),complex(1.0, 0.0))
    @test isequal(tanh(complex( Inf,-0.0)),complex(1.0,-0.0))
    @test isequal(tanh(complex( Inf, 5.0)),complex(1.0,sin(2*5.0)* 0.0))
    @test isequal(tanh(complex( Inf,-5.0)),complex(1.0,sin(2*5.0)*-0.0))
    @test isequal(tanh(complex( Inf, Inf)),complex(1.0, 0.0))
    @test isequal(tanh(complex( Inf,-Inf)),complex(1.0,-0.0))
    @test isequal(tanh(complex( Inf, NaN)),complex(1.0, 0.0))

    @test isequal(tanh(complex(-Inf, 0.0)),complex(-1.0, 0.0))
    @test isequal(tanh(complex(-Inf,-0.0)),complex(-1.0,-0.0))
    @test isequal(tanh(complex(-Inf, 5.0)),complex(-1.0,sin(2*5.0)* 0.0))
    @test isequal(tanh(complex(-Inf,-5.0)),complex(-1.0,sin(2*5.0)*-0.0))
    @test isequal(tanh(complex(-Inf, Inf)),complex(-1.0, 0.0))
    @test isequal(tanh(complex(-Inf,-Inf)),complex(-1.0,-0.0))
    @test isequal(tanh(complex(-Inf, NaN)),complex(-1.0, 0.0))

    @test isequal(tanh(complex( NaN, 0.0)),complex(NaN, 0.0))
    @test isequal(tanh(complex( NaN,-0.0)),complex(NaN,-0.0))
    @test isequal(tanh(complex( NaN, 5.0)),complex(NaN, NaN))
    @test isequal(tanh(complex( NaN,-5.0)),complex(NaN, NaN))
    @test isequal(tanh(complex( NaN, NaN)),complex(NaN, NaN))
end

@testset "tan(z) == -i tanh(iz)" begin
    @test isequal(tan(complex( 0.0, Inf)),complex( 0.0, 1.0))
    @test isequal(tan(complex( 0.0,-Inf)),complex( 0.0,-1.0))
    @test isequal(tan(complex( 0.0, NaN)),complex( 0.0, NaN))
    @test isequal(tan(complex(-0.0,-Inf)),complex(-0.0,-1.0))
    @test isequal(tan(complex(-0.0, Inf)),complex(-0.0, 1.0))
    @test isequal(tan(complex(-0.0, NaN)),complex(-0.0, NaN))

    @test isequal(tan(complex( 5.0, Inf)),complex(sin(2*5.0)* 0.0, 1.0))
    @test isequal(tan(complex( 5.0,-Inf)),complex(sin(2*5.0)* 0.0,-1.0))
    @test isequal(tan(complex( 5.0, NaN)),complex( NaN, NaN))
    @test isequal(tan(complex(-5.0, Inf)),complex(sin(2*5.0)*-0.0, 1.0))
    @test isequal(tan(complex(-5.0,-Inf)),complex(sin(2*5.0)*-0.0,-1.0))
    @test isequal(tan(complex(-5.0, NaN)),complex( NaN, NaN))

    @test_throws DomainError  tan(complex( Inf, 5.0))
    @test isequal(tan(complex( Inf, Inf)),complex( 0.0, 1.0))
    @test isequal(tan(complex( Inf,-Inf)),complex( 0.0,-1.0))
    @test isequal(tan(complex(-Inf, Inf)),complex(-0.0, 1.0))
    @test isequal(tan(complex(-Inf,-Inf)),complex(-0.0,-1.0))

    @test isequal(tan(complex( NaN, 5.0)),complex( NaN, NaN))
    @test isequal(tan(complex( NaN, Inf)),complex( 0.0, 1.0))
    @test isequal(tan(complex( NaN,-Inf)),complex( 0.0,-1.0))
    @test isequal(tan(complex( NaN, NaN)),complex( NaN, NaN))
end

@testset "acosh(conj(z)) == conj(acosh(z))" begin
    @test isequal(acosh(complex( 0.0, 0.0)), complex( 0.0, pi/2))
    @test isequal(acosh(complex( 0.0,-0.0)), complex( 0.0,-pi/2))
    @test isequal(acosh(complex( 0.0, Inf)), complex( Inf, pi/2))
    @test isequal(acosh(complex( 0.0,-Inf)), complex( Inf,-pi/2))
    @test isequal(acosh(complex(-0.0, 0.0)), complex( 0.0, pi/2))
    @test isequal(acosh(complex(-0.0,-0.0)), complex( 0.0,-pi/2))
    @test isequal(acosh(complex( 5.0, Inf)), complex( Inf, pi/2))
    @test isequal(acosh(complex( 5.0,-Inf)), complex( Inf,-pi/2))
    @test isequal(acosh(complex( 5.0, NaN)), complex( NaN, NaN))

