**Question Corner and Discussion Area**

I am interested in knowing whatThe first question to address is what it means to raise one complex number to the power of another. There is a basic definition of what it means to raiseito the power ofiis.

What are the possible values for *z*? Well, if we write *z *= *a *+ *bi*,
then . By
de Moivre's theorem (explained in the answer to an earlier question), , so
. This expression equals *i*
exactly when *a*=0, cos(*b*)=0, and sin(*b*)=1. This occurs when
for some integer *n*, so the possible values of
*z* are .

Therefore, the values of are

for any integer *n*.

Note that there is more than one value
for , just as 2 and -2 are both square roots of 4.
(However, while the square roots of a number always have the same
magnitude even if they differ in sign, the values of have
different magnitudes).
The
principal value of would be --the case where
*n*=0.

It's also interesting to note that all these values of are real numbers.

This part of the site maintained by (No Current Maintainers)

Last updated: April 19, 1999

Original Web Site Creator / Mathematical Content Developer: Philip Spencer

Current Network Coordinator and Contact Person: Joel Chan - mathnet@math.toronto.edu

Go backward to What is the Square Root of i?

Go up to Question Corner Index

Go forward to Raising a Number to a Complex Power

Switch to text-only version (no graphics)

Access printed version in PostScript format (requires PostScript printer)

Go to University of Toronto Mathematics Network
Home Page