the lines η = ±a become the two circles expressed, if z = x + iy, by
| (x + c)² + y² = ± | c(c + 1) | y, |
| a |
the points (η = 0, ξ = 1 − a), (η = 0, ξ = 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 = −∞). The circle (x + c)² + y² = c(c + 1)y/a can be written
| y = | (x + c)² | + | (x + c)4 | {μ + √[μ² − (x + c)²]}−2, |
| 2μ | 2μ |
where μ = ½c(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 ⋜ ∞, we can take a quantity a, and hence, with an arbitrary c, determine a number r; then corresponding to an arbitrary εs, we can determine a polynomial Ps, such that, for all points interior to the region, we have
|(1 − z−1) − Ps| < εs;
thus the series of polynomials
P1 + (P2 − P1) + (P3 − P2) + ...,
constructed with an arbitrary aggregate of real positive numbers ε1, ε2, ε3, ... with zero as their limit, converges uniformly and represents (1 − z)−1 for the whole region considered.