Fuchs's 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 ${\displaystyle p(x)}$, ${\displaystyle q(x)}$ 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

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