# Fuchs' theorem

(Redirected from Fuchs's 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 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 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 $g(x)$.

## References

• Asmar, Nakhlé H. (2005), Partial differential equations with Fourier series and boundary value problems, Upper Saddle River, NJ: Pearson Prentice Hall, ISBN 0-13-148096-0.