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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1Have a look here: https://www.math.toronto.edu/mathnet/questionCorner/complexexp.html – Bilbottom Jul 01 '16 at 23:56
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2Provided 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.) – hardmath Jul 02 '16 at 00:00
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1See also here: http://www.milefoot.com/math/complex/exponentofi.htm It is not really a secret. – callculus42 Jul 02 '16 at 00:01
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This is question rally “off-topic”? – ThomasMcLeod Sep 04 '21 at 15:26
2 Answers
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The basic formula is $e^{i\theta}=\cos\theta+i\sin\theta$, so your example would be
$$3^i=e^{i\ln3}=\cos(\ln3)+i\sin(\ln3)$$
Barry Cipra
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Using euler's formula :
$$c^{a+bi}=c^ac^{bi}=c^ae^{bi \ln (c)}$$
$$=c^a \left((\cos (b \ln (c))+ i \sin(b \ln (c) \right))$$
Ahmed S. Attaalla
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