# Real Numbers Raised to Imaginary Powers? [closed]

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.

## closed as off-topic by user296602, Daniel W. Farlow, M. Vinay, Claude Leibovici, user223391 Jul 2 '16 at 22:33

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## 2 Answers

The basic formula is $e^{i\theta}=\cos\theta+i\sin\theta$, so your example would be

$$3^i=e^{i\ln3}=\cos(\ln3)+i\sin(\ln3)$$

Using euler's formula :

$$c^{a+bi}=c^ac^{bi}=c^ae^{bi \ln (c)}$$

$$=c^a \left((\cos (b \ln (c))+ i \sin(b \ln (c) \right))$$