Complex Exponentiation

A complex number may be taken to the power of another complex number. In particular, complex exponentiation satisfies

(1)

where  is the complex argument. Written explicitly in terms of real and imaginary parts,

(2)

An explicit example of complex exponentiation is given by

(3)

A complex number taken to a complex number can be real. In fact, the famous example

(4)

shows that the power of the purely imaginary  to itself is real.

Min   Max       Re       Im

In fact, there is a family of values  such that  is real, as can be seen by writing

(5)

This will be real when , i.e., for

(6)

for  an integer. For positive , this gives roots  or

(7)

where  is the Lambert W-function. For , this simplifies to

(8)

For , 2, ..., these give the numeric values 1, 2.92606 (OEIS A088928), 4.30453, 5.51798, 6.63865, 7.6969, ....

REFERENCES:

Sloane, N. J. A. Sequence A088928 in "The On-Line Encyclopedia of Integer Sequences."