Plane trigonometry, by S.L. Loney.

384

TRIGONOMETRY.

By equating real and imaginary parts, we have
el cos y = r cos (2n7r + 0),
and ex sin y = r sin (2n7r + 0).
Hence e = r, and y = 2nrr + 0.
Since x and r are both real, x is the ordinary algebraic
Napierian logarithm of r, so that
x = log, r.
Hence a logarithm of a + /3i is
loge r + i (2nr + 0),
%.e. loge /a2 + /2 + i (2n7r + tan-l ).
Since n is any integer we see that there are therefore
an infinite number of logarithms of a + /3i, and that these
only differ by multiples of 27ri.
329. With the extended definition of a logarithm
given in Art. 327, it follows by the last article that the
logarithm of any number is many-valued.
When this many-valuedness is taken into consideration
we write the logarithm of a + f/i as Log (a +,i).
Hence
Log (a + /i) = log /a2 + 2 + i (2nr + tan-1 ).
If we put n equal to zero in the value of Log (a + 3i)
the result is called the principal value of the logarithm
and is denoted by log (a + 83i), so that
log (a + 3i) = log, V(a2 + 32) + i tan-l I
and
Log (a + 13i) = 2n7ri + log (a + /3i).