For what values of $\alpha$ does $1^{\alpha}$ does $1^{\alpha} = 1$. What are the possible values of $1^{\alpha}$? What are the values of $1^{\frac{1}{2}}$? (Hint: use the definition of $z^{\alpha}$.)
Attempt: Recall the definition of a complex $\alpha$ constant where $z \neq 0$, then $z^{\alpha} = e^{\alpha \log z}$.
Then, the possible values for $1^{\alpha}$ using the definition are: $1^{\alpha} = e^{\alpha \log 1} = e^{\alpha [\log 1 + i\arg 1]} = e^{\alpha i 2 k\pi } $.
And when $\alpha = \frac{1}{2}$ we have $ 1^ {\frac {1}{2}} = e^{\frac{1}{2} i 2 k\pi } = e^{ik\pi} $.
I don't know how to continue.
I dont know for what values of $\alpha$ does $1^{\alpha}$ does $1^{\alpha} = 1$. Can someone please help me? I would really appreciate it. Thank you in advance.
Hint: You want $e^{2\pi k \alpha i} = 1$. What values of $z$ give you $e^{iz} = 1$? You'll want to distinguish between the cases $k = 0$ and $k \ne 0$.