wrong sign of zero for (x+0im)^(negative real)

[stevengj]

With Julia 0.5, (2.0+0im)^-2 correctly gives 0.25 - 0.0im, but (2.0+0im)^-2.1 gives 0.23325824788420185 + 0.0im. Notice that the sign of the imaginary part is incorrect, in that (2.0+δ*im)^-2.1 should give a negative imaginary part as δ ⟶ 0⁺.

cc: @jiahao

[jiahao]

Related to #10000?

[stevengj]

@jiahao, I don't think so, because negative-integer powers work.

[jiahao]

Negative integer powers use a different method. What gets called for (2.0+0.0im)^-2 is

Base.power_by_squaring(inv(2.0+0.0im), +2)

whereas (2.0+0.0im)^-2.1 calls exp((-2.1+0.0im)*log(2.0+0.0im)).

The problem appears to be the premature promotion of (2.0+0.0im)^-2.1 to (2.0+0.0im)^(-2.1+0.0im), as

julia> exp((-2.1+0.0im)log(2.0+0.0im))
0.23325824788420185 + 0.0im

julia> exp((-2.1)log(2.0+0.0im))
0.23325824788420185 - 0.0im

We can solve this by defining ^(z::Complex, w::Real) as its own special method, but perhaps there is a cleverer solution.

[stevengj]

^(z::Complex, w::Real) should probably be its own method anyway, since I would be surprised if you couldn't compute this more efficiently than Complex^Complex.