Weierstrass $\zeta$ and $\sigma$ functions

Let $\mathrm{\Lambda }\subseteq \mathbb{C}$ be a full complex lattice, and let the sum over $\mathrm{\Lambda }\setminus \{0\}$ and the product over $\mathrm{\Lambda }\setminus \{0\}$ be the unordered finite-subset limits of Weierstrass p function: the net of partial sums, respectively partial products, over the finite subsets ordered by inclusion, when it converges independently of the exhaustion. The Weierstrass $\zeta$-function and $\sigma$-function of $\mathrm{\Lambda }$ are

${\zeta }_{\mathrm{\Lambda }}(z):=\frac{1}{z}+{\sum }_{\omega \in \mathrm{\Lambda }\setminus \{0\}}(\frac{1}{z-\omega }+\frac{1}{\omega }+\frac{z}{{\omega }^2}),z\in \mathbb{C}\setminus \mathrm{\Lambda },$

${\sigma }_{\mathrm{\Lambda }}(z):=z{\prod }_{\omega \in \mathrm{\Lambda }\setminus \{0\}}{E}_2\text{\hspace{0.17em}⁣}\left(\frac{z}{\omega }\right)=z{\prod }_{\omega \in \mathrm{\Lambda }\setminus \{0\}}(1-\frac{z}{\omega })\exp \text{\hspace{0.17em}⁣}(\frac{z}{\omega }+\frac{z^2}{2{\omega }^2}),z\in \mathbb{C},$

where $E_2(w)=(1-w)e^{w+w^2\mathrm{/}2}$ is the second Weierstrass elementary factor (Weierstrass elementary factors). The following theorem of this page proves that the defining net converges normally on $\mathbb{C}\setminus \mathrm{\Lambda }$ for ${\zeta }_{\mathrm{\Lambda }}$ and on $\mathbb{C}$ for ${\sigma }_{\mathrm{\Lambda }}$, so that both functions are well defined and depend only on the set $\mathrm{\Lambda }$.

A lattice element $\omega \in \mathrm{\Lambda }$ is primitive when it is part of a $\mathbb{Z}$-basis of $\mathrm{\Lambda }$, equivalently when $\omega \mathrm{/}n\mathrm{\notin }\mathrm{\Lambda }$ for every integer $n\ge 2$. For a primitive period $\omega$ put

${\eta }_{\omega }:=2{\zeta }_{\mathrm{\Lambda }}\text{\hspace{0.17em}⁣}\left(\frac{\omega }{2}\right)\mathrm{.}$

The quasi-period laws

${\zeta }_{\mathrm{\Lambda }}(z+\omega )={\zeta }_{\mathrm{\Lambda }}(z)+{\eta }_{\omega },{\sigma }_{\mathrm{\Lambda }}(z+\omega )=-\exp \text{\hspace{0.17em}⁣}\left({\eta }_{\omega }(z+\frac{\omega }{2})\right){\sigma }_{\mathrm{\Lambda }}(z)$

for every primitive $\omega \in \mathrm{\Lambda }$ are proved in Convergence, zeros and quasi-periods of the Weierstrass zeta and sigma functions ↗, together with ${\zeta }_{\mathrm{\Lambda }}^{\mathrm{\prime }}=-{\mathrm{\wp }}_{\mathrm{\Lambda }}$, ${\sigma }_{\mathrm{\Lambda }}^{\mathrm{\prime }}\mathrm{/}{\sigma }_{\mathrm{\Lambda }}={\zeta }_{\mathrm{\Lambda }}$, ${\sigma }_{\mathrm{\Lambda }}^{\mathrm{\prime }}(0)=1$, the oddness of ${\zeta }_{\mathrm{\Lambda }}$ and ${\sigma }_{\mathrm{\Lambda }}$, and the fact that ${\sigma }_{\mathrm{\Lambda }}$ is entire with simple zeros exactly at the lattice points.

