Fuchs's theorem

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

y ″ + p ( x ) y ′ + q ( x ) y = g ( x ) {\displaystyle y''+p(x)y'+q(x)y=g(x)}

has a solution expressible by a generalised Frobenius series when p ( x ) {\displaystyle p(x)} , q ( x ) {\displaystyle q(x)} and g ( x ) {\displaystyle g(x)} are analytic at x = a {\displaystyle x=a} or a {\displaystyle a} is a regular singular point. That is, any solution to this second-order differential equation can be written as

y = ∑ n = 0 ∞ a n ( x − a ) n + s , a 0 ≠ 0 {\displaystyle y=\sum _{n=0}^{\infty }a_{n}(x-a)^{n+s},\quad a_{0}\neq 0}

for some positive real s, or

y = y 0 ln ⁡ ( x − a ) + ∑ n = 0 ∞ b n ( x − a ) n + r , b 0 ≠ 0 {\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 p ( x ) {\displaystyle p(x)} , q ( x ) {\displaystyle q(x)} and g ( x ) {\displaystyle g(x)} .

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