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complex.jl
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complex.jl
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# sqrt:
# 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))
# log:
# log(conj(z)) = conj(log(z))
@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))
# exp:
# exp(conj(z)) = conj(exp(z))
@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))
# ^ (cpow)
# 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))
# sinh: has properties
# sinh(conj(z)) = conj(sinh(z))
# sinh(-z) = -sinh(z)
@test isequal(sinh(complex( 0.0, 0.0)), complex( 0.0, 0.0))
@test isequal(sinh(complex( 0.0,-0.0)), complex( 0.0,-0.0))
@test isequal(sinh(complex( 0.0, Inf)), complex( 0.0, NaN))
@test isequal(sinh(complex( 0.0,-Inf)), complex( 0.0, NaN))
@test isequal(sinh(complex( 0.0, NaN)), complex( 0.0, NaN))
@test isequal(sinh(complex(-0.0, 0.0)), complex(-0.0, 0.0))
@test isequal(sinh(complex(-0.0,-0.0)), complex(-0.0,-0.0))
@test isequal(sinh(complex(-0.0, Inf)), complex(-0.0, NaN))
@test isequal(sinh(complex(-0.0,-Inf)), complex(-0.0, NaN))
@test isequal(sinh(complex(-0.0, NaN)), complex(-0.0, NaN))
@test isequal(sinh(complex( 5.0, Inf)), complex( NaN, NaN))
@test isequal(sinh(complex( 5.0, NaN)), complex( NaN, NaN))
@test isequal(sinh(complex( Inf, 0.0)), complex( Inf, 0.0))
@test isequal(sinh(complex( Inf,-0.0)), complex( Inf,-0.0))
@test isequal(sinh(complex( Inf, 5.0)), complex(cos(5.0)*Inf,sin(5.0)*Inf))
@test isequal(sinh(complex( Inf, Inf)), complex( Inf, NaN))
@test isequal(sinh(complex( Inf,-Inf)), complex( Inf, NaN))
@test isequal(sinh(complex( Inf, NaN)), complex( Inf, NaN))
@test isequal(sinh(complex(-Inf, 0.0)), complex(-Inf, 0.0))
@test isequal(sinh(complex(-Inf,-0.0)), complex(-Inf,-0.0))
@test isequal(sinh(complex(-Inf, 5.0)), complex(cos(5.0)*-Inf,sin(5.0)*Inf))
@test isequal(sinh(complex(-Inf, Inf)), complex(-Inf, NaN))
@test isequal(sinh(complex(-Inf,-Inf)), complex(-Inf, NaN))
@test isequal(sinh(complex(-Inf, NaN)), complex(-Inf, NaN))
@test isequal(sinh(complex( NaN, 0.0)), complex( NaN, 0.0))
@test isequal(sinh(complex( NaN,-0.0)), complex( NaN,-0.0))
@test isequal(sinh(complex( NaN, 5.0)), complex( NaN, NaN))
@test isequal(sinh(complex( NaN, NaN)), complex( NaN, NaN))
# 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( 0.0,-0.0)),complex( 0.0,-0.0))
@test isequal(sin(complex( 0.0, 0.0)),complex( 0.0, 0.0))
@test isequal(sin(complex( 0.0, Inf)),complex( 0.0, Inf))
@test isequal(sin(complex( 0.0,-Inf)),complex( 0.0,-Inf))
@test isequal(sin(complex( 0.0, NaN)),complex( 0.0, NaN))
@test isequal(sin(complex(-0.0, 0.0)),complex(-0.0, 0.0))
@test isequal(sin(complex(-0.0,-0.0)),complex(-0.0,-0.0))
@test isequal(sin(complex(-0.0, Inf)),complex(-0.0, Inf))
@test isequal(sin(complex(-0.0,-Inf)),complex(-0.0,-Inf))
@test isequal(sin(complex(-0.0, NaN)),complex(-0.0, NaN))
@test isequal(sin(complex( 5.0, Inf)),complex(sin(5.0)*Inf,cos(5.0)* Inf))
@test isequal(sin(complex( 5.0,-Inf)),complex(sin(5.0)*Inf,cos(5.0)*-Inf))
