(For other notation see Notation for the Special Functions.)
|lattice in .|
|integer, except in §23.20(ii).|
|complex variable, except in §§23.20(ii), 23.21(iii).|
|or||closed, or open, straight-line segment joining and , whether or not and are real.|
|primes||derivatives with respect to the variable, except where indicated otherwise.|
|,||complete elliptic integrals (§19.2(i)).|
|lattice generators ().|
|lattice parameter ().|
|set of all integer multiples of .|
|set of all elements of , modulo elements of . Thus two elements of are equivalent if they are both in and their difference is in . (For an example see §20.12(ii).)|
|Cartesian product of groups and , that is, the set of all pairs of elements with group operation .|
The main functions treated in this chapter are the Weierstrass -function ; the Weierstrass zeta function ; the Weierstrass sigma function ; the elliptic modular function ; Klein’s complete invariant ; Dedekind’s eta function .
Whittaker and Watson (1927) requires only , instead of . Abramowitz and Stegun (1964, Chapter 18) considers only rectangular and rhombic lattices (§23.5); , are replaced by , for the former and by , for the latter. Silverman and Tate (1992) and Koblitz (1993) replace and by and , respectively. Walker (1996) normalizes , , and uses homogeneity (§23.10(iv)). McKean and Moll (1999) replaces and by and , respectively.