Weierstrass p-function

Intuition [Beginner]

A periodic function repeats its values at regular intervals: $\sin (x)$ has period $2\pi$ because $\sin (x+2\pi )=\sin (x)$. A doubly-periodic function repeats in two independent directions simultaneously. On the complex plane, this means the function repeats when you shift by ${\omega }_1$ in one direction and by ${\omega }_2$ in another, where ${\omega }_1$ and ${\omega }_2$ are two complex numbers that are not on the same line through the origin.

The Weierstrass $\wp$-function is the simplest doubly-periodic function with poles. It is defined by an infinite collection of terms, one for each point of a lattice $\Lambda$ (a grid of points $m{\omega }_1+n{\omega }_2$ for integers $m,n$). At each lattice point, $\wp$ has a double pole (it blows up like $1/(z-\omega {)}^2$). Between the lattice points, $\wp$ is smooth in the complex sense.

The fundamental fact is that every doubly-periodic meromorphic function can be built from $\wp$ and its derivative ${\wp }^{\prime }$ using addition, multiplication, and division. Just as every polynomial is built from $x$, and every rational function from $x$ and constants, every elliptic function (doubly-periodic meromorphic function) is built from $\wp$ and ${\wp }^{\prime }$.

Why does this concept exist? Doubly-periodic functions are the two-dimensional generalisation of periodic functions. They arise naturally in the study of elliptic integrals, the theory of elliptic curves, and applications in number theory and mathematical physics.

Visual [Beginner]

A diagram showing the complex plane with a lattice of points (a tilted grid formed by ${\omega }_1$ and ${\omega }_2$). A fundamental parallelogram is shaded (the parallelogram with vertices $0,{\omega }_1,{\omega }_1+{\omega }_2,{\omega }_2$). The $\wp$-function has double poles at each lattice point and takes every other value exactly twice in the fundamental parallelogram.

The picture shows that the $\wp$-function is fully determined by its behaviour on one fundamental parallelogram, then repeated across the entire lattice.

Worked example [Beginner]

The simplest lattice is $\Lambda =\{m+ni:m,n\in \mathbb{Z}\}$, the Gaussian integer lattice, with ${\omega }_1=1$ and ${\omega }_2=i$. Compute $\wp (1/2)$ approximately.

Step 1. The $\wp$-function collects terms $1/z^2$ from the origin and $1/(z-\omega {)}^2-1/{\omega }^2$ from each non-zero lattice point $\omega =m+ni$. At $z=1/2$, the dominant term is $1/(1/2{)}^2=4$.

Step 2. The correction terms $1/(z-\omega {)}^2-1/{\omega }^2$ are small for large $|\omega |$. The nearest non-zero lattice points are $\omega =1,-1,i,-i$. For $\omega =1$: $1/(1/2-1{)}^2-1/1^2=1/(1/4)-1=4-1=3$. For $\omega =-1$: $1/(1/2+1{)}^2-1/1^2=1/(9/4)-1=4/9-1=-5/9$. For $\omega =i$: $1/(1/2-i{)}^2-1/(i^2)=1/(1/4-i-1)-1/(-1)$. This requires complex arithmetic.

Step 3. The full computation requires many terms for accuracy. The exact value is not elementary — it involves the lattice invariants $g_2$ and $g_3$. For the Gaussian integer lattice, $g_2$ is proportional to the total of $1/{\omega }^4$ over all non-zero lattice points, and $g_3$ to the total of $1/{\omega }^6$. The key point is that $\wp (1/2)$ is a specific real number (since the lattice is symmetric under conjugation and $z=1/2$ is real) determined by the lattice geometry.

