Weierstrass elliptic function

In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions is also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script p. They play an important role in the theory of elliptic functions, i.e., meromorphic functions that are doubly periodic. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice.

A cubic of the form ${\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x-g_{3}\}}$, where ${\displaystyle g_{2},g_{3}\in \mathbb {C} }$ are complex numbers with ${\displaystyle g_{2}^{3}-27g_{3}^{2}\neq 0}$, cannot be rationally parameterized.^[1] Yet one still wants to find a way to parameterize it.

For the quadric ${\displaystyle K=\left\{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}=1\right\}}$; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function: $${\displaystyle \psi :\mathbb {R} /2\pi \mathbb {Z} \to K,\quad t\mapsto (\sin t,\cos t).}$$ Because of the periodicity of the sine and cosine ${\displaystyle \mathbb {R} /2\pi \mathbb {Z} }$ is chosen to be the domain, so the function is bijective.

In a similar way one can get a parameterization of ${\displaystyle C_{g_{2},g_{3}}^{\mathbb {C} }}$ by means of the doubly periodic ${\displaystyle \wp }$-function (see in the section "Relation to elliptic curves"). This parameterization has the domain ${\displaystyle \mathbb {C} /\Lambda }$, which is topologically equivalent to a torus.^[2]

There is another analogy to the trigonometric functions. Consider the integral function $${\displaystyle a(x)=\int _{0}^{x}{\frac {dy}{\sqrt {1-y^{2}}}}.}$$ It can be simplified by substituting ${\displaystyle y=\sin t}$ and ${\displaystyle s=\arcsin x}$: $${\displaystyle a(x)=\int _{0}^{s}dt=s=\arcsin x.}$$ That means ${\displaystyle a^{-1}(x)=\sin x}$. So the sine function is an inverse function of an integral function.^[3]

Elliptic functions are the inverse functions of elliptic integrals. In particular, let: $${\displaystyle u(z)=\int _{z}^{\infty }{\frac {ds}{\sqrt {4s^{3}-g_{2}s-g_{3}}}}.}$$ Then the extension of ${\displaystyle u^{-1}}$ to the complex plane equals the ${\displaystyle \wp }$-function.^[4] This invertibility is used in complex analysis to provide a solution to certain nonlinear differential equations satisfying the Painlevé property, i.e., those equations that admit poles as their only movable singularities.^[5]

Let ${\displaystyle \omega _{1},\omega _{2}\in \mathbb {C} }$ be two complex numbers that are linearly independent over ${\displaystyle \mathbb {R} }$ and let ${\displaystyle \Lambda :=\mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}:=\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}}$ be the period lattice generated by those numbers. Then the ${\displaystyle \wp }$-function is defined as follows:

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

This series converges locally uniformly absolutely in the complex torus ${\displaystyle \mathbb {C} /\Lambda }$.

It is common to use ${\displaystyle 1}$ and ${\displaystyle \tau }$ in the upper half-plane ${\displaystyle \mathbb {H} :=\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}}$ as generators of the lattice. Dividing by ${\textstyle \omega _{1}}$ maps the lattice ${\displaystyle \mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}}$ isomorphically onto the lattice ${\displaystyle \mathbb {Z} +\mathbb {Z} \tau }$ with ${\textstyle \tau ={\tfrac {\omega _{2}}{\omega _{1}}}}$. Because ${\displaystyle -\tau }$ can be substituted for ${\displaystyle \tau }$, without loss of generality we can assume ${\displaystyle \tau \in \mathbb {H} }$, and then define ${\displaystyle \wp (z,\tau ):=\wp (z,1,\tau )}$. With that definition, we have ${\displaystyle \wp (z,\omega _{1},\omega _{2})=\omega _{1}^{-2}\wp (z/\omega _{1},\omega _{2}/\omega _{1})}$.

• ${\displaystyle \wp }$ is a meromorphic function with a pole of order 2 at each period ${\displaystyle \lambda }$ in ${\displaystyle \Lambda }$.
• ${\displaystyle \wp }$ is a homogeneous function in that:

${\displaystyle \wp (\lambda z,\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-2}\wp (z,\omega _{1},\omega _{2}).}$

• ${\displaystyle \wp }$ is an even function. That means ${\displaystyle \wp (z)=\wp (-z)}$ for all ${\displaystyle z\in \mathbb {C} \setminus \Lambda }$, which can be seen in the following way:

${\displaystyle {\begin{aligned}\wp (-z)&={\frac {1}{(-z)^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(-z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)\\&={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z+\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)\\&={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)=\wp (z).\end{aligned}}}$

The second last equality holds because ${\displaystyle \{-\lambda :\lambda \in \Lambda \}=\Lambda }$. Since the sum converges absolutely this rearrangement does not change the limit.