Why the corrections. The summand of ${\zeta }_{\mathrm{\Lambda }}$, $\frac{1}{z-\omega }+\frac{1}{\omega }+\frac{z}{{\omega }^2}=\frac{z^2}{(z-\omega ){\omega }^2},$ vanishes to second order at $z=0$, so the single term $1\mathrm{/}z$ carries the whole principal part there; at a general lattice point the translation $z\mapsto z+\lambda$ exhibits the principal part $(z-\lambda {)}^{-1}$. Similarly the factor $E_2(z\mathrm{/}\omega )$ has a simple zero at $z=\omega$ and no other zero, and its expansion $\log E_2(w)=-w^3\mathrm{/}3-w^4\mathrm{/}4-\cdots$ shows that the correction $e^{z\mathrm{/}\omega +z^2\mathrm{/}(2{\omega }^2)}$ is exactly what makes the product converge on compact sets. Both facts are proved in the items named above.

Normalisation. The factor $z$ in front of ${\sigma }_{\mathrm{\Lambda }}$ is chosen so that $z=0$ is a simple zero and ${\sigma }_{\mathrm{\Lambda }}^{\mathrm{\prime }}(0)=1$; the constants ${\eta }_{\omega }$ are the analogues of the half-period values $\mathrm{\wp }(\omega \mathrm{/}2)$, and for an oriented basis $({\omega }_1,{\omega }_2)$ the Legendre relation ${\eta }_{{\omega }_1}{\omega }_2-{\eta }_{{\omega }_2}{\omega }_1=2\pi i$ holds; it is proved in Convergence, zeros and quasi-periods of the Weierstrass zeta and sigma functions ↗. Extend the quasi-period constants additively from the chosen basis by ${\eta }_{m{\omega }_1+n{\omega }_2}:=m{\eta }_1+n{\eta }_2$. For an arbitrary period $\omega =m{\omega }_1+n{\omega }_2$, iteration gives ${\zeta }_{\mathrm{\Lambda }}(z+\omega )={\zeta }_{\mathrm{\Lambda }}(z)+{\eta }_{\omega },{\sigma }_{\mathrm{\Lambda }}(z+\omega )=(-1{)}^{m+n+mn}\exp \text{\hspace{0.17em}⁣}\left({\eta }_{\omega }(z+\frac{\omega }{2})\right){\sigma }_{\mathrm{\Lambda }}(z)\mathrm{.}$ The sign is $-1$ when $\omega$ is primitive: then $\gcd (m,n)=1$, so $m,n$ are not both even and $m+n+mn$ is odd. For a primitive period, the additive constant agrees with the displayed definition ${\eta }_{\omega }=2{\zeta }_{\mathrm{\Lambda }}(\omega \mathrm{/}2)$, by applying the zeta translation law at $z=-\omega \mathrm{/}2$ and using oddness.

Primitive-period criterion. Fix a $\mathbb{Z}$-basis $({\omega }_1,{\omega }_2)$ of $\mathrm{\Lambda }$ and write $\omega =a{\omega }_1+b{\omega }_2$ with $a,b\in \mathbb{Z}$. For $\omega =0$, both basis membership and the no-divisor condition fail. If $\omega \mathrm{\ne }0$, put $d=\gcd (a,b)>0$. Uniqueness of coordinates shows that $\omega \mathrm{/}n\in \mathrm{\Lambda }$ for an integer $n\ge 2$ exactly when $n$ divides both $a$ and $b$; since $d$ divides both coordinates and every positive common divisor is at most $d$ (Common divisor, and the greatest common divisor $\gcd (a,b)$, with the convention $\gcd (0,0):=0$), no such $n$ exists exactly when $d=1$. By Bézout's identity: for integers $a,b$ not both zero, $\gcd (a,b)$ is the least positive element of $\{ax+by:x,y\in \mathbb{Z}\}$; in particular $ax+by=\gcd (a,b)$ has an integer solution, in that case there are integers $x,y$ with $ax+by=1$. Then $\nu =-y{\omega }_1+x{\omega }_2$ belongs to $\mathrm{\Lambda }$, and the coordinate matrix of $(\omega ,\nu )$ has determinant $ax+by=1$, so $(\omega ,\nu )$ is a $\mathbb{Z}$-basis. Conversely, if $\omega$ is a member of a $\mathbb{Z}$-basis and $\omega \mathrm{/}n\in \mathrm{\Lambda }$, writing $\omega \mathrm{/}n$ in that basis would make its coordinate on $\omega$ equal to $1\mathrm{/}n$, not an integer.