What this tells us: evaluating $\wp$ at specific points requires summing an infinite series, but the function is well-defined and the series converges rapidly for large lattice points.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $\Lambda =\{m{\omega }_1+n{\omega }_2:m,n\in \mathbb{Z}\}$ be a lattice in $\mathbb{C}$, where ${\omega }_1,{\omega }_2$ are $\mathbb{R}$-linearly independent (i.e., ${\omega }_2/{\omega }_1\notin \mathbb{R}$). The Weierstrass $\wp$-function associated to $\Lambda$ is

$$\wp (z)=\frac{1}{z^2}+\sum _{\omega \in \Lambda \setminus \{0\}}\left[\frac{1}{(z-\omega {)}^2}-\frac{1}{{\omega }^2}\right].$$

The lattice invariants are

$$g_2=60\sum _{\omega \in \Lambda \setminus \{0\}}\frac{1}{{\omega }^4},g_3=140\sum _{\omega \in \Lambda \setminus \{0\}}\frac{1}{{\omega }^6}.$$

The Weierstrass $\zeta$-function (not to be confused with the Riemann zeta function) is

$$\zeta (z)=\frac{1}{z}+\sum _{\omega \in \Lambda \setminus \{0\}}\left[\frac{1}{z-\omega }+\frac{1}{\omega }+\frac{z}{{\omega }^2}\right],$$

and the Weierstrass $\sigma$-function is

$$\sigma (z)=z\prod _{\omega \in \Lambda \setminus \{0\}}\left(1-\frac{z}{\omega }\right)\exp \left(\frac{z}{\omega }+\frac{z^2}{2{\omega }^2}\right).$$

^{[Ahlfors Ch. 7]}

Counterexamples to common slips

• $\wp$ is not doubly-periodic for the naive sum $\sum 1/(z-\omega {)}^2$. The raw sum $\sum 1/(z-\omega {)}^2$ does converge (it is absolutely convergent since $|\omega |\sim \sqrt{m^2+n^2}$ and the terms decay like $1/|\omega {|}^2$). But the convergence of the differentiated series is delicate. The correction $-1/{\omega }^2$ in the definition of $\wp$ ensures absolute convergence without affecting the principal parts.
• The periods ${\omega }_1,{\omega }_2$ must be linearly independent over $\mathbb{R}$. If ${\omega }_2/{\omega }_1\in \mathbb{R}$, the "lattice" degenerates to a one-dimensional arithmetic progression, and $\wp$ reduces to a singly-periodic function.
• $\wp$ is even but ${\wp }^{\prime }$ is odd. The evenness $\wp (-z)=\wp (z)$ follows from the lattice being symmetric under $\omega \mapsto -\omega$. The derivative ${\wp }^{\prime }(-z)=-{\wp }^{\prime }(z)$ is odd.

Key theorem with proof [Intermediate+]

Theorem (Properties of the Weierstrass $\wp$-function). Let $\Lambda$ be a lattice. The Weierstrass $\wp$-function is:

(i) A meromorphic function on $\mathbb{C}$ with double poles at each $\omega \in \Lambda$ and no other singularities.

(ii) Even: $\wp (-z)=\wp (z)$ for all $z$.

(iii) Doubly-periodic: $\wp (z+\omega )=\wp (z)$ for all $\omega \in \Lambda$.

(iv) It satisfies the differential equation $({\wp }^{\prime }{)}^2=4{\wp }^3-g_2\wp -g_3$.

Proof of (i). The series defining $\wp$ is absolutely and uniformly convergent on compact subsets of $\mathbb{C}\setminus \Lambda$. For $|z|\le R$ and $|\omega |>2R$, the estimate

$$\left|\frac{1}{(z-\omega {)}^2}-\frac{1}{{\omega }^2}\right|=\left|\frac{2\omega z-z^2}{{\omega }^2(z-\omega {)}^2}\right|\le \frac{2R|\omega |+R^2}{|\omega {|}^2\cdot (|\omega |/2{)}^2}\le \frac{12R}{|\omega {|}^3}$$

holds. Since the lattice points satisfy ${\sum }_{|\omega |>2R}1/|\omega {|}^3<\infty$ (the lattice has $O(r^2)$ points in $|z|\le r$), the series converges uniformly on compact subsets by the Weierstrass $M$-test. Each term is meromorphic, and the uniform limit of meromorphic functions (with controlled poles) is meromorphic.

