Fuchs' theorem
# 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 $y_0$ 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 $g(x)$.

