I reviewed Chapter 3 of one of the texts I used when studying complex
variables (Ruel V. Churchill, "Complex Variables and Applications",
McGraw-Hill, 1960). There I found, for z and c any complex numbers ( z not
equal 0):
z^c = e^(c* log z) (1)
Which is a multiple-valued function, since log z is a multiple-valued
function. Churchill defines the principal value of log z as:
log z = Log r + i*theta (2)
Where z = r *e^i*theta and Log r is the usual single-valued function of r
(with r a real number > 0)
Principal values of z^c from (1) are then computed using the principal
value of the log function shown in (2). Letting z = c = -0.1 (the point
explicitly displayed in the video), yields, from (1):
z^c = (-0.1)^(-0.1) = 1.197 - 0.389 i
This agrees with the result shown on the screen of the Casio fx 9860G SD
graphics calculator in the video. I also got the same value from Wolfram
Alpha (which labeled the result as the principal value). I conclude that the
author of the video was using the above principal values for x^x when x< 0.
Don
Dr. Donald G. Polvani
Adjunct Faculty, Physics, Retired
Anne Arundel Community College
Arnold, MD 21012