International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 13, Pages 835-844
doi:10.1155/S0161171203110186
Notes on algebraic functions
1Department of Mathematics, Northern Jiaotong University, Beijing 100044, China
2Department of Mathematical Science, Tsinghua University, Beijing 100084, China
Received 11 October 2001
Copyright © 2003 Guan Ke-Ying and Lei Jinzhi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.