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[MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
This line of section will render (— x) : 1 = e (£-1)l0 * (_x > uniform, and we shall take 
that value which is real when x is real and negative. * 
Then it is known that* 
G (x) = c(qCq + a Y cpx + . . . + a,fi n x n + . . 
This function will be an integral function, for 
o ' 
L t Va„c„ = U . =U~ = 0. 
n = :o n = co \/ ~~h &) n — x: p H 
Consider now the integral 
G{xz)e-*(-x)°-'dx. 
2 sin 7T0 J x x ' 
This integral is equal to 
2 sin 7 t9 
[ S \a IL c ll x”z"'] e x (— x) e 1 dx, or N a n z n . 
J 71 = 0 71 = 0 
That is to say, when \z \ < p, the integral represents the same function as the 
original series. For all values of \z\, the integral, provided it has a meaning, repre¬ 
sents the analytic continuation of the series. And if, when the series is divergent, we 
regard it as a command to perform the processes which lead to the integral 
sin 7 t9 J 
G (xz) e * ( — x) 0 1 dx 
we shall obtain a conception of such a divergent series which is in harmony with 
Weierstrass’ theory of functions. 
* See a paper by the author, ‘ Messenger of Mathematics,’ vol, 29, p. 105. 
