Local solutions of \(w''+A(z)w=0\) and branched polymorphic functions
The authors' main theorem is the following: Let \(G\) be a simply connected domain in \(C\) such that \(A(z)\) is meromorphic in \(G\). The quotient of any two local solutions of \(W'' + A(z) W(z) = 0\) is meromorphic and admits a meromorphic continuation into all of \(G\) iff at all poles \(z_0\) of \(A(z)\), if any, the Laurent expansion of \(A(z)\) has the form \(A(z) = b_0/(z-z_0)^2 + b_1/(z-z_0)^2 + b_2 + \ldots\) where \(4b_0 = 1-m^2\), \(m\) an integer \(\geq2\) and \[ D(z_ 0) := \left|\begin{matrix} 1-m & 0 & \ldots & \ldots & 0 & b_1 \\ b_1 & 4-2m &&&& b_2 \\ b_2 & b_1 &\ddots&& (0) & b_3 \\ \vdots & \vdots && \ddots && \vdots \\ b_{m-2} & b_{m-3} & \ldots & \ldots & (m-1)^2 -(m-1)m & b_{m-1} \\ b_{m-1} & b_{m-2} & \ldots & \ldots & b_1 & b_ m \end{matrix}\right| \] Moreover, these continuations \(g\) of local quotients all satisfy the differential equation \(S_ g = 2A(z)\) in \(G\). Here \(S_ g\) is the Schwarzian derivative of \(g\), i.e. \(S_ g = (g''/g')^1 - (g''/g')^2\).
scientific article; zbMATH DE number 4063525
scientific article