H1(z) + { H2(z) - H1(z)} + { H3(z) - H2(z)} + ...
then, as before, represents [f](z) within R0. And R0 may be taken to coincide as nearly as desired with the interior of the region bounded by [Gamma].
§ 11. _Expression of (1 - z)^(-1) by means of Polynomials. Applications._--We pursue the ideas just cursorily explained in some further detail.
Let c be an arbitrary real positive quantity; putting the complex variable [zeta] = [xi] + i[eta], enclose the points [zeta] = l, [zeta] = 1 + c by means of (i.) the straight lines [eta] = ±a, from [xi] = l to [xi] = 1 + c, (ii.) a semicircle convex to [zeta] = 0 of equation ([xi] - 1)² + [eta]² = a², (iii.) a semicircle concave to [zeta] = 0 of equation ([xi] - 1 - c)² + [eta]² = a². The quantities c and a are to remain fixed. Take a positive integer r so that 1/r (c/a) is less than unity, and put [sigma] = 1/r (c/a). Now take
c1 = 1 + c/r, c2 = 1 + 2c/r, ... c_r = 1 + c;
if n1, n2, ... n_r, be positive integers, the rational function _ _ 1 | / c1 - 1 \^n1 | ---------- |1 - ( ----------- ) | 1 - [zeta] |_ \ c1 - [zeta] / _|
is finite at [zeta] = 1, and has a pole of order n1 at [zeta] = c1; the rational function _ _ _ _ 1 | / c1 - 1 \^n1 | | / c2 - c1 \^n2 |^n1 ---------- |1 - ( ------------ ) | |1 - ( ----------- ) | 1 - [zeta] |_ \ c1 - [zeta] / _| |_ \ c2 - [zeta]/ _|
is thus finite except for [zeta] = c2, where it has a pole of order n1n2; finally, writing
/ c_s - c_(s-1) \^n_s x_s = ( ------------- ), \ c_s - [zeta] /
U = (1 - [zeta])^(-1) (1 - x1)(1 - x2)^n1 (1 - x3)^n1n2 ... (1 - x_r)^(n1n2 ... n_(r - 1))
has a pole only at [zeta] = 1 + c, of order n1n2 ... n_r.
The difference (1 - [zeta])^(-1) - U is of the form (1 - [zeta])^(-1)P, where P, of the form
1 - (1 - [rho]1)(1 - [rho]2)...(1 - [rho]_k),
in which there are equalities among [rho]1, [rho]2, ... [rho]_k, is of the form
[Sigma][rho]1 - [Sigma][rho]1[rho]2 + [Sigma][rho]1[rho]2[rho]3 - ...;
|P| < [Sigma]r1 + [Sigma]r1r2 + [Sigma]r1r2r3 + ... < (1 + r1)(1 + r2)...(1 + r_k) - 1;
now, so long as [zeta] is without the closed curve above described round [zeta] = 1, [zeta] = 1 + c, we have
| 1 | 1 |c_m - c_(m-1)| c/r |----------| < ---, |-------------| < --- < [sigma], |1 - [zeta]| a |c_m - [zeta] | a