We fix the range of 𝜃 as 𝜃 ∈ [0,π]. Since the original black hole has S2 topology and no conical singularities, the functions γ(𝜃), α(𝜃) also obey regularity conditions at the north and south poles
2 γ(𝜃)--∼ 𝜃2 + O(𝜃3) ∼ (π − 𝜃)2 +O ((π − 𝜃)3). (26) α(𝜃)2
Similar regularity requirements apply for the scalar and gauge fields.