You do in fact need the complex exponential to define this correctly because the function a^x for nonintegers x is only unambiguously defined when a is a positive real number. For example, your function could be either e^(pi i x) or e^(-pi i x), which trace the circle in opposite directions as x varies over the reals. (They happen to agree when x is an integer.)
You do in fact need the complex exponential to define this correctly because the function a^x for nonintegers x is only unambiguously defined when a is a positive real number. For example, your function could be either e^(pi i x) or e^(-pi i x), which trace the circle in opposite directions as x varies over the reals. (They happen to agree when x is an integer.)
I agree. I just meant the 2D function cos(pix),sin(pix) is quite natural to work with and it can be reflected easily in the formulation of (-1)^x
That's because
That's equivalent to saying, no need for -1 because we have exp.
One can change based of the exponentiation operation. Exp happens to be a convenient base.
Yeah bad formulation on my part
Not bad at all, just equivalent.
We do need complex exp to define my formula, you’re right
If you plot it over which domain?
Complex domain