ABSTRACT
In this chapter, we determine the behavior of the zeroes of the function Sa(z) as a tends to a point in the boundary. We suppose that Ω is a bounded n-connected domain with C∞ smooth boundary. We know that, for a ∈ Ω the function
f(a)(z) = S(z, a)
L(z, a) (27.1)
is the Ahlfors mapping associated to Ω, which is a branched n-to-one covering map of Ω onto the unit disc (see Chapter 13). Notice that f(a)(a) = 0 because of the pole of L(z, a) at z = a and that f
′ (a)(a) is
equal to 2πS(a, a). The n-to-one map f(a) must have n− 1 other zeroes besides the one at a; these zeroes coincide with the zeroes of S(z, a) since L(z, a) is nonvanishing. As we have before, we list these zeroes (with multiplicity) a1, a2, . . . , an−1. When we want to emphasize the dependence of these zeroes on a we will write aj = Zj(a). As before, let γj , j = 1, . . . , n, denote the boundary curves of Ω.
