Plane trigonometry, by S.L. Loney.

LOGARITHMS OF COMPLEX QUANTITIES.

383

We say "a" logarithm because, as we shall now shew,
there are with the above definition many logarithms of a
quantity.
W e have a + /3i e yi.................. (1).
Now, by Art. 308, we have, for all integral values of n,
e2nTi = cos 2n7r + i sin 2n7r = 1............(2).
Hence from (1) and (2) we have, by Art. 305,
a +,i - ex+i. e2nri = e+ (y+27r) i
According to the above definition we see that, if x + yi
be a logarithm of a + /3i, so also is
x + yi + 2nri, i.e. x + (y + 2n7r) i.
328. We proceed to find the logarithms of the
complex quantity a + /i, where a and 3 are real.
By Art. 267, we have
a + /i = r [cos (2n7r + 0) + i sin (2n7r + 0)]
where n is any integer, r = + /a2 + /32, and 0 is that value
lying between - r and + wr such that cos 0 is - and sin 0
r
is Q, i.e. with the restriction of Art. 267,
r
0= tan- _.
a
If x + yi be a logarithm of a + /i, we have then
r [cos (2n7r + 0) + i sin (2nr + 0)] = ex+yi
= ex. eyi (Art. 305)
= ex (cos y + i sin y).