# 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.

• Have a look here: math.toronto.edu/mathnet/questionCorner/complexexp.html Commented Jul 1, 2016 at 23:56
• Provided the "real number" raised to a complex power is positive, this is well-defined. (Zero raised to a complex power is left as an exercise for the Reader.) Commented Jul 2, 2016 at 0:00
• See also here: milefoot.com/math/complex/exponentofi.htm It is not really a secret. Commented Jul 2, 2016 at 0:01
• This is question rally “off-topic”? Commented Sep 4, 2021 at 15:26

The basic formula is $e^{i\theta}=\cos\theta+i\sin\theta$, so your example would be
$$3^i=e^{i\ln3}=\cos(\ln3)+i\sin(\ln3)$$
$$c^{a+bi}=c^ac^{bi}=c^ae^{bi \ln (c)}$$
$$=c^a \left((\cos (b \ln (c))+ i \sin(b \ln (c) \right))$$