    @test isequal(acosh(complex( Inf, 0.0)), complex( Inf, 0.0))
    @test isequal(acosh(complex( Inf,-0.0)), complex( Inf,-0.0))
    @test isequal(acosh(complex( Inf, 5.0)), complex( Inf, 0.0))
    @test isequal(acosh(complex( Inf,-5.0)), complex( Inf,-0.0))
    @test isequal(acosh(complex( Inf, Inf)), complex( Inf, pi/4))
    @test isequal(acosh(complex( Inf,-Inf)), complex( Inf,-pi/4))
    @test isequal(acosh(complex( Inf, NaN)), complex( Inf, NaN))

    @test isequal(acosh(complex(-Inf, 0.0)), complex( Inf, pi))
    @test isequal(acosh(complex(-Inf,-0.0)), complex( Inf,-pi))
    @test isequal(acosh(complex(-Inf, 5.0)), complex( Inf, pi))
    @test isequal(acosh(complex(-Inf,-5.0)), complex( Inf,-pi))
    @test isequal(acosh(complex(-Inf, Inf)), complex( Inf, 3*pi/4))
    @test isequal(acosh(complex(-Inf,-Inf)), complex( Inf,-3*pi/4))
    @test isequal(acosh(complex(-Inf, NaN)), complex( Inf, NaN))

    @test isequal(acosh(complex( NaN, Inf)), complex( Inf, NaN))
    @test isequal(acosh(complex( NaN,-Inf)), complex( Inf, NaN))
    @test isequal(acosh(complex( NaN, NaN)), complex( NaN, NaN))
end

@testset "acos(conj(z)) == conj(acos(z))" begin
    @test isequal(acos(complex( 0, 0)),complex(pi/2,-0.0)) #integer fallback
    @test isequal(acos(complex( 0.0, 0.0)),complex(pi/2,-0.0))
    @test isequal(acos(complex( 0.0,-0.0)),complex(pi/2, 0.0))
    @test isequal(acos(complex( 0.0, Inf)),complex(pi/2,-Inf))
    @test isequal(acos(complex( 0.0,-Inf)),complex(pi/2, Inf))
    @test isequal(acos(complex( 0.0, NaN)),complex(pi/2, NaN))

    @test isequal(acos(complex(-0.0, 0.0)),complex(pi/2,-0.0))
    @test isequal(acos(complex(-0.0,-0.0)),complex(pi/2, 0.0))
    @test isequal(acos(complex(-0.0, NaN)),complex(pi/2, NaN))

    @test isequal(acos(complex( 5.0, Inf)),complex(pi/2,-Inf))
    @test isequal(acos(complex( 5.0,-Inf)),complex(pi/2, Inf))
    @test isequal(acos(complex( 5.0, NaN)),complex( NaN, NaN))

    @test isequal(acos(complex( Inf, 0.0)),complex( 0.0,-Inf))
    @test isequal(acos(complex( Inf,-0.0)),complex( 0.0, Inf))
    @test isequal(acos(complex( Inf, 5.0)),complex( 0.0,-Inf))
    @test isequal(acos(complex( Inf,-5.0)),complex( 0.0, Inf))
    @test isequal(acos(complex( Inf, Inf)),complex(pi/4,-Inf))
    @test isequal(acos(complex( Inf,-Inf)),complex(pi/4, Inf))
    @test isequal(acos(complex( Inf, NaN)),complex( NaN, Inf))

    @test isequal(acos(complex(-Inf, 0.0)),complex(pi,-Inf))
    @test isequal(acos(complex(-Inf,-0.0)),complex(pi, Inf))
    @test isequal(acos(complex(-Inf, 5.0)),complex(pi,-Inf))
    @test isequal(acos(complex(-Inf,-5.0)),complex(pi, Inf))
    @test isequal(acos(complex(-Inf, Inf)),complex(3*pi/4,-Inf))
    @test isequal(acos(complex(-Inf,-Inf)),complex(3*pi/4, Inf))
    @test isequal(acos(complex(-Inf, NaN)),complex( NaN, Inf))

    @test isequal(acos(complex( NaN, 0.0)),complex( NaN, NaN))
    @test isequal(acos(complex( NaN, 5.0)),complex( NaN, NaN))
    @test isequal(acos(complex( NaN, Inf)),complex( NaN,-Inf))
    @test isequal(acos(complex( NaN,-Inf)),complex( NaN, Inf))
    @test isequal(acos(complex( NaN, NaN)),complex( NaN, NaN))
end

@testset "asinh(op(z)) == op(asinh(z)) for op in (conj, -)" begin
    @test isequal(asinh(complex( 0.0, 0.0)),complex( 0.0, 0.0))
    @test isequal(asinh(complex( 0.0,-0.0)),complex( 0.0,-0.0))
    @test isequal(asinh(complex( 0.0, Inf)),complex( Inf, pi/2))
    @test isequal(asinh(complex( 0.0,-Inf)),complex( Inf,-pi/2))
    @test isequal(asinh(complex( 0.0, NaN)),complex( NaN, NaN))

