single-valued
monogenic branch of an analytical function existing and without
singularities all over this region. If, now, the values of [sigma]
that so arise be plotted on to another plane, a value p + iq of
[sigma] being represented by a point (p, q) of this [stigma]-plane, and
the value of x from which it arose being mentally associated with this
point of the [sigma]-plane, these points will fill a connected region
therein, with a continuous boundary formed of four portions
corresponding to the two sides of the two barriers of the x-plane. The
question is then, firstly, whether the same value of s can arise for
two different values of x, that is, whether the same point (p, q) of
the [sigma]-plane can arise twice, or in other words, whether the
region of the [sigma]-plane overlaps itself or not. Supposing this is
not so, a second part of the question presents itself. If in the
x-plane the barrier joining -[oo] to 0 be momentarily removed, and x
describe a small circle with centre at x = 0 starting from a point x =
-h - ik, where h, k are small, real, and positive and coming back to
this point, the original value s at this point will be changed to a
value [sigma], which in the original case did not arise for this value
of x, and possibly not at all. If, now, after restoring the barrier
the values arising by continuation from [sigma] be similarly plotted
on the s-plane, we shall again obtain a region which, while not
overlapping itself, may quite possibly overlap the former region. In
that case two values of x would arise for the same value or values of
the quotient y2/y1, arising from two different branches of this
quotient. We shall understand then, by the condition that x is to be a
single-valued function of x, that the region in the [stimga]-plane
corresponding to any branch is not to overlap itself, and that no two
of the regions corresponding to the different branches are to overlap.
Now in describing the circle about x = 0 from x = -h - ik to -h + ik,
where h is small and k evanescent,

[stigma] = x^[lambda]1 F([lambda] + [lambda]1, [mu] + [lambda]1, 1 + [lambda]1, x)/F([lambda], [mu], 1 - [lambda]1, x)

is changed to [sigma] = [stigma]e^(2[pi]i[lambda])1. Thus the two
portions of boundary of the s-region corresponding to the two sides of
the barrier (-[oo], 0) meet (at [sigmaf] = 0 if the real part of
[lambda]1 be positive) at an angle 2[pi]