Fuchs' theorem

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

+ p ( x )

+ q ( x ) y = g ( x )

has a solution expressible by a generalised Frobenius series when

are analytic at

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

ln ⁡ ( x − a ) +

{\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