Plane trigonometry, by S.L. Loney.

390

TRIGONOMETRY.

If the modulus of y be equal to unity, so that y may be put equal to
cos. + i sin 0, the expansion can be shewn to be still true, except in the
cases when 0 is equal to an odd multiple of 7r.
Since Log (1 + y) = 2n7ri + log (1 + y),
we have
Log (1 + y) = 2n7ri + y-2 Yy2 - y4......
337. To separate into its real and imaginary parts
the expression (a + /i)+yi.
Let a + 3i = r (cos a + i sin 0),
so that as in Art. 265,
r = V/a2 + 32, and 0 = tan-l
Then, by definition,
(a + i)xz+i = e(x+Yi) Log (a+3i)
= e{x+yi} {log (a+ii)+2mirit
= e{x+yi} {logr+ (O+2m7r) i
= { logr-y (0+2m7r) }+i {ylogr+x (0+2m,7) }
= exlogr. e-y(e0+2mn ) ei{ylogr+x(0+2m7r))
= rx. e-y(+2mr) [cos {y log r + x (0 + 2m7r)}
+ i sin {y log r + x (0 + 2m7r)}].
If we put m equal to zero, we obtain the principal
value of the given quantity, viz.
rxe-yO [cos (y log r + x0) + i sin (y log r + x0)].
338. Ex. 1. Find the general value of [/- 1]/-1.
We have [ / -1]^/ 1= en/-1o-Lg o l.