23 Weierstrass Elliptic and Modular Functions Notation 23 Weierstrass Elliptic and Modular Functions 23.2 Definitions and Periodic Properties

§23.1 Special Notation

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Defines:: $\times$ : $G\times H$ : Cartesian product of groups
Keywords:: Dedekind’s eta function, Dedekind’s modular function, Klein’s complete invariant, Weierstrass, Weierstrass $\wp$ -function, Weierstrass elliptic functions, Weierstrass sigma function, Weierstrass zeta function, elliptic functions, elliptic modular function, lattice, modular functions, nome, notation
Permalink:: http://dlmf.nist.gov/23.1
See also:: Annotations for Ch.23

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

$\mathbb{L}$	lattice in $\mathbb{C}$ .
$\ell,n$	integers.
$m$	integer, except in §23.20(ii).
$z=x+iy$	complex variable, except in §§23.20(ii), 23.21(iii).
$[a,b]$ or $(a,b)$	closed, or open, straight-line segment joining $a$ and $b$ , whether or not $a$ and $b$ are real.
primes	derivatives with respect to the variable, except where indicated otherwise.
$K\left(k\right)$ , $K'\left(k\right)$	complete elliptic integrals (§19.2(i)).
$2\omega_{1},2\omega_{3}$	lattice generators ( $\Im\left(\omega_{3}/\omega_{1}\right)>0$ ).
$\omega_{2}$	$-\omega_{1}-\omega_{3}$ .
$\tau=\omega_{3}/\omega_{1}$	lattice parameter ( $\Im\tau>0$ ).
$q=e^{i\pi\omega_{3}/\omega_{1}}$
$\phantom{q}=e^{i\pi\tau}$	nome.
$\Delta$	discriminant ${g_{2}}^{3}-27{g_{3}}^{2}$ .
$n\mathbb{Z}$	set of all integer multiples of $n$ .
$S_{1}/S_{2}$	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).)
$G\times H$	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

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See also:: Annotations for §23.1 and Ch.23

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.

23 Weierstrass Elliptic and Modular Functions 23.2 Definitions and Periodic Properties

NIST

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