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

§23.1 Special Notation

Defines:: $\times$ : $G\times H$ : Cartesian product of groups and
Keywords:: Dedekind’s eta function, Dedekind’s modular function, Klein’s complete invariant, Weierstrass $\mathop{\wp\/}\nolimits$ -function, Weierstrass elliptic functions, Weierstrass sigma function, Weierstrass zeta function, elliptic functions, elliptic modular function, lattice, modular functions, nome
Permalink:: http://dlmf.nist.gov/23.1

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

$\mathbb{L}$	lattice in $\Complex$ .
$\ell,n$	integers.
	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.
$\mathop{K\/}\nolimits\!\left(k\right)$ , ${\mathop{K\/}\nolimits^{{\prime}}}\!\left(k\right)$	complete elliptic integrals (§19.2(i)).
$2\omega_{1},2\omega_{3}$	lattice generators ( $\imagpart{(\omega_{3}/\omega_{1})}>0$ ).
$\omega_{2}$	$-\omega_{1}-\omega_{3}$ .
$\tau=\omega_{3}/\omega_{1}$	lattice parameter ( $\imagpart{\tau}>0$ ).
$q=e^{{i\pi\omega_{3}/\omega_{1}}}$
$\phantom{q}=e^{{i\pi\tau}}$	nome.
$g_{2},g_{3}$	lattice invariants.
$e_{1},e_{2},e_{3}$	zeros of Weierstrass normal cubic $4z^{3}-g_{2}z-g_{3}$ .
$\Delta$	discriminant $g_{2}^{3}-27g_{3}^{2}$ .
$n\Integer$	set of all integer multiples of .
$S_{1}\setmod S_{2}$	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).)
$G\times H$	Cartesian product of groups and , that is, the set of all pairs of elements 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.

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 23 Weierstrass Elliptic and Modular Functions 23.2 Definitions and Periodic Properties