Proof of (ii). Replace $z$ by $-z$ in the definition:

$$\wp (-z)=\frac{1}{z^2}+\sum _{\omega \ne 0}\left[\frac{1}{(-z-\omega {)}^2}-\frac{1}{{\omega }^2}\right].$$

The lattice is symmetric: $\omega \in \Lambda \iff -\omega \in \Lambda$. Reindexing the sum by $\omega \mapsto -\omega$:

$$\wp (-z)=\frac{1}{z^2}+\sum _{\omega \ne 0}\left[\frac{1}{(-z+\omega {)}^2}-\frac{1}{{\omega }^2}\right]=\frac{1}{z^2}+\sum _{\omega \ne 0}\left[\frac{1}{(\omega -z{)}^2}-\frac{1}{{\omega }^2}\right]=\wp (z).$$

Proof of (iii). Differentiate $\wp$:

$${\wp }^{\prime }(z)=-\frac{2}{z^3}-2\sum _{\omega \ne 0}\frac{1}{(z-\omega {)}^3}.$$

The derivative series $\sum 1/(z-\omega {)}^3$ converges absolutely since $|\omega {|}^{-3}$ is summable over the lattice (as $|\omega |\sim \sqrt{m^2+n^2}$ and $\sum 1/|m^2+n^2{|}^{3/2}$ converges in two dimensions). This series is manifestly periodic: shifting $z\mapsto z+{\omega }_0$ merely reindexes the sum. Hence ${\wp }^{\prime }(z+{\omega }_0)={\wp }^{\prime }(z)$ for all ${\omega }_0\in \Lambda$.

Since ${\wp }^{\prime }$ is periodic with periods ${\omega }_1,{\omega }_2$, the function $\wp (z+{\omega }_j)-\wp (z)$ has zero derivative for $j=1,2$, hence is constant. At $z=-{\omega }_j/2$ (the half-lattice point): $\wp ({\omega }_j/2)-\wp (-{\omega }_j/2)=0$ by evenness (part (ii)). Hence $\wp (z+{\omega }_j)=\wp (z)$.

Proof of (iv). Near $z=0$, the Laurent expansion of $\wp$ is $\wp (z)=1/z^2+g_2{z}^2/20+g_3{z}^4/28+\cdots$ (obtained by expanding each term). Differentiating: ${\wp }^{\prime }(z)=-2/z^3+g_{2}z/10+g_3{z}^3/7+\cdots$. Computing:

$${\wp }^{\prime }(z{)}^2=\frac{4}{z^6}-\frac{2g_2}{5z^2}+\frac{4g_3}{7}+\cdots$$ $$4\wp (z{)}^3=\frac{4}{z^6}+\frac{3g_2}{5z^2}+\frac{3g_3}{7}+\cdots$$ $$g_2\wp (z)=\frac{g_2}{z^2}+\cdots ,g_3=g_3.$$

Therefore ${\wp }^{\prime 2}-4{\wp }^3+g_2\wp +g_3=0+(\text{higher order terms})$. The left side is an elliptic function (combination of elliptic functions) with no poles (all poles cancel), hence bounded and entire by Liouville's theorem applied to the fundamental parallelogram. A bounded entire elliptic function is constant, and the constant is $0$ (evaluate at $z=0$ to check the leading terms cancel). $\square$

Bridge. The Weierstrass $\wp$-function builds toward the theory of elliptic curves, where it appears again as the uniformising map from $\mathbb{C}/\Lambda$ to the cubic curve $y^2=4x^3-g_{2}x-g_3$. The foundational reason $\wp$ works is that the Mittag-Leffler theorem 06.01.18 guarantees the existence of a meromorphic function with prescribed double poles on the lattice, and the symmetry of the lattice forces double periodicity. This is exactly the bridge from the discrete data (the lattice $\Lambda$) to the continuous object (the elliptic function), and the central insight is that the differential equation $({\wp }^{\prime }{)}^2=4{\wp }^3-g_2\wp -g_3$ identifies $\wp$ with the algebraic geometry of the corresponding elliptic curve. Putting these together with the Weierstrass factorisation 06.01.17 identifies the $\sigma$-function as the canonical product over the lattice, and the pattern generalises through the addition formula to the group law on the elliptic curve.

