Hypergeometric differential equation

Summary

The hypergeometric differential equation is the linear equation $z(1-z)w'' + [c - (a+b+1)z]w' - ab\,w = 0$. It is a Fuchsian equation with exactly three regular singular points, at $0$, $1$ and $\infty$. The hypergeometric function is its solution that is holomorphic at $0$ with value $1$ there. Every second-order Fuchsian equation with three singular points reduces to this form. One uses a Möbius change of variable and a change of the unknown function.