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 - ...;
therefore, if |r_i| = |[rho]_i|, we have
|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
and hence
|(1 - [zeta])^(-1) - U| < a^(-1) {(1 + [sigma]^n1) (1 + [sigma]^n2)^n1
(1 + [sigma]^n3)^n1n2 ... (1 + [sigma]^n_r)^(n1n2 ... n_(r-1)) - 1}.
Take an arbitrary real positive [epsilon], and [mu], a positive
number, so that [epsilon]^[mu] - 1 < [epsilon]a, then a value of n1
such that [sigma]^n1 < [mu]/(1 + [mu]) and therefore [sigma]^n1 / (1 -
[sigma]^n1 < [mu], and values for n2, n3 ... such that[sigma]^n2 <
1/n1 [sigma]^2n1, [sigma]^n3 < 1/n1n2 [sigma]^{3n1, ... [sigma]^n_r} <
1/(n1...n_(r - 1)} [sigma]^n_rn1; then, as 1 + x < e^x, we have
|(-[zeta])^(-1) - U| < a^-1 {exp([sigma]^n1 + n1[sigma]^n2 +
n1n2[sigma]^n3 + ... + n1n2...n_(r - 1)[sigma]^n_r) - 1}, and
therefore less than
a^(-1) {exp([sigma]^n1 + [sigma]^2n1 + ... + [sigma]^n_rn1) - 1},
which is less than
_ _
1 | / [sigma]^n1 \ |
-- |exp ( -------------- ) - 1 |
a |_ \1 - [sigma]^n1/ _|
and therefore less than [epsilon].
The rational function U, with a pole at [zeta] = 1 + c, differs
therefore from (1 - [zeta])^(-1), for all points outside the closed
region put about [zeta] = 1, [zeta] = l + c, by a quantity numerically
less than [epsilon]. So long as a remains the same, r and [sigma] will
remain the same, and a less value of [epsilon] will require at most an
increase of the numbers n1, n2, ... n_r; but if a be taken smaller it
may be necessary to increase r, and with this the complexity of the
function U.