Exercises [Intermediate+]

Advanced results [Master]

The Weierstrass $\sigma$-function and the addition formula. The Weierstrass $\sigma$-function is the canonical product (in the sense of 06.01.17 the Weierstrass factorisation theorem) with simple zeros at each lattice point $\omega \in \Lambda$. It satisfies $\wp (z)=-d^2/d{z}^2[\log \sigma (z)]$ and $\zeta (z)={\sigma }^{\prime }(z)/\sigma (z)=-d/dz[\log \sigma (z)]$. The addition formula for $\wp$ can be expressed in terms of the $\sigma$-function as $\wp (z_1+z_2)-\wp (z_1)-\wp (z_2)=[\sigma (z_1-z_2)/(\sigma (z_1)\sigma (z_2)){]}^2\cdot [{\wp }^{\prime }(z_1){\wp }^{\prime }(z_2)/(\sigma (z_1-z_2){)}^2]$, or more concisely using the $\zeta$-function. ^{[Ahlfors Ch. 7]}

Liouville's theorem for elliptic functions. A holomorphic elliptic function (doubly-periodic entire function) is constant. The proof is direct: an entire doubly-periodic function is bounded on the fundamental parallelogram (compact), hence bounded on all of $\mathbb{C}$, hence constant by the ordinary Liouville theorem. This is the foundational reason that non-constant elliptic functions must have poles.

The field of elliptic functions. The set of all elliptic functions with period lattice $\Lambda$ forms a field $\mathbb{C}(\wp ,{\wp }^{\prime })$. Every elliptic function can be written as $R_1(\wp )+R_2(\wp )\cdot {\wp }^{\prime }$ where $R_1,R_2$ are rational functions. This is the algebraic closure theorem: $\wp$ and ${\wp }^{\prime }$ are the generators of the field of elliptic functions, just as $x$ generates the field of rational functions.

Connection to elliptic curves. The differential equation $({\wp }^{\prime }{)}^2=4{\wp }^3-g_2\wp -g_3$ identifies the torus $\mathbb{C}/\Lambda$ with the elliptic curve $E:y^2=4x^3-g_{2}x-g_3$. The discriminant $\Delta =g_2^3-27g_3^2$ is non-zero if and only if the curve is smooth (no repeated roots). The $j$-invariant $j(\Lambda )=1728g_2^3/\Delta$ classifies elliptic curves up to isomorphism over $\mathbb{C}$.

Modular invariance. The $j$-invariant depends only on the lattice up to homothety: $j(\lambda \Lambda )=j(\Lambda )$ for $\lambda \ne 0$. This makes $j$ a function on the modular curve ${\mathrm{SL}}_2(\mathbb{Z})\backslash \mathbb{H}$, where $\mathbb{H}$ is the upper half-plane parametrising lattices via $\tau ={\omega }_2/{\omega }_1$. The function $j(\tau )$ is a modular function and is the starting point for the theory of modular forms.

