Classification of Singularities; Nature of the Solutions in the Neighborhood of a Regular Singularity

If z = 2, is not an ordinary point, but both (2-2,) f ( z ) and ( ~ - z , ) ~ g ( r ) are analytic there, then 2, is said to be a regularsingularity, or singularity of thefirst kind.

Lastly, if 2, is neither an ordinary point nor a regular singularity, then it is said to be an irregular singularity, or singularity of the second kind. When the singularities of f ( z ) and g(z) at zo are no worse than poles 2, is said to be a singularity of rank I - 1, where I is the least integer such that both (z-z,)'f(z) and ( ~ - z , ) ~ ' g ( z ) are analytic. Thus a regular singularity is of rank zero. If either f ( z ) or g(z ) has an essential singularity at z,, then the rank may be said to be infinite.