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.
$\endgroup$ 42 Answers
$\begingroup$The basic formula is $e^{i\theta}=\cos\theta+i\sin\theta$, so your example would be
$$3^i=e^{i\ln3}=\cos(\ln3)+i\sin(\ln3)$$
$\endgroup$ $\begingroup$Using euler's formula :
$$c^{a+bi}=c^ac^{bi}=c^ae^{bi \ln (c)}$$
$$=c^a \left((\cos (b \ln (c))+ i \sin(b \ln (c) \right))$$
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