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.
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1$\begingroup$ Have a look here: math.toronto.edu/mathnet/questionCorner/complexexp.html $\endgroup$– BilbottomCommented Jul 1, 2016 at 23:56
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2$\begingroup$ 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.) $\endgroup$– hardmathCommented Jul 2, 2016 at 0:00
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1$\begingroup$ See also here: milefoot.com/math/complex/exponentofi.htm It is not really a secret. $\endgroup$– callculus42Commented Jul 2, 2016 at 0:01
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$\begingroup$ This is question rally “off-topic”? $\endgroup$– ThomasMcLeodCommented Sep 4, 2021 at 15:26
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2 Answers
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The basic formula is $e^{i\theta}=\cos\theta+i\sin\theta$, so your example would be
$$3^i=e^{i\ln3}=\cos(\ln3)+i\sin(\ln3)$$
answered Jul 1, 2016 at 23:56
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Using euler's formula :
$$c^{a+bi}=c^ac^{bi}=c^ae^{bi \ln (c)}$$
$$=c^a \left((\cos (b \ln (c))+ i \sin(b \ln (c) \right))$$
answered Jul 2, 2016 at 0:09