On Wed, Jan 3, 2024 at 4:09 PM Aaron Meurer <asmeurer@gmail.com> wrote:
sign(z) = z/|z| is a fairly standard definition. See https://oeis.org/wiki/Sign_function and https://en.wikipedia.org/wiki/Sign_function. It's also implemented this way in MATLAB and Mathematica (see https://www.mathworks.com/help/symbolic/sign.html and https://reference.wolfram.com/language/ref/Sign.html). The function z/|z| is useful because it represents a normalization of z as a vector in the complex plane onto the unit circle.
With that being said, I'm not so sure about the suggestion about extending copysign(x1, x2) as |x1|*sign(x2). I generally think of copysign as a function to manipulate the floating-point representation of a number. It literally copies the sign *bit* from x2 into x1. It's useful because of things like -0.0, which is otherwise difficult to work with since it compares equal to 0.0. I would find it surprising for copysign to do a numeric calculation on complex numbers. Also, your suggested definition would be wrong for 0.0 and -0.0, since sign(0) is 0, and this is precisely where copysign matters.
Agreed on all points. -- Robert Kern