alt Hacker NewsEven better : did you know (-1)^x draws the unit circle in the complex plane ? No need for complex exp and i*pi
Replies
srean • today at 7:17 AM
That's because
a^b = exp (b ln a)
One can change based of the exponentiation operation. Exp happens to be a convenient base.
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.)