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 $${\displaystyle z^{2}u''+p(z)zu'+q(z)u=0}$$ with ${\textstyle u'\equiv {\frac {du}{dz}}}$ and ${\textstyle u''\equiv {\frac {d^{2}u}{dz^{2}}}}$.

in the vicinity of the regular singular point ${\displaystyle z=0}$.

One can divide by ${\displaystyle z^{2}}$ to obtain a differential equation of the form $${\displaystyle u''+{\frac {p(z)}{z}}u'+{\frac {q(z)}{z^{2}}}u=0}$$ 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).