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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 ...
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 ...
The Frobenius number exists as long as the set of coin denominations is setwise coprime. There is an explicit formula for the Frobenius number when there are only two different coin denominations, and , where the greatest common divisor of these two numbers is 1: . If the number of coin denominations is three or more, no explicit formula is known.
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 ...
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.
The roots of the characteristic polynomial () are the eigenvalues of ().If there are n distinct eigenvalues , …,, then () is diagonalizable as () =, where D is the diagonal matrix and V is the Vandermonde matrix corresponding to the λ 's: = [], = [].
In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducting.It can be used to leverage knowledge about representations of a subgroup to find and classify representations of "large" groups that contain them.