Plane trigonometry, by S.L. Loney.

386

TRIGONOMETRY.

Hence the principal value of the logarithm of a
negative quantity -x (with our extended definition) is
equal to the ordinary algebraic logarithm of x added
on to 7ri.
332. Logarithm of a quantity which is wholly imaginary. In the result of Art. 329 put a = 0, and we have
Log (3i) = 2n7ri + log,, + i r
= loge3+i (2n + \r,
so that the logarithm of any quantity which is wholly
imaginary, consists of two parts, the first of which is real,
and the second of which is imaginary and many-valued.
As a particular case, put 3= 1, and we have
Log (/-l)= i (2 + r,
so that the principal value of Log (V- 1) is i.
333. In the result of Art. 329 put
a = cos 0 and 3 = sin 0... Log (cos 0 + i sin 0)
= loge 1 + i (2n7r + 0) = i + 2n7ri,
Log e1 =0 i + 2n7ri.
The principal value of Log e~i, i.e. log e~0, is therefore
that value of (0 + 2n7r)i which is such that '0 + 2n7r lies
between - r and + 7r.