Plane trigonometry, by S.L. Loney.

COMPLEX INDICES.

389

335. Definition of ax when a and x are any
quantities, complex or real. When a and x are real
quantities we know that
ax = elogea, (Art. 253.)
When a and x are complex the ordinary algebraic
definition of ax no longer holds.
Let us so define it that
ax = exLoga
for all values of x and a, whether real or complex.
Now, by Art. 329, Log a is many-valued and complex
when a is complex. Hence ax is many-valued and complex, so that
ax = exLoga = ex (2nfri+loga)
From Art. 305 it now follows that ax x ay =ax+Y, so that ax obeys the
ordinary algebraic law of indices.
The value of ax obtained by putting n equal to zero is
called its principal value.
Hence the principal value of ax
= exloga
x2
= 1 + x log a + 2 (log a)2 +... (by Art. 304.)
336. It may now be shewn that, if y be complex,
1 y (l+)= y _1 - y4.........
log (1 +y) =-y +2 -P -.
The proof is similar to the proof when y is real.
(Art. 256.)
It is, in general, necessary that the modulus of y be < 1;
otherwise the Binomial Theorem does not hold for complex quantities. (Art. 273.)