Is there a well-known name for the multiplicative equivalent to the absolute value function?
The absolute value function f(x), for real x, is the identity operation f(x) = x for x greater than the additive identity (zero), but it's the additive inverse f(x) = -x for x less than the additive identity.
I'm interested in the function g(x) over the positive reals that is the identity operation for x greater than the *multiplicative* identity (one), but the *multiplicative* inverse x⁻¹ for x less than the multiplicative identity.
I don't need g to take on any particular value for negative x, much less complex x, but I think the most natural extension would be that the function's rotationally symmetrical about the origin (an odd function, so g(-x) = -g(x)).
@khird I’ve never seen that addresses as a function, much less given a name, but I think it would be nice to have that.
But I think for complex z it should match the symmetry of |z|, or in effect be defined as |z| or 1/|z|
If you must have an odd form, I’d go with (z or 1/|z|) cos θ
@ShadSterling for positive reals, which is my use case, I'm going with e^abs(ln(x)) for the sake of convenience - using the exponents and logarithms to map from multiplication to addition and back. It's a bit more cumbersome to discuss for lack of a recognized name, but handy enough for doing math on.
I'm not sure that the complex extension should be real-valued everywhere. I can see why that makes a nice symmetry with the additive case, but it seems more elegant to let the function take complex values for complex arguments. So: undefined at zero, 1/z on the rest of the unit disk, and z elsewhere. It does come out odd over the reals in this case.
@khird oops, I misdescribed what I was thinking of, which was (r,θ) => ( r or 1/r, θ ), which is only real when z is real, and projects over the unit circle. If nothing else, exploring the options would make an interesting blog post, especially if you can find places where each variant is useful