• The derivative of ${\displaystyle \wp }$ is given by:^[6] $${\displaystyle \wp '(z)=-2\sum _{\lambda \in \Lambda }{\frac {1}{(z-\lambda )^{3}}}.}$$
• ${\displaystyle \wp }$ and ${\displaystyle \wp '}$ are doubly periodic with the periods ${\displaystyle \omega _{1}}$ and ${\displaystyle \omega _{2}}$.^[6] This means: $${\displaystyle {\begin{aligned}\wp (z+\omega _{1})&=\wp (z)=\wp (z+\omega _{2}),\ {\textrm {and}}\\[3mu]\wp '(z+\omega _{1})&=\wp '(z)=\wp '(z+\omega _{2}).\end{aligned}}}$$ It follows that ${\displaystyle \wp (z+\lambda )=\wp (z)}$ and ${\displaystyle \wp '(z+\lambda )=\wp '(z)}$ for all ${\displaystyle \lambda \in \Lambda }$.

Let ${\displaystyle r:=\min\{{|\lambda }|:0\neq \lambda \in \Lambda \}}$. Then for ${\displaystyle 0<|z|<r}$ the ${\displaystyle \wp }$-function has the following Laurent expansion $${\displaystyle \wp (z)={\frac {1}{z^{2}}}+\sum _{n=1}^{\infty }(2n+1)G_{2n+2}z^{2n}}$$ where $${\displaystyle G_{n}=\sum _{0\neq \lambda \in \Lambda }\lambda ^{-n}}$$ for ${\displaystyle n\geq 3}$ are so called Eisenstein series.^[6]

Set ${\displaystyle g_{2}=60G_{4}}$ and ${\displaystyle g_{3}=140G_{6}}$. Then the ${\displaystyle \wp }$-function satisfies the differential equation^[6] $${\displaystyle \wp '^{2}(z)=4\wp ^{3}(z)-g_{2}\wp (z)-g_{3}.}$$ This relation can be verified by forming a linear combination of powers of ${\displaystyle \wp }$ and ${\displaystyle \wp '}$ to eliminate the pole at ${\displaystyle z=0}$. This yields an entire elliptic function that has to be constant by Liouville's theorem.^[6]

The coefficients of the above differential equation ${\displaystyle g_{2}}$ and ${\displaystyle g_{3}}$ are known as the invariants. Because they depend on the lattice ${\displaystyle \Lambda }$ they can be viewed as functions in ${\displaystyle \omega _{1}}$ and ${\displaystyle \omega _{2}}$.

The series expansion suggests that ${\displaystyle g_{2}}$ and ${\displaystyle g_{3}}$ are homogeneous functions of degree ${\displaystyle -4}$ and ${\displaystyle -6}$. That is^[7] $${\displaystyle g_{2}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-4}g_{2}(\omega _{1},\omega _{2})}$$ $${\displaystyle g_{3}(\lambda \omega _{1},\lambda \omega _{2})=\lambda ^{-6}g_{3}(\omega _{1},\omega _{2})}$$ for ${\displaystyle \lambda \neq 0}$.

If ${\displaystyle \omega _{1}}$ and ${\displaystyle \omega _{2}}$ are chosen in such a way that ${\displaystyle \operatorname {Im} \left({\tfrac {\omega _{2}}{\omega _{1}}}\right)>0}$, ${\displaystyle g_{2}}$ and ${\displaystyle g_{3}}$ can be interpreted as functions on the upper half-plane ${\displaystyle \mathbb {H} :=\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}}$.

Let ${\displaystyle \tau ={\tfrac {\omega _{2}}{\omega _{1}}}}$. One has:^[8] $${\displaystyle g_{2}(1,\tau )=\omega _{1}^{4}g_{2}(\omega _{1},\omega _{2}),}$$ $${\displaystyle g_{3}(1,\tau )=\omega _{1}^{6}g_{3}(\omega _{1},\omega _{2}).}$$ That means g₂ and g₃ are only scaled by doing this. Set $${\displaystyle g_{2}(\tau ):=g_{2}(1,\tau )}$$ and $${\displaystyle g_{3}(\tau ):=g_{3}(1,\tau ).}$$ As functions of ${\displaystyle \tau \in \mathbb {H} }$, ${\displaystyle g_{2}}$ and ${\displaystyle g_{3}}$ are so called modular forms.

