Method of Frobenius

Method of Frobenius

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 u ′ ≡ d u d z {\textstyle u'\equiv {\frac {du}{dz}}} and u ″ ≡ d 2 u d z 2 {\textstyle u''\equiv {\frac {d^{2}u}{dz^{2}}}} . in the vicinity of the regular singular point z = 0 {\displaystyle z=0} . One can divide by z 2 {\displaystyle z^{2}} 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).

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