# §22.11 Fourier and Hyperbolic Series

Throughout this section and are defined as in §22.2.

If , then

Next, if , then

In (22.11.7)–(22.11.12) the left-hand sides are replaced by their limiting values at the poles of the Jacobian functions.

Next, with denoting the complete elliptic integral of the second kind (§19.2(ii)) and ,

Similar expansions for and follow immediately from (22.6.1).

For further Fourier series see Oberhettinger (1973, pp. 23–27).

A related hyperbolic series is

where is defined by §19.2.9. Again, similar expansions for and may be derived via (22.6.1). See Dunne and Rao (2000).