Fuchs' theorem

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In mathematics, Fuchs' theorem, named after Lazarus Fuchs, states that a second-order differential equation of the form $${\displaystyle y''+p(x)y'+q(x)y=g(x)}$$ has a solution expressible by a generalised Frobenius series when ${\displaystyle p(x)}$, ${\displaystyle q(x)}$ and ${\displaystyle g(x)}$ are analytic at ${\displaystyle x=a}$ or ${\displaystyle a}$ is a regular singular point. That is, any solution to this second-order differential equation can be written as $${\displaystyle y=\sum _{n=0}^{\infty }a_{n}(x-a)^{n+s},\quad a_{0}\neq 0}$$ for some positive real s, or $${\displaystyle y=y_{0}\ln(x-a)+\sum _{n=0}^{\infty }b_{n}(x-a)^{n+r},\quad b_{0}\neq 0}$$ 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 ${\displaystyle p(x)}$, ${\displaystyle q(x)}$ and ${\displaystyle g(x)}$.