Fuchs' theorem

In mathematics, Fuchs' theorem, named after Lazarus Fuchs, states that a second-order differential equation of the form [math]\displaystyle{ y'' + p(x)y' + q(x)y = g(x) }[/math] has a solution expressible by a generalised Frobenius series when [math]\displaystyle{ p(x) }[/math], [math]\displaystyle{ q(x) }[/math] and [math]\displaystyle{ g(x) }[/math] are analytic at [math]\displaystyle{ x = a }[/math] or [math]\displaystyle{ a }[/math] is a regular singular point. That is, any solution to this second-order differential equation can be written as [math]\displaystyle{ y = \sum_{n=0}^\infty a_n (x - a)^{n + s}, \quad a_0 \neq 0 }[/math] for some positive real s, or [math]\displaystyle{ y = y_0 \ln(x - a) + \sum_{n=0}^\infty b_n(x - a)^{n + r}, \quad b_0 \neq 0 }[/math] 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 [math]\displaystyle{ p(x) }[/math], [math]\displaystyle{ q(x) }[/math] and [math]\displaystyle{ g(x) }[/math].

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 .