Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 111

as /3j. For suppose that we put ai = 1, which may be done without
loss of generality. The principal part of +(x) at the pole a, is then
AgA _ - Ai (1 + + X2 +..-),
and the composition of this with +(x) gives for the corresponding
component of f(x)
Ai (b0 + b-o +- b2.2.* ).
Hence the singularities 7ij and f/j differ by a multiplicative
constant.
Only one other general fact concerning the composition of singularities seems to be known. Borel proves that if the functions
+(x) and +r(x) are one-valued at a and /j respectively, f(x) is
also one-valued at yi,. Thus when two one-valued functions are
compounded, the resultant function is also one-valued. But
this statement, as he himself points out, must be correctly construed and will not necessarily hold true when the singular points
of the two given functions are not sets of isolated points but condense in infinite number along curves. To construct an example
in which f(x) in not one-valued, Borel makes use of the fact,
now so well known, that the decision whether the circle of convergence is or is not a natural boundary of a given series depends
upon the arguments of its coefficients. If, for instance, we take
the series
1 +~elx + e0-x2 +..
which has a radius of convergence equal to 1, by a proper choice
of the arguments 0n the circle of convergence can be made a natural boundary. Put now
(6) / - x = c + c + c22 +...,
in which the coefficients are necessarily real. Clearly the unit
circle will be a natural boundary for

(x) = c + c-eli1 + c e2iO 2 -+ '.