Series Solutions and Special Functions

When the coefficients vary, a solution can be built as a power series. This unit finds series solutions at ordinary and regular singular points and studies the Bessel and Legendre equations.

At an ordinary point, matching powers in a power series gives a coefficient recurrence, and the initial values fix the first two. Analytic coefficients guarantee convergence at least out to the nearest complex singularity of the normalized equation, though a solution may converge farther. At a regular singular point the Frobenius method allows a leading power set by the indicial equation, and a repeated root or roots that differ by an integer can lead to a logarithmic second solution. Bessel's equation comes from radial problems, and Legendre's equation has orthogonal polynomial solutions.