# §19.5 Maclaurin and Related Expansions

For Jacobi’s nome :

Also,

19.5.6,

where

19.5.7

Coefficients of terms up to are given in Lee (1990), along with tables of fractional errors in and , , obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9).

An infinite series for is equivalent to the infinite product

where and

Series expansions of and are surveyed and improved in Van de Vel (1969), and the case of is summarized in Gautschi (1975, §1.3.2). For series expansions of when see Erdélyi et al. (1953b, §13.6(9)). See also Karp et al. (2007).