Synthesis. The Weierstrass $\wp$-function is the foundational reason that the theory of doubly-periodic functions reduces to the study of a single function and its derivative, and the central insight is that the lattice symmetry forces both the double periodicity and the algebraic relation $({\wp }^{\prime }{)}^2=4{\wp }^3-g_2\wp -g_3$. This is exactly the structure that identifies the analytic object $\wp$ with the algebraic object (the elliptic curve $y^2=4x^3-g_{2}x-g_3$), and the bridge is between the complex torus $\mathbb{C}/\Lambda$ and the algebraic curve $E$. Putting these together with the Mittag-Leffler theorem 06.01.18 (which guarantees existence of the prescribed-pole function) and the Weierstrass factorisation 06.01.17 (which produces the $\sigma$-function), the entire theory of elliptic functions emerges from the interplay of these two constructions. The pattern recurs in the addition formula (which encodes the group law on the elliptic curve), the $j$-invariant (which classifies elliptic curves), and the modular theory (which parametrises lattices), and the pattern generalises to abelian varieties (higher-dimensional analogues of elliptic curves) and the Weierstrass preparation theorem (which extends the $\sigma$-function machinery to several variables).

Full proof set [Master]

Proposition (Convergence of the $\wp$ series). The series $\wp (z)=1/z^2+{\sum }_{\omega \ne 0}[1/(z-\omega {)}^2-1/{\omega }^2]$ converges absolutely and uniformly on compact subsets of $\mathbb{C}\setminus \Lambda$.

Proof. Let $K$ be a compact set with $K\cap \Lambda =\varnothing$, and let $R={\max }_{z\in K}|z|$. For $|\omega |>2R$ and $z\in K$:

$$\left|\frac{1}{(z-\omega {)}^2}-\frac{1}{{\omega }^2}\right|=\left|\frac{{\omega }^2-(z-\omega {)}^2}{{\omega }^2(z-\omega {)}^2}\right|=\left|\frac{2z\omega -z^2}{{\omega }^2(z-\omega {)}^2}\right|\le \frac{2R|\omega |+R^2}{|\omega {|}^2(|\omega |-R{)}^2}.$$

For $|\omega |>2R$: $|\omega |-R>|\omega |/2$, so the bound becomes $(2R|\omega |+R^2)/(|\omega {|}^2\cdot |\omega {|}^2/4)\le 12R/|\omega {|}^3$.

The number of lattice points with $n\le |\omega |\le n+1$ is $O(n)$ (area of an annulus divided by the area of a fundamental parallelogram). Hence ${\sum }_{\omega \ne 0,|\omega |>2R}|\omega {|}^{-3}$ converges by comparison with $\sum n\cdot n^{-3}=\sum n^{-2}$. By the Weierstrass $M$-test, the series converges uniformly on $K$. $\square$

Proposition (The $\sigma$-function as canonical product). The Weierstrass $\sigma$-function $\sigma (z)=z{\prod }_{\omega \ne 0}(1-z/\omega )\exp (z/\omega +z^2/(2{\omega }^2))$ converges to an entire function with simple zeros exactly at the lattice points.

Proof. The lattice $\Lambda$ has counting function $n(r)=O(r^2)$ (the number of lattice points in $|z|\le r$ grows quadratically). The exponent of convergence of the zeros is $\inf \{\alpha :\sum 1/|\omega {|}^{\alpha }<\infty \}$. Since ${\sum }_{|\omega |>N}1/|\omega {|}^{\alpha }$ converges for $\alpha >2$ and diverges for $\alpha \le 2$, the exponent of convergence is $2$.

By the Weierstrass factorisation theorem 06.01.17, the canonical product of genus $p=2$ (using primary factors $E_2(z/\omega )=(1-z/\omega )\exp (z/\omega +z^2/(2{\omega }^2))$) converges uniformly on compact sets when $\sum 1/|\omega {|}^{p+1}=\sum 1/|\omega {|}^3<\infty$, which holds. Hence $\sigma$ is entire with simple zeros at each $\omega \in \Lambda$. $\square$

Connections [Master]

• Weierstrass factorisation theorem 06.01.17. The $\sigma$-function is the canonical Weierstrass product over the lattice: $\sigma (z)=z{\prod }_{\omega \ne 0}{E}_2(z/\omega )$, where $E_2$ is the primary factor of genus $2$. The factorisation theorem guarantees convergence, and the $\sigma$-function is the building block from which $\zeta$ and $\wp$ are derived by differentiation and negation of the logarithmic derivative.

