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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 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.
Since Kummer's equation is second order there must be another, independent, solution. The indicial equation of the method of Frobenius tells us that the lowest power of a power series solution to the Kummer equation is either 0 or 1 − b. If we let w(z) be = then the differential equation gives
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 ...
Then the Frobenius method based on the indicial equation may be applied to find possible solutions that are power series times complex powers (z − a) r near any given a in the complex plane where r need not be an integer; this function may exist, therefore, only thanks to a branch cut extending out from a, or on a Riemann surface of some ...
Latysheva developed an effective method for the construction of solutions of linear ordinary differential equations around regular and irregular points by expanding on Poincaré's concept of rank and L. Tome's concept of anti-rank. This is now known as the Frobenius-Latysheva method.
In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducting. It ...