Hi, mongolia-ga:
The seemingly simple operation of raising a complex number to a
complex power is not well-defined in general. This is a bit like the
fact that a nonzero complex number has two square roots, three cube
roots, etc., but in your case there turn out to be infinitely many
"answers".
One can raise a positive real number to a complex power, and the
result is uniquely defined.
One can raise a nonzero complex number to a (real) integer power, and
the result is again uniquely defined.
Here we have a nonzero (complex) Gaussian integer raised to a like
power, but the result is not uniquely defined. To see why, let's
write (1+i) as a power of e:
1 + i = e^( ln(2)/2 + i*(pi/4) )
But for any integer k, e^(i*(2pi*k)) = 1. In other words the
exponential function (base e) has "imaginary" period 2pi*i, and any
multiple of 2pi*i can be added to an exponent of e without changing
the result. Therefore the general way of expressing 1 + i as a power
of e is the following:
1 + i = e^( ln(2)/2 + i*(pi/4) + i*(2pi*k) )
for any integer k.
Therefore when one attempts to raise both sides to the power (1+i):
(1 + i)^(1 + i)
= e^( (1+i)*(ln(2)/2 + i*(pi/4) + i*(2pi*k)) )
= e^( (ln(2)/2 - pi/4 - 2pi*k) + i*(ln(2)/2 + pi/4 + 2pi*k) )
= e^( (ln(2)/2 - pi/4 - 2pi*k) + i*(ln(2)/2 + pi/4) )
Even after disposing of the term i*(2pi*k) in the exponent, which
would not affect the result of exponentiation, we still have the term
-2pi*k involving an infinite number of possibilities for k.
In other words there is no unique answer, but instead a countably
infinite number of possible results.
For the particular choice k = 0, the result would be:
e^( (ln(2)/2 - pi/4) + i*(ln(2)/2 + pi/4) )
which is approximately:
0.27395725383... + i * 0.58370075876...
Taken together all these possibilites form a doubly-infinite geometric
progression whose common ratio is:
e^(2pi) ~ 535.49165552...
Plotted in the complex plane these points all lie on a ray extending
from the origin in the first quadrant at angle ln(2)/2 + pi/4 radians,
or 64.857204014... degrees, to the positive real axis.
regards, mathtalk-ga |