On the existence of analytic mappings II.
Genkō Hiromi, Hideo Mutō · Kodai Mathematical Journal · 1967
§ 2. In this section we assume that R and S have an infinite number of branch points.Put h(z)=^s°φ^R\z).Let E be the projection of all the branch points of R. Let Zo$E be an arbitrary but fixed point in the z-plane.Let U(z 0 \ U(zo)Γ}E=φ be a disk whose center is z 0 .In U(z 0 ) there exist n analytic branches of ^\z): ^(z) l9 •••, $feXz) n .Put Aι(«)=φ 5 ^^1 Wι, -, h n (z)=%o φo ^-\ z ) n .For these functions we define the fundamental symmetric polynomials:We can extend these functions over the 2-plane except E. The resulting functions denoting with the same symbols are single-valued regular functions except E. Hence h(z) satisfies the equation