    @test isequal(asinh(complex(-0.0, 0.0)),complex(-0.0, 0.0))
    @test isequal(asinh(complex(-0.0,-0.0)),complex(-0.0,-0.0))
    @test isequal(asinh(complex(-0.0, Inf)),complex(-Inf, pi/2))
    @test isequal(asinh(complex(-0.0,-Inf)),complex(-Inf,-pi/2))

    @test isequal(asinh(complex( 5.0, Inf)),complex( Inf, pi/2))
    @test isequal(asinh(complex( 5.0,-Inf)),complex( Inf,-pi/2))
    @test isequal(asinh(complex( 5.0, NaN)),complex( NaN, NaN))
    @test isequal(asinh(complex(-5.0, Inf)),complex(-Inf, pi/2))
    @test isequal(asinh(complex(-5.0,-Inf)),complex(-Inf,-pi/2))

    @test isequal(asinh(complex( Inf, Inf)),complex( Inf, pi/4))
    @test isequal(asinh(complex( Inf,-Inf)),complex( Inf,-pi/4))
    @test isequal(asinh(complex( Inf, NaN)),complex( Inf, NaN))
    @test isequal(asinh(complex(-Inf, Inf)),complex(-Inf, pi/4))
    @test isequal(asinh(complex(-Inf,-Inf)),complex(-Inf,-pi/4))
    @test isequal(asinh(complex(-Inf, NaN)),complex(-Inf, NaN))

    @test isequal(asinh(complex( NaN, 0.0)),complex( NaN, 0.0))
    @test isequal(asinh(complex( NaN,-0.0)),complex( NaN,-0.0))
    @test isequal(asinh(complex( NaN, 5.0)),complex( NaN, NaN))
    @test isequal(asinh(complex( NaN, Inf)),complex( Inf, NaN))
    @test isequal(asinh(complex( NaN,-Inf)),complex( Inf, NaN))
    @test isequal(asinh(complex( NaN, NaN)),complex( NaN, NaN))
end

@testset "asin(z) == -i*asinh(iz)" begin
    @test isequal(asin(complex( 0.0, 0.0)),complex( 0.0, 0.0))
    @test isequal(asin(complex( 0.0,-0.0)),complex( 0.0,-0.0))
    @test isequal(asin(complex(-0.0, 0.0)),complex(-0.0, 0.0))
    @test isequal(asin(complex( 0.0, NaN)),complex( 0.0, NaN))
    @test isequal(asin(complex(-0.0,-0.0)),complex(-0.0,-0.0))
    @test isequal(asin(complex(-0.0, NaN)),complex(-0.0, NaN))
    @test isequal(asin(complex( 5.0, NaN)),complex( NaN, NaN))

    @test isequal(asin(complex( Inf, 0.0)),complex( pi/2, Inf))
    @test isequal(asin(complex( Inf,-0.0)),complex( pi/2,-Inf))
    @test isequal(asin(complex( Inf, 5.0)),complex( pi/2, Inf))
    @test isequal(asin(complex( Inf,-5.0)),complex( pi/2,-Inf))
    @test isequal(asin(complex( Inf, Inf)),complex( pi/4, Inf))
    @test isequal(asin(complex( Inf,-Inf)),complex( pi/4,-Inf))
    @test isequal(asin(complex( Inf, NaN)),complex( NaN, Inf))

    @test isequal(asin(complex(-Inf, 0.0)),complex(-pi/2, Inf))
    @test isequal(asin(complex(-Inf,-0.0)),complex(-pi/2,-Inf))
    @test isequal(asin(complex(-Inf, 5.0)),complex(-pi/2, Inf))
    @test isequal(asin(complex(-Inf,-5.0)),complex(-pi/2,-Inf))
    @test isequal(asin(complex(-Inf, Inf)),complex(-pi/4, Inf))
    @test isequal(asin(complex(-Inf,-Inf)),complex(-pi/4,-Inf))
    @test isequal(asin(complex(-Inf, NaN)),complex( NaN, Inf))

    @test isequal(asin(complex( NaN, 0.0)),complex( NaN, NaN))
    @test isequal(asin(complex( NaN, 5.0)),complex( NaN, NaN))
    @test isequal(asin(complex( NaN, Inf)),complex( NaN, Inf))
    @test isequal(asin(complex( NaN,-Inf)),complex( NaN,-Inf))
    @test isequal(asin(complex( NaN, NaN)),complex( NaN, NaN))
end

