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)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 $z=0$ .

One can divide by $z^{2}$ to obtain a differential equation of the form

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 $q (z)/ z 2$ 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).

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.