in which there are equalities among ρ1, ρ2, ... ρk, is of the form
Σρ1 − Σρ1ρ2 + Σρ1ρ2ρ3 − ...;
therefore, if |ri| = |ρi|, we have
| |P| < Σ r1 + Σ r1r2 + Σ r1r2r3 + ... < (1 + r1) (1 + r2)...(1 + rk) − 1; |
now, so long as ζ is without the closed curve above described round ζ = 1, ζ = 1 + c, we have
| | | 1 | | < | 1 | , | | cm − cm−1 | | < | c/r | < σ, |
| 1 − ζ | a | cm − ζ | a |
and hence
| |(1 − ζ)−1 − U| < a−1 {(1 + σn1) (1 + σn2)n1 (1 + σn3)n1n2 ... (1 + σnr)n1n2 ... nr−1 − 1}. |
Take an arbitrary real positive ε, and μ, a positive number, so that εmu − 1 < εa, then a value of n1 such that σn1 < μ/(1 + μ) and therefore σn1/(1 − σn1 < μ, and values for n2, n3 ... such that σn2 < 1/n1 σ2n1, σn3 < 1/n1n2 σ3n1, ... σnr < 1/(n1 ... nr−1) σnr n1; then, as 1 + x < ex, we have
| |(−ζ)−1 − U| < a−1 {exp (σn1 + n1σn2 + n1n2σn3 + ... + n1n2 ... nr−1σnr) − 1}, |