@testset "atanh(op(z)) == op(atanh(z)) for op in (conj, -)" begin
    @test isequal(atanh(complex( 0, 0)),complex( 0.0, 0.0)) #integer fallback
    @test isequal(atanh(complex( 0.0, 0.0)),complex( 0.0, 0.0))
    @test isequal(atanh(complex( 0.0,-0.0)),complex( 0.0,-0.0))
    @test isequal(atanh(complex( 0.0, NaN)),complex( 0.0, NaN))
    @test isequal(atanh(complex( 0.0, Inf)),complex( 0.0, pi/2))
    @test isequal(atanh(complex( 0.0,-Inf)),complex( 0.0,-pi/2))

    @test isequal(atanh(complex(-0.0, NaN)),complex(-0.0, NaN))
    @test isequal(atanh(complex(-0.0, 0.0)),complex(-0.0, 0.0))
    @test isequal(atanh(complex(-0.0,-0.0)),complex(-0.0,-0.0))
    @test isequal(atanh(complex(-0.0, Inf)),complex(-0.0, pi/2))
    @test isequal(atanh(complex(-0.0,-Inf)),complex(-0.0,-pi/2))

    @test isequal(atanh(complex( 1.0, 0.0)),complex( Inf, 0.0))
    @test isequal(atanh(complex(-1.0, 0.0)),complex(-Inf, 0.0))
    @test isequal(atanh(complex( 5.0, Inf)),complex( 0.0, pi/2))
    @test isequal(atanh(complex( 5.0,-Inf)),complex( 0.0,-pi/2))
    @test isequal(atanh(complex( 5.0, NaN)),complex( NaN, NaN))
    @test isequal(atanh(complex(-5.0, Inf)),complex(-0.0, pi/2))
    @test isequal(atanh(complex(-5.0,-Inf)),complex(-0.0,-pi/2))
    @test isequal(atanh(complex(-5.0, NaN)),complex( NaN, NaN))

    @test isequal(atanh(complex( Inf, 0.0)),complex(0.0, pi/2))
    @test isequal(atanh(complex( Inf,-0.0)),complex(0.0,-pi/2))
    @test isequal(atanh(complex( Inf, 5.0)),complex(0.0, pi/2))
    @test isequal(atanh(complex( Inf,-5.0)),complex(0.0,-pi/2))
    @test isequal(atanh(complex( Inf, Inf)),complex(0.0, pi/2))
    @test isequal(atanh(complex( Inf,-Inf)),complex(0.0,-pi/2))
    @test isequal(atanh(complex( Inf, NaN)),complex(0.0, NaN))

    @test isequal(atanh(complex(-Inf, 0.0)),complex(-0.0, pi/2))
    @test isequal(atanh(complex(-Inf,-0.0)),complex(-0.0,-pi/2))
    @test isequal(atanh(complex(-Inf, 5.0)),complex(-0.0, pi/2))
    @test isequal(atanh(complex(-Inf,-5.0)),complex(-0.0,-pi/2))
    @test isequal(atanh(complex(-Inf, Inf)),complex(-0.0, pi/2))
    @test isequal(atanh(complex(-Inf,-Inf)),complex(-0.0,-pi/2))
    @test isequal(atanh(complex(-Inf, NaN)),complex(-0.0, NaN))

    @test isequal(atanh(complex( NaN, 0.0)),complex( NaN, NaN))
    @test isequal(atanh(complex( NaN,-0.0)),complex( NaN, NaN))
    @test isequal(atanh(complex( NaN, 5.0)),complex( NaN, NaN))
    @test isequal(atanh(complex( NaN,-5.0)),complex( NaN, NaN))
    @test isequal(atanh(complex( NaN, Inf)),complex( 0.0, pi/2))
    @test isequal(atanh(complex( NaN,-Inf)),complex( 0.0,-pi/2))
    @test isequal(atanh(complex( NaN, NaN)),complex( NaN, NaN))
end

@testset "atan(z) == -i*atanh(iz)" begin
    @test isequal(atan(complex( 0.0, 0.0)),complex( 0.0, 0.0))
    @test isequal(atan(complex( 0.0,-0.0)),complex( 0.0,-0.0))
    @test isequal(atan(complex( 0.0, 1.0)),complex( 0.0, Inf))
    @test isequal(atan(complex( 0.0, Inf)),complex( pi/2, 0.0))
    @test isequal(atan(complex( 0.0,-Inf)),complex( pi/2,-0.0))
    @test isequal(atan(complex( 0.0, NaN)),complex( NaN, NaN))

    @test isequal(atan(complex(-0.0, 0.0)),complex(-0.0, 0.0))
    @test isequal(atan(complex(-0.0,-0.0)),complex(-0.0,-0.0))
    @test isequal(atan(complex(-0.0, Inf)),complex(-pi/2, 0.0))
    @test isequal(atan(complex(-0.0,-Inf)),complex(-pi/2,-0.0))
    @test isequal(atan(complex(-0.0, NaN)),complex( NaN, NaN))

