Fuchs's theorem

{{Short description|Mathematical theorem}}

{{no footnotes|date=June 2017}}

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

$$y\text{'}\text{'}\; +\; p(x)y\text{'}\; +\; q(x)y\; =\; g(x)$$

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

$$y\; =\; \backslash sum\_\{n=0\}^\backslash infty\; a\_n\; (x\; -\; a)^\{n\; +\; s\},\; \backslash quad\; a\_0\; \backslash neq\; 0$$

for some positive real s, or

$$y\; =\; y\_0\; \backslash ln(x\; -\; a)\; +\; \backslash sum\_\{n=0\}^\backslash infty\; b\_n(x\; -\; a)^\{n\; +\; r\},\; \backslash quad\; b\_0\; \backslash neq\; 0$$

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 $g(x)$.