In general has multiple values for a given complex z.
The multiple values are found as follows:
for
This comes from a definition for the complex power of a complex number given below. This definition uses the multi-valued complex natural logarithm which is itself defined by:
for
where is our principal branch defined by:
were is our principal branch of the argument function and is the real nature logarithm.
The complex power is defined by:
where
which gives
for
The s slider moves the complex point around the circle radius center the origin. Our principal value of is and the other values are show in red and green.
The red point are for positive k and the green for negative k.