Plane trigonometry, by S.L. Loney.

CHAPTER XXV.

EXPONENTIAL SERIES FOR COMPLEX QUANTITIES. CIRCULAR FUNCTIONS FOR COMPLEX ANGLES. HYPERBOLIC
FUNCTIONS.
302. WHEN x is a real quantity we have proved in
Art. 253 that
X2 X3
ex=1 + ~++ 3 + ad inf...............(l).
When x is not real but is complex, i.e. of the form
a + b ^ -1, the expression ex has no meaning at present.
Let us so define it that for all values of x (whether
real or complex) it shall mean the series
X2 X3
1+x X 2+ 3 + ad inf..............(2).
303. We can easily shew that this series is convergent
when x is complex.
For let x = r (cos 0 + /-1 sin 8).