and therefore less than
a−1 {exp (σn1 + σ2n1 + ... + σnr n1) − 1},
which is less than
| 1 | [ exp ( | σn1 | ) − 1 ] |
| a | 1 − σn1 |
and therefore less than ε.
The rational function U, with a pole at ζ = 1 + c, differs therefore from (1 − ζ)−1, for all points outside the closed region put about ζ = 1, ζ = l + c, by a quantity numerically less than ε. So long as a remains the same, r and σ will remain the same, and a less value of ε will require at most an increase of the numbers n1, n2, ... nr; but if a be taken smaller it may be necessary to increase r, and with this the complexity of the function U.
Now put
| z = | cζ | , ζ = | (c + 1)z | ; |
| c + 1 − ζ | c + z |
thereby the points ζ = 0, 1, 1 + c become the points z = 0, 1, ∞, the function (1 − z)−1 being given by (1 − z)−1 = c(c + 1)−1 (1 − ζ)−1 + (c + 1)−1; the function U becomes a rational function of z with a pole only at z = ∞, 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 | − H = | c | [ | 1 | − U ]; |
| 1 − z | c + 1 | 1 − ζ |