• Mittag-Leffler theorem 06.01.18. The $\wp$-function is a Mittag-Leffler prescription: double poles at each lattice point with principal part $1/(z-\omega {)}^2$. The Mittag-Leffler theorem guarantees the existence of a meromorphic function with these prescribed poles, and the lattice symmetry forces the result to be doubly periodic. The correction $-1/{\omega }^2$ in the $\wp$ series is the Mittag-Leffler convergence-producing term.

• Schwarz-Christoffel formula 06.01.19. The Schwarz-Christoffel map from the upper half-plane to a rectangle is an elliptic integral, and the inverse map is a ratio of $\sigma$-functions (an elliptic function). The connection between conformal mapping to rectangles and the Weierstrass $\wp$-function is the bridge between the geometric theory of conformal maps and the algebraic theory of elliptic functions.

Historical & philosophical context [Master]

Abel 1827 ^{[Abel 1827]}, in Recherches sur les fonctions elliptiques in the Journal fur die reine und angewandte Mathematik, introduced elliptic functions as inverses of elliptic integrals, opening the field that bears their name. Abel's approach was through the inversion of the integral that gives the arc length of an ellipse. Jacobi 1829 ^{[Jacobi 1829]}, in Fundamenta Nova Theoriae Functionum Ellipticarum, developed the theory further through theta functions and the Jacobi elliptic functions $\mathrm{sn},\mathrm{cn},\mathrm{dn}$.

Weierstrass, in his 1862 lectures on elliptic functions (published in Mathematische Werke Vol. 5) ^{[Weierstrass 1862]}, introduced the $\wp$-function as a more symmetric approach. While Abel and Jacobi worked with specific elliptic integrals and their inverses, Weierstrass showed that a single function $\wp$ (defined by a lattice sum) generates the entire theory. The $\wp$-approach has the advantage of making the connection to algebraic geometry (the elliptic curve $y^2=4x^3-g_{2}x-g_3$) manifest. The canonical modern treatments are Ahlfors Complex Analysis Ch. 7 and Jones-Singerman Complex Functions ^{[Jones-Singerman]}.

Bibliography [Master]

@article{Abel1827,
  author = {Abel, Niels Henrik},
  title = {Recherches sur les fonctions elliptiques},
  journal = {Journal f\"ur die reine und angewandte Mathematik},
  volume = {2},
  year = {1827},
  pages = {101--181},
  note = {Foundation of elliptic function theory via inversion of elliptic integrals}
}

@book{Jacobi1829,
  author = {Jacobi, Carl Gustav Jacob},
  title = {Fundamenta Nova Theoriae Functionum Ellipticarum},
  publisher = {Borntr\"ager, K\"onigsberg},
  year = {1829},
  note = {Jacobi theta functions and elliptic functions}
}

@book{WeierstrassWerke5,
  author = {Weierstrass, Karl},
  title = {Vorlesungen \"uber elliptische Functionen},
  booktitle = {Mathematische Werke},
  publisher = {Mayer \& M\"uller, Berlin},
  year = {1915},
  volume = {5},
  note = {Lectures from circa 1862; Weierstrass p-function}
}

@book{Ahlfors1979,
  author = {Ahlfors, Lars V.},
  title = {Complex Analysis},
  publisher = {McGraw-Hill},
  year = {1979},
  edition = {3rd},
  note = {Chapter 7: elliptic functions, Weierstrass p-function}
}

@book{JonesSingerman1987,
  author = {Jones, Gareth A. and Singerman, David},
  title = {Complex Functions: An Algebraic and Geometric Viewpoint},
  publisher = {Cambridge University Press},
  year = {1987},
  note = {Chapters 3--4: elliptic functions and modular forms}
}