@test isequal(sin(complex( 5.0, NaN)),complex( NaN, NaN))
@test isequal(sin(complex( Inf, 0.0)), complex( NaN, 0.0))
@test isequal(sin(complex( Inf,-0.0)), complex( NaN,-0.0))
@test isequal(sin(complex( Inf, 5.0)), complex( NaN, NaN))
@test isequal(sin(complex( Inf, Inf)), complex( Inf, NaN))
@test isequal(sin(complex( Inf,-Inf)), complex( Inf, NaN))
@test isequal(sin(complex(-Inf, 0.0)), complex( NaN, 0.0))
@test isequal(sin(complex(-Inf,-0.0)), complex( NaN,-0.0))
@test isequal(sin(complex(-Inf, 5.0)), complex( NaN, NaN))
@test isequal(sin(complex(-Inf, Inf)), complex(-Inf, NaN))
@test isequal(sin(complex(-Inf,-Inf)), complex(-Inf, NaN))
@test isequal(sin(complex( NaN, 0.0)),complex( NaN, 0.0))
@test isequal(sin(complex( NaN,-0.0)),complex( NaN,-0.0))
@test isequal(sin(complex( NaN, 5.0)),complex( NaN, NaN))
@test isequal(sin(complex( NaN, Inf)),complex( NaN, Inf))
@test isequal(sin(complex( NaN,-Inf)),complex( NaN,-Inf))
@test isequal(sin(complex( NaN, NaN)),complex( NaN, NaN))
# cosh: has properties
# cosh(conj(z)) = conj(cosh(z))
# coshh(-z) = cosh(z)
@test isequal(cosh(complex( 0.0, 0.0)), complex( 1.0, 0.0))
@test isequal(cosh(complex( 0.0,-0.0)), complex( 1.0,-0.0))
@test isequal(cosh(complex( 0.0, Inf)), complex( NaN, 0.0))
@test isequal(cosh(complex( 0.0,-Inf)), complex( NaN,-0.0))
@test isequal(cosh(complex( 0.0, NaN)), complex( NaN, 0.0))
@test isequal(cosh(complex(-0.0,-0.0)), complex( 1.0, 0.0))
@test isequal(cosh(complex(-0.0, 0.0)), complex( 1.0,-0.0))
@test isequal(cosh(complex(-0.0, NaN)), complex( NaN, 0.0))
@test isequal(cosh(complex( 5.0, Inf)), complex( NaN, NaN))
@test isequal(cosh(complex( 5.0, NaN)), complex( NaN, NaN))
@test isequal(cosh(complex( Inf, 0.0)), complex( Inf, 0.0))
@test isequal(cosh(complex( Inf,-0.0)), complex( Inf,-0.0))
@test isequal(cosh(complex( Inf, 5.0)), complex(cos(5.0)*Inf,sin(5.0)* Inf))
@test isequal(cosh(complex( Inf,-5.0)), complex(cos(5.0)*Inf,sin(5.0)*-Inf))
@test isequal(cosh(complex( Inf, Inf)), complex( Inf, NaN))
@test isequal(cosh(complex( Inf,-Inf)), complex( Inf, NaN))
@test isequal(cosh(complex( Inf, NaN)), complex( Inf, NaN))
@test isequal(cosh(complex(-Inf,-0.0)), complex( Inf, 0.0))
@test isequal(cosh(complex(-Inf, 0.0)), complex( Inf,-0.0))
@test isequal(cosh(complex(-Inf,-5.0)), complex(cos(5.0)*Inf,sin(5.0)* Inf))
@test isequal(cosh(complex(-Inf, 5.0)), complex(cos(5.0)*Inf,sin(5.0)*-Inf))
@test isequal(cosh(complex(-Inf, Inf)), complex( Inf, NaN))
@test isequal(cosh(complex(-Inf,-Inf)), complex( Inf, NaN))
@test isequal(cosh(complex(-Inf, NaN)), complex( Inf, NaN))
@test isequal(cosh(complex( NaN, 0.0)), complex( NaN, 0.0))
@test isequal(cosh(complex( NaN, 5.0)), complex( NaN, NaN))
@test isequal(cosh(complex( NaN,-0.0)), complex( NaN,-0.0))
@test isequal(cosh(complex( NaN,-5.0)), complex( NaN, NaN))
@test isequal(cosh(complex( NaN, NaN)), complex( NaN, NaN))
# 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)
@test isequal(cos(complex( 0.0, 0.0)), complex( 1.0,-0.0))
@test isequal(cos(complex( 0.0,-0.0)), complex( 1.0, 0.0))
