450 
MR. E. \Y. BARNES OX INTEGRAL FUNCTIONS. 
terms which correspond to the first n zeros, we can expand log (1 — in the form 
log- 2 
log(- O - ~ -i[~) ~ ■ 
We thus obtain F ( z) = e H(:> <ji (z) — e p<: \ where, when \a H \ < 1 2; ] < | a,, +l |, P (2) is 
an absolutely convergent double series of positive and negative powers of 2, together 
with logarithmic terms. 
Now, unless 2 be in the immediate vicinity of the zeros of the function, this 
expression, considered from the point of view of divergent summable series, will be 
valid for all values of |z|. For, when \z\ > a n , the expression of log (1 — ~ j in the 
form 
L t( n 
+ • • ■ + 
SCL, 
still exists as a divergent summable series outside the circle of radius | a„ | for all points 
except those near a n . Therefore the form of P (2) exists continuously as \z\ increases, 
provided we do not cross the line of, or come within the immediate vicinity of, the 
zeros of the function. And thus, if we treat the series entering into the expression 
of P (:) as series which are summable though divergent, the expansion will he 
independent of n. 
.Now the expansion may be written 
X (p r (to) 
n 
where </>,. (to) is a function of m which depends also on r. Expand X (f>,. (to) 
m — 1 
asymptotically in a series of successive differentials of <■/>,. (n) by the Maclaurin sum 
formula, and rearrange the series. 
We shall get 
(A) a certain series of positive and negative powers of 2, each multiplying terms 
like | </v (to) dm ; and 
(B) an expansion consisting of a finite number of positive and an infinite number 
of negative powers of 2, each associated with a constant arising from a 
corresponding Maclaurin expansion. 
The other terms depend upon n and vanish identically ; the coefficient of each 
BernouiIlian number is zero. 
When we apply the processes of divergent summation which have been previously 
developed, the series which forms the group (A) of terms will reduce to a definite 
