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Name: lord byron
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995

How do you raise a number to an imaginary power? How about raising a number to a complex power?

Actually, the way to understand how this is done is to represent raising a number to a power in a different way:
        x^y = e^(y * log(x))

That is, as long as we can take the logarithm of the number, and as long as we know what e to any power is (complex included) then we can raise that number to any complex power. Well, there is no problem taking the log of any positive real number - you cannot do it for zero though, and there are some problems with negative numbers - the logarithm is itself complex to start with! So we can at least do it for the positive reals. The exponen- tial of a complex number is also easy to evaluate:
  e^(a + i b) = e^a (cos(b) + i sin(b))

so for positive real x, with y = a + ib complex, the answer is:
x^(a + i b) = e^((a + i b) log(x))=x^a (cos(b log(x)) + i sin (b log(x)))

So in general, raising a number to an imaginary power leads to trigonometric functions of the logarithm of that number.


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