@test isequal(cos(complex( 0.0,-Inf)), complex( Inf, 0.0))
@test isequal(cos(complex( 0.0, Inf)), complex( Inf,-0.0))
@test isequal(cos(complex( 0.0, NaN)), complex( NaN, 0.0))
@test isequal(cos(complex(-0.0, 0.0)), complex( 1.0, 0.0))
@test isequal(cos(complex(-0.0,-0.0)), complex( 1.0,-0.0))
@test isequal(cos(complex(-0.0,-Inf)), complex( Inf,-0.0))
@test isequal(cos(complex(-0.0, Inf)), complex( Inf, 0.0))
@test isequal(cos(complex(-0.0, NaN)), complex( NaN,-0.0))
@test isequal(cos(complex( 5.0,-Inf)), complex(cos(5.0)*Inf,sin(5.0)* Inf))
@test isequal(cos(complex( 5.0, Inf)), complex(cos(5.0)*Inf,sin(5.0)*-Inf))
@test isequal(cos(complex( 5.0, NaN)), complex( NaN, NaN))
@test isequal(cos(complex(-5.0,-Inf)), complex(cos(5.0)*Inf,sin(5.0)*-Inf))
@test isequal(cos(complex(-5.0, Inf)), complex(cos(5.0)*Inf,sin(5.0)* Inf))
@test isequal(cos(complex(-5.0, NaN)), complex( NaN, NaN))
@test isequal(cos(complex( Inf, 0.0)), complex( NaN,-0.0))
@test isequal(cos(complex( Inf, 5.0)), complex( NaN, NaN))
@test isequal(cos(complex( Inf, Inf)), complex( Inf, NaN))
@test isequal(cos(complex( Inf,-Inf)), complex( Inf, NaN))
@test isequal(cos(complex(-Inf, 0.0)), complex( NaN, 0.0))
@test isequal(cos(complex(-Inf, Inf)), complex( Inf, NaN))
@test isequal(cos(complex(-Inf,-Inf)), complex( Inf, NaN))
@test isequal(cos(complex( NaN, 0.0)), complex( NaN, 0.0))
@test isequal(cos(complex( NaN,-0.0)), complex( NaN, 0.0))
@test isequal(cos(complex( NaN, 5.0)), complex( NaN, NaN))
@test isequal(cos(complex( NaN, Inf)), complex( Inf, NaN))
@test isequal(cos(complex( NaN,-Inf)), complex( Inf, NaN))
@test isequal(cos(complex( NaN, NaN)), complex( NaN, NaN))
# tanh
# tanh(conj(z)) = conj(tanh(z))
# tanh(-z) = -tanh(z)
@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 tanh(complex( 0.0, Inf)) #complex(NaN,0.0)
@test_throws tanh(complex( 0.0,-Inf)) #complex(NaN,-0.0)
@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 tanh(complex( 5.0, Inf)) #complex(NaN,NaN)
@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))
# tan
# tan(z) = -i tanh(iz)
@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 tan(complex( Inf, 5.0)) #complex(NaN,NaN)
@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))
# acosh
# acosh(conj(z)) = conj(acosh(z))
@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))
#
## acos
## acos(conj(z)) = conj(acos(z))
@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))
## asinh
## asinh(conj(z)) = conj(asinh(z))
## asinh(-z) = -asinh(z)
@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))
# asin
# asin(z) = -i*asinh(iz)
@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))
# atanh
# atanh(conj(z)) = conj(atanh(z))
# atanh(-z) = -atanh(z)
@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))
# atan
# atan(z) = -i*atanh(iz)
@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))
# misc.
@test complex(1//2,1//3)^2 === complex(5//36, 1//3)
@test complex(2,2)^2 === complex(0,8)
@test_throws complex(2,2)^(-2)
@test complex(2.0,2.0)^(-2) === complex(0.0, -0.125)