Elliptic function

An elliptic function is a meromorphic function with two real-linearly independent periods ${\omega }_1,{\omega }_2$. A fundamental cell is a half-open parallelogramOpposite boundary integrals cancel. Consequently a nonconstant elliptic function has equally many zeros and poles in a cell, counted with multiplicity, and the sum of its pole residues in a cell is zero.

For a nonzero elliptic function with period lattice $\Lambda$, its zeros of a function $a_j$ and poles $b_j$, repeated with their multiplicities, have equal total number and satisfy ${\sum }_j{a}_j-{\sum }_j{b}_j\in \Lambda$. Choose a fundamental parallelogram avoiding all zeros and poles on its boundary. The argument principle applied to $f^{\prime }/f$ gives equality of the numbers. Applying the residue theorem to $z{f}^{\prime }/f$ gives their difference of sums. Pairing opposite edges, the extra factors are the two lattice generators multiplied by integrals of $f^{\prime }/f$ along an edge. These integrals are integer multiples of $2\pi i$, because the endpoints have the same nonzero function value. Thus the difference of sums is a lattice element. This links zeros of a pulled-back line to the chord-and-tangent group law.

For a nonzero elliptic function, the weighted sum of its zero positions minus its pole positions belongs to the period lattice. Integrating $z{f}^{\prime }/f$ around a fundamental parallelogram proves this: translating opposite edges gives a lattice-linear combination of integrals of $f^{\prime }/f$, each an integer multiple of $2\pi i$. The unweighted integral proves that the zero and pole multiplicities have equal total.

Two real-linearly independent periods ${\omega }_1,{\omega }_2$ generate the period lattice

A fundamental parallelogram for $\Lambda =\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$ is a translate ofIts translates tile the complex plane.

If an elliptic function had no poles, it would be entire. It is bounded on the closure of a fundamental parallelogram, and periodicity then makes it bounded on the whole complex plane. The Liouville theorem would make it constant. Thus every nonconstant elliptic function has a pole.

The Jacobi elliptic sine is the local inverse of the elliptic integral of the first kind: ifthen $z=\mathrm{sn}(u,k)$. Its analytic continuation is a meromorphic doubly periodic function.

For real $0<k<1$, the period lattice of $\mathrm{sn}(u,k)$ is generated byThe periods arise by composing the monodromy reflections of its inverse elliptic integral of the first kind.

For any finite value $a$, apply the argument principle to $f-a$ around a fundamental parallelogram whose boundary avoids zeros and poles. Periodicity cancels the opposite-edge integrals, so the number of solutions of $f(z)=a$ equals the fixed number of poles of $f$, counting multiplicities.

For a period lattice $\Lambda \subset \mathbb{C}$, the Weierstrass elliptic function is the normally convergent lattice sumIt is an even elliptic function with a double pole of principal part $(z-\omega {)}^{-2}$ and zero residue at every point $\omega$ of the period lattice.

Every elliptic function has a unique expression $A(\wp )+B(\wp ){\wp }^{\prime }$ with rational functions $A,B$. Its even part is rational in the Weierstrass elliptic function; its odd part divided by ${\wp }^{\prime }$ is even and meromorphic, so is also rational in $\wp$. The local even Laurent series at a half-period and at the pole justify meromorphic descent through the double cover of the Riemann sphere.

The Weierstrass elliptic function is the quotient map from the complex torus by $z\mapsto -z$. An even elliptic function descends through this map; its even Laurent series at each branch point makes the descended function meromorphic there. A meromorphic function on the Riemann sphere is rational, so $f=Q(\wp )$.

On every compact set disjoint from the lattice, the summandis bounded by $C|\omega {|}^{-3}$ for all sufficiently large $|\omega |$. Since the lattice sum of $|\omega {|}^{-3}$ converges, the defining series for $\wp$ converges normally.

At the three nonzero two-torsion points $z_i$ of $\mathbb{C}/\Lambda$, put $e_i=\wp (z_i)$. The values are distinct because $\wp (z)=\wp (w)$ exactly when $z\equiv \pm w(\mathrm{mod}\Lambda )$.

A two-torsion point satisfies $2z\in \Lambda$. The three nonzero classes are represented by half-periods $\lambda /2$, $\mu /2$, and $(\lambda +\mu )/2$.

The equianharmonic triangular lattice is preserved by multiplication by the cube root of unity $\omega =e^{2\pi i/3}$. On its three nonzero two-torsion classes, multiplication by $\omega$ acts as a three-cycle.

The Weierstrass zeta function satisfies ${\zeta }^{\prime }=-\wp$ and is quasi-periodic: for every period $\omega$, the difference $\zeta (z+\omega )-\zeta (z)$ is constant. It has a simple pole of residue one at each lattice point. Consequently, if ${\sum }_j{c}_j=0$, thenis an elliptic function.

Writing $G_{2r}={\sum }_{\omega \ne 0}{\omega }^{-2r}$ for the lattice Eisenstein sums, expansion about zero givesThus the constant coefficient vanishes and, for $k\ge 1$, the coefficient of $z^{2k}$ is $(2k+1)G_{2k+2}$.

The Weierstrass elliptic function satisfiesWriting $G_{2r}={\sum }_{\omega \in \Lambda \setminus \{0\}}{\omega }^{-2r}$, the Laurent coefficients of the Weierstrass elliptic function giveThus $g_2=60G_4$ and $g_3=140G_6$ cancel every nonremovable term in the Laurent series of ${\wp }^{\prime 2}-4{\wp }^3+g_2\wp +g_3$ at each point of the period lattice. The resulting elliptic function is an entire function and is bounded, hence is zero by Liouville theorem.