The Nineteenth Week: Fuchsian Differential Equations

[A] Let D be the region obtained by removing the points a l, a 2,…, a n from the complex plane: D = C - {a l, a 2,…, a n }. It can also be obtained by removing the n + 1 points {a 1, a 2,…, a n, a n+1 = ∞} from the Riemann sphere C U {∞ }. Let z: $$\tilde D \to D$$ be the universal covering surface of D. We take a “5-yen coin” Ua: around each a i (defined at the beginning of Week 18).Let $${\tilde U_{ai,1}},{\tilde U_{ai,2}},{\tilde U_{ai,3}}, \cdots $$ be the connected components of the open subset $${z^{ - 1}}\left( {{U_{ai}}} \right)$$ of $${\tilde D}$$ .Then each z : $${\tilde U_{ai,j}} \to {U_{ai}}$$ is a covering of U ai ,.We call it a spiral staircase which covers U ai .