The Fourier series for ${\displaystyle g_{2}}$ and ${\displaystyle g_{3}}$ are given as follows:^[9] $${\displaystyle g_{2}(\tau )={\frac {4}{3}}\pi ^{4}\left[1+240\sum _{k=1}^{\infty }\sigma _{3}(k)q^{2k}\right]}$$ $${\displaystyle g_{3}(\tau )={\frac {8}{27}}\pi ^{6}\left[1-504\sum _{k=1}^{\infty }\sigma _{5}(k)q^{2k}\right]}$$ where $${\displaystyle \sigma _{m}(k):=\sum _{d\mid {k}}d^{m}}$$ is the divisor function and ${\displaystyle q=e^{\pi i\tau }}$ is the nome.

The modular discriminant ${\displaystyle \Delta }$ is defined as the discriminant of the characteristic polynomial of the differential equation $${\displaystyle \wp '^{2}(z)=4\wp ^{3}(z)-g_{2}\wp (z)-g_{3}}$$ as follows: $${\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}.}$$ The discriminant is a modular form of weight ${\displaystyle 12}$. That is, under the action of the modular group, it transforms as $${\displaystyle \Delta \left({\frac {a\tau +b}{c\tau +d}}\right)=\left(c\tau +d\right)^{12}\Delta (\tau )}$$ where ${\displaystyle a,b,d,c\in \mathbb {Z} }$ with ${\displaystyle ad-bc=1}$.^[10]

Note that ${\displaystyle \Delta =(2\pi )^{12}\eta ^{24}}$ where ${\displaystyle \eta }$ is the Dedekind eta function.^[11]

For the Fourier coefficients of ${\displaystyle \Delta }$, see Ramanujan tau function.

${\displaystyle e_{1}}$, ${\displaystyle e_{2}}$ and ${\displaystyle e_{3}}$ are usually used to denote the values of the ${\displaystyle \wp }$-function at the half-periods. $${\displaystyle e_{1}\equiv \wp \left({\frac {\omega _{1}}{2}}\right)}$$ $${\displaystyle e_{2}\equiv \wp \left({\frac {\omega _{2}}{2}}\right)}$$ $${\displaystyle e_{3}\equiv \wp \left({\frac {\omega _{1}+\omega _{2}}{2}}\right)}$$ They are pairwise distinct and only depend on the lattice ${\displaystyle \Lambda }$ and not on its generators.^[12]

${\displaystyle e_{1}}$, ${\displaystyle e_{2}}$ and ${\displaystyle e_{3}}$ are the roots of the cubic polynomial ${\displaystyle 4\wp (z)^{3}-g_{2}\wp (z)-g_{3}}$ and are related by the equation: $${\displaystyle e_{1}+e_{2}+e_{3}=0.}$$ Because those roots are distinct the discriminant ${\displaystyle \Delta }$ does not vanish on the upper half plane.^[13] Now we can rewrite the differential equation: $${\displaystyle \wp '^{2}(z)=4(\wp (z)-e_{1})(\wp (z)-e_{2})(\wp (z)-e_{3}).}$$ That means the half-periods are zeros of ${\displaystyle \wp '}$.

The invariants ${\displaystyle g_{2}}$ and ${\displaystyle g_{3}}$ can be expressed in terms of these constants in the following way:^[14] $${\displaystyle g_{2}=-4(e_{1}e_{2}+e_{1}e_{3}+e_{2}e_{3})}$$ $${\displaystyle g_{3}=4e_{1}e_{2}e_{3}}$$ ${\displaystyle e_{1}}$, ${\displaystyle e_{2}}$ and ${\displaystyle e_{3}}$ are related to the modular lambda function: $${\displaystyle \lambda (\tau )={\frac {e_{3}-e_{2}}{e_{1}-e_{2}}},\quad \tau ={\frac {\omega _{2}}{\omega _{1}}}.}$$

For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of Jacobi's elliptic functions.

The basic relations are:^[15] $${\displaystyle \wp (z)=e_{3}+{\frac {e_{1}-e_{3}}{\operatorname {sn} ^{2}w}}=e_{2}+(e_{1}-e_{3}){\frac {\operatorname {dn} ^{2}w}{\operatorname {sn} ^{2}w}}=e_{1}+(e_{1}-e_{3}){\frac {\operatorname {cn} ^{2}w}{\operatorname {sn} ^{2}w}}}$$ where ${\displaystyle e_{1},e_{2}}$ and ${\displaystyle e_{3}}$ are the three roots described above and where the modulus k of the Jacobi functions equals $${\displaystyle k={\sqrt {\frac {e_{2}-e_{3}}{e_{1}-e_{3}}}}}$$ and their argument w equals $${\displaystyle w=z{\sqrt {e_{1}-e_{3}}}.}$$

The function ${\displaystyle \wp (z,\tau )=\wp (z,1,\omega _{2}/\omega _{1})}$ can be represented by Jacobi's theta functions: $${\displaystyle \wp (z,\tau )=\left(\pi \theta _{2}(0,q)\theta _{3}(0,q){\frac {\theta _{4}(\pi z,q)}{\theta _{1}(\pi z,q)}}\right)^{2}-{\frac {\pi ^{2}}{3}}\left(\theta _{2}^{4}(0,q)+\theta _{3}^{4}(0,q)\right)}$$ where ${\displaystyle q=e^{\pi i\tau }}$ is the nome and ${\displaystyle \tau }$ is the period ratio ${\displaystyle (\tau \in \mathbb {H} )}$.^[16] This also provides a very rapid algorithm for computing ${\displaystyle \wp (z,\tau )}$.

