Fuchs' theorem

In mathematics, Fuchs' theorem, named after Lazarus Fuchs, states that a second order differential equation of the form

has a solution expressible by a generalised Frobenius series when p ( x ) , q ( x ) and g ( x ) are analytic at x = a or a is a regular singular point. That is, any solution to this second order differential equation can be written as

for some real s, or

for some real r, where y₀ is a solution of the first kind.

Its radius of convergence is at least as large as the minimum of the radii of convergence of p ( x ) , q ( x ) and g ( x ) .