International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 13, Pages 835-844

Notes on algebraic functions

Guan Ke-Ying1 and Lei Jinzhi2

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.


Consideration of the monodromy group of the hypergeometric equation z(1z)w+[γ(1+α+β)z]wαβw=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.