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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 $k \in \mathbb{Z}$?

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    @Dan You're right, of course.2012-11-09

2 Answers 2

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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 $n$.