# §28.30 Expansions in Series of Eigenfunctions

## §28.30(i) Real Variable

Let , , be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let , , be the eigenfunctions, that is, an orthonormal set of -periodic solutions; thus

Then every continuous -periodic function whose second derivative is square-integrable over the interval can be expanded in a uniformly and absolutely convergent series

where

## §28.30(ii) Complex Variable

For analogous results to those of §28.19, see Schäfke (1960, 1961b), and Meixner et al. (1980, §1.1.11).