c[zeta] (c + 1)z
z = --------------, [zeta] = --------;
c + 1 - [zeta] c + z
thereby the points [zeta] = 0, 1, 1 + c become the points z = 0, 1,
[oo], the function (1 - z)^(-1) being given by (1-z)^(-1) = c(c +
1)^(-1)(1 - [zeta])^(-1) + (c + 1)^(-1); the function U becomes a
rational function of z with a pole only at z = [oo], that is, it
becomes a polynomial in z, say [(c + 1)/c]H - 1/c, where H is also a
polynomial in z, and
_ _
1 c | 1 |
----- - H = ----- | ---------- - U |;
1 - z c + 1 |_1 + [zeta] _|
the lines [eta] = [+-]a become the two circles expressed, if z = x +
iy, by
c(c + 1)
(x + c)^2 + y^2 = [+-] -------- y,
a
the points ([eta] = 0, [xi] = 1 - a), ([eta] = 0, [xi] = 1 + c + a)
become respectively the points (y = 0, x = c(1 - a)/(c + a), (y = 0, x
= -c(l + c + a)/a), whose limiting positions for a = 0 are
respectively (y = 0, x = 1), (y = 0, x = -[oo]). The circle (x + c)^2 +
y^2 = c(c + 1)y/a can be written
(x + c)^2 (x + c)^4
y = --------- + --------- {[mu] + [root][[mu]^2 - (x + c)^2]}^(-2),
2[mu] 2[mu]
where [mu] = 1/2c(c + 1)/a; its ordinate y, for a given value of x,
can therefore be supposed arbitrarily small by taking a sufficiently
small.
We have thus proved the following result; taking in the plane of z any
finite region of which every interior and boundary point is at a
finite distance, however short, from the points of the real axis for
which 1 =< x =< [oo], we can take a quantity a, and hence, with an
arbitrary c, determine a number r; then corresponding to an arbitrary
[epsilon]_s, we can determine a polynomial P_s, such that, for all
points interior to the region, we have
|(1 - z^(-1)) - P_s| < [epsilon]_s;
thus the series of polynomials