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
$y = \sum_^\infty a_n (x - a)^, \quad a_0 \neq 0$
for some positive real ''s'', or
$y = y_0 \ln(x - a) + \sum_^\infty b_n(x - a)^, \quad b_0 \neq 0$
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 $g(x)$.

See also

* Frobenius method

References

* .
* {{Citation , last=Butkov , first=Eugene , title=Mathematical Physics , location=Reading, MA , publisher=Addison-Wesley , year=1995 , isbn=0-201-00727-4 .

Differential equations
Theorems in analysis