Fuchs' theorem

Last updated August 23, 2020

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 positive 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 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 $p(x)$ , $q(x)$ and $g(x)$ .

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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 .

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Fuchs' theorem

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