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Q: How are imaginary exponents defined?

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Posted in Cool

There isn’t really anything to base xi on from previous rules of exponents as it is a completely new idea. Even more important, this technique is used because it recovers the usual rules for real exponents (exponents that don’t involve ), or at the very least doesn’t mess them up. Euler’s equation is an “analytic continuation” of the exponential function ( ) from the real numbers, to the complex ones. It’s not obvious, but it turns out that Euler’s equation is the only “nice” way to define complex exponents. You’ll find (at least, those people who are so inclined will find) that Euler’s equation, and in particular the method for finding imaginary exponents above, is consistent with all the rules of exponentiation.