Colloquium publications.

132

THE BOSTON COLLOQUIUM.

Since the series is uniformly convergent, it can be integrated
term by term. Clearly also the numerators A, in (6) can be so
conditioned that the term-by-term derivative of (6) shall be
uniformly convergent. Then the derivative of >(z) is coincident
with the derivative of the series. It is even possible to so choose
the A, that the series will be unlimitedly differentiable.
I may add that in any region of the plane there will be an
infinite or, more specifically, a non-enumerable set of points,
through each of which passes an infinite number of lines of convergence. If a closed curve is given it will be possible to
approximate as closely as desired to this curve by a rectilinear
polygon, along whose entire length the series converges and defines
a continuous function. Integration around such a polygon gives
for the value of the integral the product of 2i7r into the sum of
the residues of those fractions whose poles lie in the interior of
the polygon. Finally, if we take for axes of x and y two perpendicular lines of continuity of O(z), all the lines of uniform continuity
which meet at their intersection will give a common value for +'(z),
and the real and imaginary parts of 4(z) will satisfy Laplace's
equation:
a2U 102U
ax2 + Vy2 = 0.
Thus we have in +(z) a species of quasi-monogenic function.
One question Borel has as yet found himself unable to resolve.
If O(z) = 0 along a finite portion of any line, will the series in
consequence vanish identically? If this question be answered in
the affirmative, the analogy with an ordinary analytic function
will be still more complete.
Let us now return to the case in which two or more functions
with natural boundaries are defined by (7). The lines of continuity just described form an infinitely thick mesh-work along
which +(z) can be carried continuously from the one analytic
function into the others. Suppose again that the origin is not a
point of condensation of the poles a. so that +(z) can be expanded