    @test isequal(atan(complex( 5.0, Inf)),complex( pi/2, 0.0))
    @test isequal(atan(complex( 5.0,-Inf)),complex( pi/2,-0.0))
    @test isequal(atan(complex( 5.0, NaN)),complex( NaN, NaN))

    @test isequal(atan(complex(-5.0, Inf)),complex(-pi/2, 0.0))
    @test isequal(atan(complex(-5.0,-Inf)),complex(-pi/2,-0.0))
    @test isequal(atan(complex(-5.0, NaN)),complex( NaN, NaN))

    @test isequal(atan(complex( Inf, 0.0)),complex( pi/2, 0.0))
    @test isequal(atan(complex( Inf,-0.0)),complex( pi/2,-0.0))
    @test isequal(atan(complex( Inf, 5.0)),complex( pi/2, 0.0))
    @test isequal(atan(complex( Inf,-5.0)),complex( pi/2,-0.0))
    @test isequal(atan(complex( Inf, Inf)),complex( pi/2, 0.0))
    @test isequal(atan(complex( Inf,-Inf)),complex( pi/2,-0.0))
    @test isequal(atan(complex( Inf, NaN)),complex( pi/2, 0.0))

    @test isequal(atan(complex(-Inf, 0.0)),complex(-pi/2, 0.0))
    @test isequal(atan(complex(-Inf,-0.0)),complex(-pi/2,-0.0))
    @test isequal(atan(complex(-Inf, 5.0)),complex(-pi/2, 0.0))
    @test isequal(atan(complex(-Inf,-5.0)),complex(-pi/2,-0.0))
    @test isequal(atan(complex(-Inf, Inf)),complex(-pi/2, 0.0))
    @test isequal(atan(complex(-Inf,-Inf)),complex(-pi/2,-0.0))
    @test isequal(atan(complex(-Inf, NaN)),complex(-pi/2, 0.0))

    @test isequal(atan(complex( NaN, 0.0)),complex( NaN, 0.0))
    @test isequal(atan(complex( NaN,-0.0)),complex( NaN,-0.0))
    @test isequal(atan(complex( NaN, 5.0)),complex( NaN, NaN))
    @test isequal(atan(complex( NaN,-5.0)),complex( NaN, NaN))
    @test isequal(atan(complex( NaN, Inf)),complex( NaN, 0.0))
    @test isequal(atan(complex( NaN,-Inf)),complex( NaN,-0.0))
    @test isequal(atan(complex( NaN, NaN)),complex( NaN, NaN))
end

@testset "lexcmp" begin
    @test lexcmp(1.0-1.0im, 1.0+0.0im) == -1
    @test lexcmp(0.0+0.0im, 0.0+0.0im) == 0
    @test lexcmp(1.0-1.0im, 0.0+0.0im) == 1
end

# misc.

@test complex(1//2,1//3)^2 === complex(5//36, 1//3)
@test complex(2,2)^2 === complex(0,8)
let p = -2
    @test_throws DomainError complex(2,2)^p
end
@test complex(2,2)^(-2) === complex(2.0,2.0)^(-2) === complex(0.0, -0.125)

@test complex.(1.0, [1.0, 1.0]) == [complex(1.0, 1.0), complex(1.0, 1.0)]
@test complex.([1.0, 1.0], 1.0) == [complex(1.0, 1.0), complex(1.0, 1.0)]
# robust division of Float64
# hard complex divisions from Fig 6 of arxiv.1210.4539
z7 = Complex{Float64}(3.898125604559113300e289, 8.174961907852353577e295)
z9 = Complex{Float64}(0.001953125, -0.001953125)
z10 = Complex{Float64}( 1.02951151789360578e-84, 6.97145987515076231e-220)
harddivs = ((1.0+im*1.0, 1.0+im*2^1023.0, 2^-1023.0-im*2^-1023.0), #1
      (1.0+im*1.0, 2^-1023.0+im*2^-1023.0, 2^1023.0+im*0.0), #2
      (2^1023.0+im*2^-1023.0, 2^677.0+im*2^-677.0, 2^346.0-im*2^-1008.0), #3
      (2^1023.0+im*2^1023.0, 1.0+im*1.0, 2^1023.0+im*0.0), #4
      (2^1020.0+im*2^-844., 2^656.0+im*2^-780.0, 2^364.0-im*2^-1072.0), #5
      (2^-71.0+im*2^1021., 2^1001.0+im*2^-323.0, 2^-1072.0+im*2^20.0), #6
      (2^-347.0+im*2^-54., 2^-1037.0+im*2^-1058.0, z7), #7
      (2^-1074.0+im*2^-1074., 2^-1073.0+im*2^-1074., 0.6+im*0.2), #8
      (2^1015.0+im*2^-989., 2^1023.0+im*2^1023.0, z9), #9
      (2^-622.0+im*2^-1071., 2^-343.0+im*2^-798.0, z10) #10
      )

