MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
433 
Now 2 “" G, i+1 ( 2 , x) is an absolutely convergent series, and | z~ n G J)+1 (zx)\ tends to 
zero as n tends to infinity. 
Moreover z~ n G n+1 (zx) and G (xz) are functions which, while n has any finite value, 
have the same character near \xz\ — oo , for they only differ by the polynomial 
a 0 c 0 + a x c^zx + . . . + (zx) . 
Therefore the 
integral 
: a [ Gri + 1 e x (— xY~ n 1 dx will represent an 
2 sin 7T0 J z n v ’ r 
analytic function of z whenever 0 ^ j G (xz) e * (— x) e 1 dx does so; and the 
two functions will have the same character near z = 0. 
Therefore, within those areas for which the second integral represents the “ sum” 
of the given asymptotic series, the first integral is finite, and | z~ n {f(z) — 5,,} j tends 
to zero as \z\ tends to zero. 
Thus the asymptotic equality satisfies Poincare’s arithmetic definition. The 
reader must note very carefully that this theorem does not apply to divergent series 
which have a finite radius of convergence. It is necessary that 1 2 1 should tend to 
zero. No computer, for instance, could make 1 — 2 -j- 2 3 — 2 3 + . . . tend to ' 
§ 31. Suppose now that we differentiate the series 
« 0 + « x 2 + • . . + Cl n Z’ 1 + . . . 
in which 
L t y/a n = co , and L£ — o. 
?l = co n—zo % 
/ 
We shall obtain the series 
cq + 2 a 2 z + . . . + na n z n ~ l . . . 
If we “ sum ” this series by the exponential process (the name which it is convenient 
to give to the process employed in the preceding paragraphs) we obtain the integral 
^ ~—- Gi (xz) e~- c (— x) e ~ l dx, in which 
2 sm 7 tu J 1 v v ' 
G x (xz) — a l (\ + 2 a. 2 c. 2 xz + . . . + na n c n (xz) d 1 + . 
We thus see that, since this series is an integral function, 
G x (xz) = G (xz). 
Therefore the “ sum ” of the series ct l + 2 ct. 2 z + • • • + na u z n 1 + . . 
3 K 
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