Colloquium publications.

130

THE BOSTON COLLOQUIUM.

further conditions, shows that when the function defined by (6)
within some one of the curves is zero, the functions defined within
the other curves must also vanish.* Take m = 1, so that
(7) () = A
z - a
By a linear transformation, az + b
cz + d
any interior point of one curve may be taken as the origin and
any interior point of a second curve may be transformed simultaneously into the point at infinity without changing the character
of the series to be investigated. Now at the origin the successive
coefficients in the expansion of +O(z) into a Taylor's series are the
negative of
(A Vn V _n...
W(8) ^^na ' a' a3
n n n
while those in the expansion for z oo are
(9) An, A,!An 2,..a
Borel proves that when
lim 17A,- 0,
the coefficients (9) must vanish if those given in (8) do. Any one
of the analytic functions under discussion is therefore completely
determined by any other, the expression (7) being the intermediary
by which we pass from one to the other.
So far as yet appears, this method of continuing an analytic
function across a natural boundary is of very limited applicability.
Its significance has been made clearer by Borel's later memoir in
the Acta Mathematica. Here the rational fractions are of a less highly
specialized character, but the essential nature of the investigation
can still be exhibited without abandoning the expression (6). Let
i A < u2+1, where it denotes the nth term of a convergent series
* Cf. pp. 32-33 of his thesis or pp. 94-98 of his Theorie des fonctions.