thumb|350px|Some solutions of a differential equation having a regular singular point with indicial roots <math>r = \frac{1}{2}</math> and <math>-1</math>

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

<math display="block">z^2 u + p(z)z u'+ q(z) u = 0</math>

with <math display="inline">u' \equiv \frac{du}{dz}</math> and <math display="inline">u \equiv \frac{d^2 u}{dz^2}</math>.

in the vicinity of the regular singular point <math>z=0</math>.

One can divide by <math>z^2</math> to obtain a differential equation of the form

<math display="block">u + \frac{p(z)}{z}u' + \frac{q(z)}{ z^2}u = 0</math>

which will not be solvable with regular power series methods if either or is not analytic at&nbsp;. 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).

History

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 Lazarus Fuchs. The indicial polynomial (see below) and its role had also been established by Fuchs. with respect to the parameter r, mentioned above.

A large part of Frobenius' 1873 publication