# Is $1$ raised to any complex power equal to $1$?

I just saw a solution that says it is since, for any complex number $z$

$$1^z = e^{z\log1} = e^{z(0)} = 1$$

However, isn't this only true for the principle branch of $1^z$, since by definiton, (letting capital L denote the princple value of log

$$\log1 = \operatorname{Log}1 + i2\pi k$$

for any $k \in \mathbb{Z}$?

-
@martini, I think the definition of complex powers is $z^\alpha = e^{\alpha \log z}$. You just did $z^\alpha = e^{\log z}$ –  Dan Nov 8 '12 at 22:56
The real number $1$ raised to any complex power is equal to $1$. The complex number $1$ raised to any complex power may not be. –  Micah Nov 8 '12 at 23:23
@Dan You're right, of course. –  martini Nov 9 '12 at 6:29

You are correct: $1$ is not the only value of $1^z$ if $z$ is not an integer. For example, $-1$ is also one of the values of $1^{1/2}$. In general the possible values of $1^z$ are $e^{2 \pi i n z}$ for integers $n$.