λ(z − a)−1 + μ + Σ n=1 μn (z − a)n,
the expansion near z = b is
−λ (z − b)− 1 + μ + Σ n=1 (−1)n μn (z − b)n,
as we see by remarking that if z′ − b = −(z − a) the function has the same value at z and z′; hence the differential equation satisfied by the function is easily calculated in terms of the coefficients in the expansions.
From the function ℜ(z) we can obtain another function, termed the Zeta-function; it is usually denoted by ζ(z), and defined by
| ζ(z) − | 1 | = ∫ π0 [ | 1 | − ℜ(z) ] dz = Σ′ ( | 1 | + | 1 | + | z | ), |
| z | z2 | z − Ω | Ω | Ω2 |
for which as before we have equations
| ζ(z + ω) = ζ(z) + 2πiη, ζ(z + ω′) = ζ(z) + 2πiη′, |
where 2η, 2η′ are certain constants, which in this case do not both vanish, since else ζ(z) would be a doubly periodic function with only one pole of the first order. By considering the integral
∫ ζ(z)dz