Frobenius method

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 z 2 u ″ + p ( z ) z u ′ + q ( z ) u = 0 {\displaystyle z^{2}u''+p(z)zu'+q(z)u=0} 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 u ″ + p ( z ) z u ′ + q ( z ) z 2 u = 0 {\displaystyle u''+{\frac {p(z)}{z}}u'+{\frac {q(z)}{z^{2}}}u=0} which will not be solvable with regular power series methods if either p(z)/z or q(z)/z2 is 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).