| 1 | { 1 − ( | c1 − 1 | ) n1 } |
| 1 − ζ | c1 − ζ |
is finite at ζ = 1, and has a pole of order n1 at ζ = c1; the rational function
| 1 | { 1 − ( | c1 − 1 | ) n1 } { 1 − ( | c2 − c1 | ) n2 } n1 |
| 1 − ζ | c1 − ζ | c2 − ζ |
is thus finite except for ζ = c2, where it has a pole of order n1n2; finally, writing
| xs = ( | cs − cs−1 | ) ns, |
| cs − ζ |
the rational function
| U = (1 − ζ)−1 (1 − x1) (1 − x2)n1 (1 − x3)n1n2 ... (1 − xr)n1n2 ... nr − 1 |
has a pole only at ζ = 1 + c, of order n1n2 ... nr.
The difference (1 − ζ)−1 − U is of the form (1 − ζ)−1P, where P, of the form
1 − (1 − ρ1) (1 − ρ2)...(1 − ρk),