Fuchs' theorem

Fuchs' theorem (Redirected from Fuchs's theorem) Jump to navigation Jump to search This article includes a list of references, leitura relacionada ou links externos, mas suas fontes permanecem obscuras porque faltam citações em linha. Ajude a melhorar este artigo introduzindo citações mais precisas. (Junho 2017) (Saiba como e quando remover esta mensagem de modelo) Na matemática, 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 {estilo de exibição p(x)} , {estilo de exibição q(x)} e {estilo de exibição g(x)} are analytic at {estilo de exibição x=a} ou {estilo de exibição a} is a regular singular point. Aquilo é, any solution to this second-order differential equation can be written as {displaystyle y=sum _{n=0}^{infty }uma_{n}(x-a)^{n+s},quad a_{0}neq 0} for some positive real s, ou {displaystyle y=y_{0}ln(x-a)+soma _{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 {estilo de exibição p(x)} , {estilo de exibição q(x)} e {estilo de exibição g(x)} .

See also Frobenius method References Asmar, Nakhlé H. (2005), Partial differential equations with Fourier series and boundary value problems, Rio Sela Superior, Nova Jersey: Salão Pearson Prentice, ISBN 0-13-148096-0. Butkov, Eugênio (1995), Mathematical Physics, Leitura, MA: Addison-Wesley, ISBN 0-201-00727-4. Categorias: Differential equationsTheorems in analysis