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

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

Replies:
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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