A complex number may be taken to the power of another complex number. In particular, complex exponentiation satisfies
(1)
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where is the complex argument. Written explicitly in terms of real and imaginary parts,
(2)
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An explicit example of complex exponentiation is given by
(3)
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A complex number taken to a complex number can be real. In fact, the famous example
(4)
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shows that the power of the purely imaginary to itself is real.
In fact, there is a family of values such that is real, as can be seen by writing
(5)
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This will be real when , i.e., for
(6)
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for an integer. For positive , this gives roots or
(7)
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where is the Lambert W-function. For , this simplifies to
(8)
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For , 2, ..., these give the numeric values 1, 2.92606 (OEIS A088928), 4.30453, 5.51798, 6.63865, 7.6969, ....