Fuchs' theorem

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

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

has a solution expressible by a generalised Frobenius series when ${\displaystyle p(x)}$, ${\displaystyle q(x)}$ and ${\displaystyle g(x)}$ are analytic at ${\displaystyle x=a}$ or ${\displaystyle a}$ 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

${\displaystyle y=y_{0}\ln(x-a)+\sum _{n=0}^{\infty }b_{n}(x-a)^{n+r},\quad b_{0}\neq 0}$

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 ${\displaystyle p(x)}$, ${\displaystyle q(x)}$ and ${\displaystyle g(x)}$.

See also

• Frobenius method

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.
• Butkov, Eugene (1995), Mathematical Physics, Reading, MA: Addison-Wesley, ISBN 0-201-00727-4.