International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 13, Pages 835-844
doi:10.1155/S0161171203110186

Notes on algebraic functions

Abstract

Consideration of the monodromy group of the hypergeometric equation z(1−z)w″+[γ−(1+α+β)z]w′−αβw=0, in the case of α=1/6, β=5/6, γ=7/6, shows that the global hypergeometric function solution F(1/6;5/6;7/6;z) is nonalgebraic although it has only algebraic singularities. Therefore, the proposition given in [2,4] that a function is algebraic if it has only the algebraic singularities on the extended z-plane is not true. Through introduction of the concept of singular element criterion for deciding when a function is algebraic on the basis of properties of its singularities is given.