# §23.1 Special Notation

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

$\mathbb{L}$ lattice in $\Complex$. 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 ($\imagpart{(\omega_{3}/\omega_{1})}>0$). $-\omega_{1}-\omega_{3}$. lattice parameter ($\imagpart{\tau}>0$). nome. lattice invariants. zeros of Weierstrass normal cubic $4z^{3}-g_{2}z-g_{3}$. discriminant $g_{2}^{3}-27g_{3}^{2}$. set of all integer multiples of $n$. set of all elements of $S_{1}$, modulo elements of $S_{2}$. Thus two elements of $S_{1}\setmod S_{2}$ 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 $\mathop{\wp\/}\nolimits$-function $\mathop{\wp\/}\nolimits\!\left(z\right)=\mathop{\wp\/}\nolimits\!\left(z|% \mathbb{L}\right)=\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$; the Weierstrass zeta function $\mathop{\zeta\/}\nolimits\!\left(z\right)=\mathop{\zeta\/}\nolimits\!\left(z|% \mathbb{L}\right)=\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$; the Weierstrass sigma function $\mathop{\sigma\/}\nolimits\!\left(z\right)=\mathop{\sigma\/}\nolimits\!\left(z% |\mathbb{L}\right)=\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$; the elliptic modular function $\mathop{\lambda\/}\nolimits\!\left(\tau\right)$; Klein’s complete invariant $\mathop{J\/}\nolimits\!\left(\tau\right)$; Dedekind’s eta function $\mathop{\eta\/}\nolimits\!\left(\tau\right)$.

# Other Notations

Whittaker and Watson (1927) requires only $\imagpart{(\omega_{3}/\omega_{1})}\neq 0$, instead of $\imagpart{(\omega_{3}/\omega_{1})}>0$. 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.