# calculate "accurate bits" in range 0:53 by algorithm given in arxiv.1210.4539
function sb_accuracy(x,expected)
    min(logacc(real(x),real(expected)),
        logacc(imag(x),imag(expected)))
end
relacc(x,expected) = abs(x-expected)/abs(expected)
function logacc(x::Float64,expected::Float64)
    x == expected && (return 53)
    expected == 0 && (return 0)
    (x == Inf || x == -Inf) && (return 0)
    isnan(x) && (return 0)
    ra = relacc(BigFloat(x),BigFloat(expected))
    max(floor(Int,-log2(ra)),0)
end
# the robust division algorithm should have 53 or 52
# bits accuracy for each of the hard divisions
@test 52 <= minimum([sb_accuracy(h[1]/h[2],h[3]) for h in harddivs])

# division of non-Float64
function cdiv_test(a,b)
    c=convert(Complex{Float64},a)/convert(Complex{Float64},b)
    50 <= sb_accuracy(c,convert(Complex{Float64},a/b))
end
@test cdiv_test(complex(1//2, 3//4), complex(17//13, 4//5))
@test cdiv_test(complex(1,2), complex(8997,2432))

@testset "inv" begin
    @test inv(1e300+0im) == 1e-300 - 0.0im
    @test inv(0+1e300im) == 0.0 - 1e-300im
end

@testset "issue #7904" begin
    @test log10(10+0im) === 1.0 + 0.0im
    @test log2(2+0im) === 1.0 + 0.0im
end

@testset "sign" begin
    for T in (Float32, Float64)
        z = Complex{T}(1)
        @test typeof(sign(z)) == typeof(z)
        z = Complex{T}(0)
        @test typeof(sign(z)) == typeof(z)
    end
    for T in (Int32, Int64)
        z = Complex{T}(1)
        @test typeof(sign(z)) == typeof(float(z))
        z = Complex{T}(0)
        @test typeof(sign(z)) == typeof(float(z))
    end

    @test sign(0 + 0im) == 0
    @test sign(2 + 0im) == 1
    @test sign(-2 + 0im) == -1
    @test sign(1 + im) ≈ (1 + im) / sqrt(2)
    @test sign(1 - im) ≈ (1 - im) / sqrt(2)

    for T in (Float16, Float32, Float64)
        z = Complex(zero(T), zero(T))
        @test sign(z) === z
        @test sign(-z) === -z
        @test sign(conj(z)) === conj(z)
        @test sign(-conj(z)) === -conj(z)
    end
end

@testset "cis" begin
    @test cis(0.0+1.0im) ≈ 0.367879441171442321595523770161460867445811131031767834507836+0.0im
    @test cis(1.0+0.0im) ≈ 0.54030230586813971740093660744297660373231042061+0.84147098480789650665250232163029899962256306079im
    @test cis(pi) ≈ -1.0+0.0im
    @test cis(pi/2) ≈ 0.0+1.0im
end

@testset "exp2" begin
    @test exp2(0.0+0.0im) == 1.0+0.0im
    @test exp2(1.0+0.0im) == 2.0+0.0im
    #wolframalpha
    @test exp2(1.0+3.0im) ≈ -0.9739888359315627962096198412+1.74681016354974281701922im
    @test exp2(im) ≈ 0.7692389013639721 + 0.6389612763136348im
end

@testset "exp10" begin
    @test exp10(0.0+0.0im) == 1.0+0.0im
    @test exp10(1.0+0.0im) == 10.0+0.0im
    #wolframalpha
    @test exp10(1.0+2.0im) ≈ -1.0701348355877020772086517528518239460495529361-9.9425756941378968736161937190915602112878340717im
    @test exp10(im) ≈ -0.6682015101903132 + 0.7439803369574931im
end

@testset "round and float, PR #8291" begin
    @test round(Complex(1.125, 0.875), 2) == Complex(1.12, 0.88)
    @test round(Complex(1.5, 0.5), RoundDown, RoundUp) == Complex(1.0, 1.0)
    @test round.([1:5;] .+ im) == [1:5;] .+ im
    @test round.([1:5;] .+ 0.5im) == [1.0:5.0;]

    @test float(Complex(1, 2)) == Complex(1.0, 2.0)
    @test round(float(Complex(π, ℯ)),3) == Complex(3.142, 2.718)
end

@testset "ComplexF16 arithmetic, PR #10003" begin
    @test Float16(1)+Float16(1)im === ComplexF16(1, 1)
    @test Float16(1)-Float16(1)im === Float16(1)+Float16(-1)im === ComplexF16(1, -1)
    @test Float16(1)*im === ComplexF16(im)
    @test Float16(1)/im === ComplexF16(0,-1)
    @test Float16(1)^im === ComplexF16(1) === Float16(1)+Float16(0)im
end

