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DIVERGENT SERIES AND CONTINUED FRACTIONS. 149

vergent, the two sets of alternate convergents have limits which
are distinct. The conclusion is next extended by Stieltjes to the
half of the complex plane for which the real part of z is positive.
This brings him to the difficult part of his problem, the extension of the result to the other half-plane but with exclusion of the
real axis. Here, particularly, Stieltjes [26, a, ~ 30] shows his
ingenuity. He overcomes the difficulty by establishing first a
preliminary theorem which is of vital importance for sequences
of polynomials or rational fractions. The theorem is as follows.
Let fi(z), f2(z), * be a sequence of functions which are holomorphic within a given region T, and suppose that =if,(z) is uniformly convergent in some part T' of the interior of T. Then
if f1(z) + f2(z) + *... + f() has an upper limit independent of n in
any arbitrary region T' which includes T" but is contained in the
interior of T, the series IEf(z) will converge uniformly in T' and
therefore has as its limit a function which is holomorphic over the
whole interior of T*.
In the application of this theorem Stieltjes decomposes each
convergent N(z)/Dn(D ) into partial fractions,
1 + 2 +...+ M
z + a za2 z + ar
in which
M.>0, ai-, EM =c0.
From this it follows that N /Dn has an upper limit independent
of n in any closed region of the plane which does not contain a
point of the negative half-axis. If now in either the sequence
of the odd convergents or of the even convergents we denote the
nth term of the sequence by N /Dn and place
f(z) +()... + fn=)
the series 1n=lfn(z) converges uniformly in any portion of the plane
* For a further extension of this line of work, see Osgood, Annals of Math.,
ser. 2, vol. 3 (1901), p. 25.