Complex exponentiation raises a nonzero complex number 
to a complex number 
. Its principal value is
,
where 
uses the principal branch of the natural
logarithm and 
. For real 
, 
, 
, and 
with 
, this gives
 |
(1)
|
where 
is the complex argument. Written explicitly in
terms of real parts and imaginary
parts,
![(a+bi)^(c+di)=(a^2+b^2)^(c/2)e^(-darg(a+ib)){cos[carg(a+ib)+1/2dln(a^2+b^2)]+isin[carg(a+ib)+1/2dln(a^2+b^2)]}.](/images/equations/ComplexExponentiation/NumberedEquation2.svg) |
(2)
|
An explicit example of complex exponentiation is given by
![(1+i)^(1+i)=sqrt(2)e^(-pi/4)[cos(1/4pi+1/2ln2)+isin(1/4pi+1/2ln2)].](/images/equations/ComplexExponentiation/NumberedEquation3.svg) |
(3)
|
A complex number taken to a complex number can be real. For example,
 |
(4)
|
shows that the principal value of the imaginary unit 
raised to itself is real.
For 
,
the principal value of 
can be real, as can
be seen by writing
![(ik)^(ik)=e^(-kpi/2)[cos(klnk)+isin(klnk)].](/images/equations/ComplexExponentiation/NumberedEquation5.svg) |
(5)
|
This will be real when 
, i.e., for
 |
(6)
|
for 
an integer. Since 
for 
, only 
is possible. These values give
 |
(7)
|
where 
is the principal branch of the Lambert
W-function. For 
, this simplifies to
 |
(8)
|
For 
,
1, 2, ..., these give the numeric values 1, 2.92606 (OEIS A088928),
4.30453, 5.51798, 6.63865, 7.6969, ....
