Fuchs' theorem

In mathematics, Fuchs' theorem, named after Lazarus Fuchs, states that a second-order differential equation of the form $$y'' + p(x)y' + q(x)y = g(x)$$ 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 $$ y = \sum_{n=0}^\infty a_n (x - a)^{n + s}, \quad a_0 \neq 0$$ for some positive real s, or $$ y = y_0 \ln(x - a) + \sum_{n=0}^\infty b_n(x - a)^{n + r}, \quad b_0 \neq 0$$ for some positive 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 $$g(x)$$.