| - ΣΣ | cλ, μ | , |
| aλν |
which proves the lemma.
Take F(x) = 1 / [xn (1 - xa) (1 - xb) ... (1 - xl)] = ∫(x) / xn, since the sought number is its constant term.
Let ρ be a root of unity which makes ∫(x) infinite when substituted for x. The function of which we have to take the residue is
| Σ ρ-nenx ∫(ρe-x) =Σ | ρ-nenx | . |
| (1 - ρae-ax) (1 - ρbe-bx) ... (1 - ρle-lx) |
We may divide the calculation up into sections by considering separately that portion of the summation which involves the primitive qth roots of unity, q being a divisor of one of the numbers a, b, ... l. Thus the qth wave is
| Σ | ρ-nq enx | , |
| (1 - ρaq e-ax) (1 - ρbq e-bx) ... (1 - ρlq e-lx) |
which, putting 1 / ρq for ρq and ν = ½(a + b + ... + l), may be written
| Σ | ρνq eνx | , |
| (ρ½aq e½ax - ρ-½aq e-½ax) (ρ½bq e½bx - ρ-½bq e-½bx) ... (ρ½lq e½lx - ρ-½lq e-½lx) |
and the calculation in simple cases is practicable.