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Some solutions of a differential equation having a regular singular point with indicial roots = and .. In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form ″ + ′ + = with ′ and ″.
In the following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand Georg Frobenius. This is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series. This is usually the method we use for ...
The Frobenius formula states that if χ is the character of the representation σ, given by χ(h) = Tr σ(h), then the character ψ of the induced representation is given by ψ ( g ) = ∑ x ∈ G / H χ ^ ( x − 1 g x ) , {\displaystyle \psi (g)=\sum _{x\in G/H}{\widehat {\chi }}\left(x^{-1}gx\right),}
The power series method calls for the construction of a power series solution = =. If a 2 is zero for some z, then the Frobenius method, a variation on this method, is suited to deal with so called "singular points". The method works analogously for higher order equations as well as for systems.
In this case the recursive calculation of the Frobenius series' coefficients stops for some roots and the Frobenius series method does not give an -dimensional solution space. The following can be shown independent of the distance between roots of the indicial polynomial: Let α ∈ C {\displaystyle \alpha \in \mathbb {C} } be a μ ...
Frobenius' theorem is one of the basic tools for the study of vector fields and foliations. There are thus two forms of the theorem: one which operates with distributions , that is smooth subbundles D of the tangent bundle TM ; and the other which operates with subbundles of the graded ring Ω( M ) of all forms on M .
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In mathematics, Fuchs' theorem, named after Lazarus Fuchs, states that a second-order differential equation of the form ″ + ′ + = has a solution expressible by a generalised Frobenius series when (), () and () are analytic at = or is a regular singular point.