ME. E. W. BARNES ON INTEGRAL FUNCTIONS. 
429 
§ 25. We now enquire whether the domain of existence of the integral is coextensive 
with the domain of existence of the analytic function defined by the original series. 
Just as the series ceases to define the function by becoming divergent, so the integral 
may cease to be an adequate expression by becoming infinite. 
Consider the series 1 + 2 + z 2 + . . . + z n + . . . 
The “ sum ” of this series, when divergent, is represented by the integral 
2 sin Ve f 6 W < - *r ‘ dx ’ in which G 04 = j, 
and the integral is taken round some contour embracing an axis in the positive half 
of the 2 -plane, 
Make now 6 tend to unity. Then G (xz) becomes e x ~, and the integral becomes 
e ~* (1_j) dx, taken along some line in the positive half of the 2 -plane. 
J 0 
Suppose now that x = pe L \ z = 1 -f- rd* where 6 and <j) are both in absolute value 
not greater than tt. Since the axis of the integral lies in the positive half of the 
77* 77* 
z-plane, — — e > d > — ( -fie, where e is a positive quantity as small as we please. 
The amplitude of x (z — 1) is 6 -f- <f>, and that the integral may be finite this quantity 
must be such that (z — 1) is negative. Therefore — > 6 </>>^ or — *> . 
These conditions can always be satisfied by values of 6 within the assigned range, 
if </> does not lie between or at the limits of the range bounded by e and — e. 
We thus see that the function 1/(1 — z) is represented by the series 
1 + 2 + 2 2 + . . . + Z» + . . . 
within a circle of radius unity; and by the integral 
sin 7 t6 
(xzY 
i=o T(n + 6) 
e x (— x)° 1 dx 
for all values of 2 except those which lie on that part of the real axis between the 
points 1 and oo. 
00 yll 
§ 2G. Similarly the series S -or its integral equivalent when it is divergent, 
will represent -- z~ l log (1 — z), provided 2 does not lie on that part of the real axis 
between 1 and oo . And the same is true of (l — z)~ m and its equivalent series, when 
m is not necessarily an integer. These statements form easy examples which the 
reader can at once work out for himself. 
It is interesting to notice that the lines from the singularities to infinity intervene 
to give uniformity to the non-uniform functions to which divergent series may 
