73] LIMIT« OF FUNCTIONS OF AN INTEGRAL VARIABLE 151 
21. <£„(#) = (« cos 2 a’tt-f 6 sin 2 .t-Tr)’ 1 . [Here 0 (x) = 0 if |a cos 2 xn + 5 sin 2 xn \ 
<1, 0 (x) = 1 if a cos 2 xn + 5 sin 2 xn = 1, and 0 (x) is otherwise undefined. 
For what values of x these respective conditions are satisfied depends on 
the values of a and 5. Thus if a and 5 are both numerically less than unity, 
0 (x) = 0 for all values of x. Consider, e.g., the cases « = 5 = 1; a = h = \ ; 
« = 5 = 2; « = 1, 5 = 2; « = 2, 5=1; « = 2, 5=^.] 
22. If iF^1752, the number of days in the year N A.D. is 
lim (365 + (cos 2 \ jVtt ) u — (cos 2 T ^jiV7r) n + (cos 2 
73. Limits of Complex functions and series of Complex 
terms. In this chapter we have, up to the present, concerned 
ourselves only with real functions of n and series all of whose 
terms are real. There is however no difficulty in extending our 
ideas and definitions to the case in which the functions or the 
terms of the series are complex. 
Suppose that <£ {n) is complex and equal to 
R (n) + iS (n), 
where R (n), S (n) are real functions of n. Then if, as n -*■ oo ,R (n) 
and S (n) converge respectively to limits r and s, we shall say that 
cf) (n) converges to the limit r + is, and write 
lim cf) (n) = r + is. 
Similarly if u n is complex and equal to v n + iw n we shall say that 
the series 
iii + + u 3 -f ... 
is convergent and has the sum r + is, if the series 
Vi + V 2 + V 3 + ..., Wj + w 2 + w 3 + ... 
are convergent and have the sums r, s respectively. 
To say that u { +u. 2 +u 3 +... is convergent and has the sum 
r + is is of course the same as to say that the sum 
s n = Ui + u 2 + ... + u n = {Vi + v 2 + ... 4- v n ) + i (w } + w 2 + ... + w n ) 
converges to the limit r + is as n -*■ oo . 
In the case of real functions and series we also gave definitions 
of divergence and oscillation (finite or infinite). But in the case 
of complex functions and series there are so many possibilities— 
e.g. R (n) may tend to + oo and S (n) oscillate—that this is 
hardly worth while. When it is necessary to make further dis 
tinctions of this kind, we shall make them by stating the way in 
which the real or imaginary parts behave when taken separately.