# issue/PR #10148
@test typeof(Int8(1) - im) == Complex{Int8}

# issue #10926
@test typeof(π - 1im) == Complex{Float64}

@testset "issue #15969" begin
    # specialized muladd for complex types
    for x in (3, 3+13im), y in (2, 2+7im), z in (5, 5+11im)
        @test muladd(x,y,z) === x*y + z
    end
end

@testset "issue #11839" begin
    # type stability for Complex{Int64}
    let x = 1+im
        @inferred sin(x)
        @inferred cos(x)
        @inferred norm(x)
        @inferred vecnorm(x)
    end
end

@testset "issue #18785" begin
    # type stability for exp, expm1 for Complex{Int64}
    let x = 2*im
        @inferred exp(x)
        @inferred expm1(x)
    end
end

# issue #19240
@test big(1)/(10+10im) ≈ (5-5im)/big(100) ≈ big"0.05" - big"0.05"*im

@testset "Complex Irrationals, issue #21204" begin
    for x in (pi, ℯ, Base.MathConstants.catalan) # No need to test all of them
        z = Complex(x, x)
        @test typeof(z) == Complex{typeof(x)}
        @test exp(z) ≈ exp(x) * cis(x)
        @test log1p(z) ≈ log(1 + z)
        @test exp2(z) ≈ exp(z * log(2))
        @test exp10(z) ≈ exp(z * log(10))
    end
end

@testset "expm1 type stability" begin
    x = @inferred expm1(0.1im)
    @test x isa ComplexF64
    x = @inferred expm1(0.1f0im)
    @test x isa ComplexF32
end

@testset "array printing with exponent format" begin
    a = [1.0 + 1e-10im, 2.0e-15 - 2.0e-5im, 1.0e-15 + 2im, 1.0 + 2e-15im]
    @test sprint((io, x) -> show(io, MIME("text/plain"), x), a) ==
        join([
            "4-element Array{Complex{Float64},1}:",
            "     1.0 + 1.0e-10im",
            " 2.0e-15 - 2.0e-5im ",
            " 1.0e-15 + 2.0im    ",
            "     1.0 + 2.0e-15im"], "\n")
end

@testset "corner cases of division, issue #22983" begin
    # These results abide by ISO/IEC 10967-3:2006(E) and
    # mathematical definition of division of complex numbers.
    for T in (Float32, Float64, BigFloat)
        @test isequal(one(T) / zero(Complex{T}), one(Complex{T}) / zero(Complex{T}))
        @test isequal(one(T) / zero(Complex{T}), Complex{T}(NaN, NaN))
        @test isequal(one(Complex{T}) / zero(T), Complex{T}(Inf, NaN))
        @test isequal(one(Complex{T}) / one(Complex{T}), one(Complex{T}))
        @test isequal(one(T) / complex(one(T),  zero(T)), Complex(one(T), -zero(T)))
        @test isequal(one(T) / complex(one(T), -zero(T)), Complex(one(T),  zero(T)))
    end
end

@testset "complex^real, issue #14342" begin
    for T in (Float32, Float64, BigFloat), p in (T(-21//10), -21//10)
        z = T(2)+0im
        @test real(z^p) ≈ 2^p
        @test signbit(imag(z^p))
    end
    @test (2+0im)^(-21//10) === (2//1+0im)^(-21//10) === 2^-2.1 - 0.0im
end

@testset "more cpow" begin
    # for testing signs of zeros, it is useful to convert ±0.0 to ±1e-15
    zero2small(r::Real) = iszero(r) ? copysign(1e-15, r) : r
    zero2small(z::Complex) = complex(zero2small(real(z)), zero2small(imag(z)))
    ≋(x::Real, y::Real) = x*y == 0 ? abs(x) < 1e-8 && abs(y) < 1e-8 && signbit(x)==signbit(y) : isfinite(x) ? x ≈ y : isequal(x, y)
    ≋(x::Complex, y::Complex) = real(x) ≋ real(y) && imag(x) ≋ imag(y)
    ≟(x,y) = isequal(x,y)

    # test z^p for positive/negative/zero real and imaginary parts of z and p:
    v=(-2.7,-3.0,-2.0,-0.0,+0.0,2.0,3.0,2.7)
    for zr=v, zi=v, pr=v, pi=v
        z = complex(zr,zi)
        p = iszero(pi) ? pr : complex(pr,pi)
        if isinteger(p)
            c = zero2small(z)^Integer(pr)
        else
            c = exp(zero2small(p) * log(zero2small(z)))
        end
        if !iszero(z*p) # z==0 or p==0 is tricky, check it separately
            @test z^p ≋ c
            if isreal(p)
                @test z^(p + 1e-15im) ≈ z^(p - 1e-15im) ≈ c
                if isinteger(p)
                    @test isequal(z^Integer(pr), z^p)
                end
            elseif (zr != 0 || !signbit(zr)) && (zi != 0 || !signbit(zi))
                @test isequal((Complex{Int}(z*10)//10)^p, z^p)
            end
        end
    end

