# §23.1 Special Notation

(For other notation see Notation for the Special Functions.)

$\mathbb{L}$ lattice in $\mathbb{C}$. integers. integer, except in §23.20(ii). complex variable, except in §§23.20(ii), 23.21(iii). closed, or open, straight-line segment joining $a$ and $b$, whether or not $a$ and $b$ are real. derivatives with respect to the variable, except where indicated otherwise. complete elliptic integrals (§19.2(i)). lattice generators ($\Im\left(\omega_{3}/\omega_{1}\right)>0$). $-\omega_{1}-\omega_{3}$. lattice parameter ($\Im\tau>0$). nome. discriminant ${g_{2}}^{3}-27{g_{3}}^{2}$. set of all integer multiples of $n$. set of all elements of $S_{1}$, modulo elements of $S_{2}$. Thus two elements of $S_{1}/S_{2}$ are equivalent if they are both in $S_{1}$ and their difference is in $S_{2}$. (For an example see §20.12(ii).) Cartesian product of groups $G$ and $H$, that is, the set of all pairs of elements $(g,h)$ with group operation $(g_{1},h_{1})+(g_{2},h_{2})=(g_{1}+g_{2},h_{1}+h_{2})$.

The main functions treated in this chapter are the Weierstrass $\wp$-function $\wp\left(z\right)=\wp\left(z|\mathbb{L}\right)=\wp\left(z;g_{2},g_{3}\right)$; the Weierstrass zeta function $\zeta\left(z\right)=\zeta\left(z|\mathbb{L}\right)=\zeta\left(z;g_{2},g_{3}\right)$; the Weierstrass sigma function $\sigma\left(z\right)=\sigma\left(z|\mathbb{L}\right)=\sigma\left(z;g_{2},g_{3}\right)$; the elliptic modular function $\lambda\left(\tau\right)$; Klein’s complete invariant $J\left(\tau\right)$; Dedekind’s eta function $\eta\left(\tau\right)$.

## Other Notations

Whittaker and Watson (1927) requires only $\Im\left(\omega_{3}/\omega_{1}\right)\neq 0$, instead of $\Im\left(\omega_{3}/\omega_{1}\right)>0$. Abramowitz and Stegun (1964, Chapter 18) considers only rectangular and rhombic lattices (§23.5); $\omega_{1}$, $\omega_{3}$ are replaced by $\omega$, $\omega^{\prime}$ for the former and by $\omega_{2}$, $\omega^{\prime}$ for the latter. Silverman and Tate (1992) and Koblitz (1993) replace $2\omega_{1}$ and $2\omega_{3}$ by $\omega_{1}$ and $\omega_{3}$, respectively. Walker (1996) normalizes $2\omega_{1}=1$, $2\omega_{3}=\tau$, and uses homogeneity (§23.10(iv)). McKean and Moll (1999) replaces $2\omega_{1}$ and $2\omega_{3}$ by $\omega_{1}$ and $\omega_{2}$, respectively.