Frobenius method

In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a second-order ordinary differential equation of the form with and . in the vicinity of the regular singular point . One can divide by to obtain a differential equation of the form which will not be solvable with regular power series methods if either p(z)/z or q(z)/z2 are not analytic at z = 0. The Frobenius method enables one to create a power series solution to such a differential equation, provided that p(z) and q(z) are themselves analytic at 0 or, being analytic elsewhere, both their limits at 0 exist (and are finite). Frobenius' contribution was not so much in all the possible forms of the series solutions involved (see below). These forms had all been established earlier, by Fuchs. The indicial polynomial (see below) and its role had also been established by Fuchs. A first contribution by Frobenius to the theory was to show that - as regards a first, linearly independent solution, which then has the form of an analytical power series multiplied by an arbitrary power r of the independent variable (see below) - the coefficients of the generalized power series obey a recurrence relation, so that they can always be straightforwardly calculated. A second contribution by Frobenius was to show that, in cases in which the roots of the indicial equation differ by an integer, the general form of the second linearly independent solution (see below) can be obtained by a procedure which is based on differentiation with respect to the parameter r, mentioned above. A large part of Frobenius' 1873 publication was devoted to proofs of convergence of all the series involved in the solutions, as well as establishing the radii of convergence of these series. The method of Frobenius is to seek a power series solution of the form Differentiating: Substituting the above differentiation into our original ODE: The expression is known as the indicial polynomial, which is quadratic in r.