    @test 2 ^ (0.3 + 0.0im) === 2.0 ^ (0.3 + 0.0im) === conj(2.0 ^ (0.3 - 0.0im)) ≋  2.0 ^ (0.3 + 1e-15im)
    @test 0.2 ^ (0.3 + 0.0im) === conj(0.2 ^ (0.3 - 0.0im)) ≋  0.2 ^ (0.3 + 1e-15im)
    @test (0.0 - 0.0im)^2.0 === (0.0 - 0.0im)^2 === (0.0 - 0.0im)^1.1 === (0.0 - 0.0im) ^ (1.1 + 2.3im) === 0.0 - 0.0im
    @test (0.0 - 0.0im)^-2.0 ≟ (0.0 - 0.0im)^-2 ≟ (0.0 - 0.0im)^-1.1 ≟ (0.0 - 0.0im) ^ (-1.1 + 2.3im) ≟ NaN + NaN*im
    @test (1.0+0.0)^(1.2+0.7im) === 1.0 + 0.0im
    @test (-1.0+0.0)^(2.0+0.7im) ≈ exp(-0.7π)
    @test (-4.0+0.0im)^1.5 === (-4.0)^(1.5+0.0im) === (-4)^(1.5+0.0im) === (-4)^(3//2+0im) === 0.0 - 8.0im

    # issue #24515:
    @test (Inf + Inf*im)^2.0 ≟ (Inf + Inf*im)^2 ≟ NaN + Inf*im
    @test (0+0im)^-3.0 ≟ (0+0im)^-3 ≟ NaN + NaN*im
    @test (1.0+0.0im)^1e300 === 1.0 + 0.0im
    @test Inf^(-Inf + 0.0im) == (Inf + 0.0im)^(-Inf - 0.0im) == (Inf - 0.0im)^(-Inf - 0.0im) == (Inf - 0.0im)^-Inf == 0

    # NaN propagation
    @test (0 + NaN*im)^1 ≟ (0 + NaN*im)^1.0 ≟ (0 + NaN*im)^(1.0+0im) ≟ 0.0 + NaN*im
    @test (0 + NaN*im)^2 ≟ (0 + NaN*im)^2.0 ≟ (0 + NaN*im)^(2.0+0im) ≟ NaN + NaN*im
    @test (NaN + 0im)^2.0 ≟ (NaN + 0im)^(2.0+0im) ≟ (2+0im)^NaN ≟ NaN + 0im
    @test (NaN + 0im)^2.5 ≟ NaN^(2.5+0im) ≟ (NaN + NaN*im)^2.5 ≟ (-2+0im)^NaN ≟ (2+0im)^(1+NaN*im) ≟ NaN + NaN*im

    # more Inf cases:
    @test (Inf + 0im)^Inf === Inf^(Inf + 0im) === (Inf + 0im)^(Inf + 0im) == Inf + 0im
    @test (-Inf + 0im)^(0.7 + 0im) === (-Inf + 1im)^(0.7 + 0im) === conj((-Inf - 1im)^(0.7 + 0im)) === -Inf + Inf*im
    @test (-Inf + 0.0im) ^ 3.1 === conj((-Inf - 0.0im) ^ 3.1) === -Inf - Inf*im
    @test (3.0+0.0im)^(Inf + 1im) === (3.0-0.0im)^(Inf + 1im) === conj((3.0+0.0im)^(Inf - 1im)) === Inf + Inf*im

    # The following cases should arguably give Inf + Inf*im, but currently
    # give partial NaNs instead.  Marking as broken for now (since Julia 0.4 at least),
    # in the hope that someday we can fix these corner cases.  (Python gets them wrong too.)
    @test_broken (Inf + 1im)^3 === (Inf + 1im)^3.0 === (Inf + 1im)^(3+0im) === Inf + Inf*im
    @test_broken (Inf + 1im)^3.1 === (Inf + 1im)^(3.1+0im) === Inf + Inf*im

    # cases where phase angle is non-finite yield NaN + NaN*im:
    @test NaN + NaN*im ≟ Inf ^ (2 + 3im) ≟ (Inf + 1im) ^ (2 + 3im) ≟ (Inf*im) ^ (2 + 3im) ≟
          3^(Inf*im) ≟ (-3)^(Inf + 0im) ≟ (-3)^(Inf + 1im) ≟ (3+1im)^Inf ≟
          (3+1im)^(Inf + 1im) ≟ (1e200+1e-200im)^Inf ≟ (1e200+1e-200im)^(Inf+1im)

    @test @inferred(2.0^(3.0+0im)) === @inferred((2.0+0im)^(3.0+0im)) === @inferred((2.0+0im)^3.0) === 8.0+0.0im
end