 Both Γ ∪ Ω′ and σ("T") are compact.So there is some open set "U" containing Γ ∪ Ω′ such that σ("T") lies in the complement of "U".Any "a" in the complement of "U" has winding number "n"(Γ ∪ Ω′, "a") = 0 and the function

is holomorphic on "U".