Equations of Fuchsian type

Again, we focus on the standard form of the ordinary differential equation of the second order, viz., equation (2.1) 2.1 $$ \frac{{{\rm{d}}^2 u(z)}}{{{\rm{d}}z^2 }} + p(z)\frac{{{\rm{d}}u(z)}}{{{\rm{d}}z}} + q(z)u(z) = 0 $$ If all its singular points are regular singular points (including the point at infinity), see Definitions 2.4 and 2.5 on pages 12 and 24, respectively, the equation is of Fuchsian type.1 An equation of Fuchsian type therefore only has regular and regular singular points in the complex plane (including the point at infinity), see Definitions 2.4 and 2.5 on pages 12 and 24, respectively, the equation is of Fuchsian type.1 An equation of Fuchsian type therefore only has regular and regular singular points in the complex plane (including the point at infinity). This implies — as we shall see soon —that the functions p(z) and q(z) are rational functions. We divide the analysis below into two different cases, Section 3.1 and Section 3.2, depending on whether the point at infinity is a singular regular point or a regular point, respectively. The chapter ends with the displacement theorem in Section 3.3.