TITLE: Real Numbers Raised to Imaginary Powers?
QUESTION [12 upvotes]: What is a real number to the power of an imaginary or complex number? e.g. 3i. I have searched through sites about imaginary numbers, but none seem to say anything about imaginary indices. Examples and explanations would be appreciated.

REPLY [8 votes]: Using euler's formula :
$$c^{a+bi}=c^ac^{bi}=c^ae^{bi \ln (c)}$$
$$=c^a \left((\cos (b \ln (c))+ i \sin(b \ln (c) \right))$$