Consider the embedding of the cubic curve in the complex projective plane

${\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }=\{(x,y)\in \mathbb {C} ^{2}:y^{2}=4x^{3}-g_{2}x-g_{3}\}\cup \{O\}\subset \mathbb {C} ^{2}\cup \mathbb {P} _{1}(\mathbb {C} )=\mathbb {P} _{2}(\mathbb {C} ).}$

where ${\displaystyle O}$ is a point lying on the line at infinity ${\displaystyle \mathbb {P} _{1}(\mathbb {C} )}$. For this cubic there exists no rational parameterization, if ${\displaystyle \Delta \neq 0}$.^[1] In this case it is also called an elliptic curve. Nevertheless there is a parameterization in homogeneous coordinates that uses the ${\displaystyle \wp }$-function and its derivative ${\displaystyle \wp '}$:^[17]

${\displaystyle \varphi (\wp ,\wp '):\mathbb {C} /\Lambda \to {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} },\quad z\mapsto {\begin{cases}\left[\wp (z):\wp '(z):1\right]&z\notin \Lambda \\\left[0:1:0\right]\quad &z\in \Lambda \end{cases}}}$

Now the map ${\displaystyle \varphi }$ is bijective and parameterizes the elliptic curve ${\displaystyle {\bar {C}}_{g_{2},g_{3}}^{\mathbb {C} }}$.

${\displaystyle \mathbb {C} /\Lambda }$ is an abelian group and a topological space, equipped with the quotient topology.

It can be shown that every Weierstrass cubic is given in such a way. That is to say that for every pair ${\displaystyle g_{2},g_{3}\in \mathbb {C} }$ with ${\displaystyle \Delta =g_{2}^{3}-27g_{3}^{2}\neq 0}$ there exists a lattice ${\displaystyle \mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}}$, such that

${\displaystyle g_{2}=g_{2}(\omega _{1},\omega _{2})}$ and ${\displaystyle g_{3}=g_{3}(\omega _{1},\omega _{2})}$.^[18]

The statement that elliptic curves over ${\displaystyle \mathbb {Q} }$ can be parameterized over ${\displaystyle \mathbb {Q} }$, is known as the modularity theorem. This is an important theorem in number theory. It was part of Andrew Wiles' proof (1995) of Fermat's Last Theorem.

The addition theorem states^[19] that if ${\displaystyle z,w,}$ and ${\displaystyle z+w}$ do not belong to ${\displaystyle \Lambda }$, then $${\displaystyle \det {\begin{bmatrix}1&\wp (z)&\wp '(z)\\1&\wp (w)&\wp '(w)\\1&\wp (z+w)&-\wp '(z+w)\end{bmatrix}}=0.}$$ This states that the points ${\displaystyle P=(\wp (z),\wp '(z)),}$ ${\displaystyle Q=(\wp (w),\wp '(w)),}$ and ${\displaystyle R=(\wp (z+w),-\wp '(z+w))}$ are collinear, the geometric form of the group law of an elliptic curve.

This can be proven^[20] by considering constants ${\displaystyle A,B}$ such that $${\displaystyle \wp '(z)=A\wp (z)+B,\quad \wp '(w)=A\wp (w)+B.}$$ Then the elliptic function $${\displaystyle \wp '(\zeta )-A\wp (\zeta )-B}$$ has a pole of order three at zero, and therefore three zeros whose sum belongs to ${\displaystyle \Lambda }$. Two of the zeros are ${\displaystyle z}$ and ${\displaystyle w}$, and thus the third is congruent to ${\displaystyle -z-w}$.

The addition theorem can be put into the alternative form, for ${\displaystyle z,w,z-w,z+w\not \in \Lambda }$:^[21] $${\displaystyle \wp (z+w)={\frac {1}{4}}\left[{\frac {\wp '(z)-\wp '(w)}{\wp (z)-\wp (w)}}\right]^{2}-\wp (z)-\wp (w).}$$

As well as the duplication formula:^[21] $${\displaystyle \wp (2z)={\frac {1}{4}}\left[{\frac {\wp ''(z)}{\wp '(z)}}\right]^{2}-2\wp (z).}$$

This can be proven from the addition theorem shown above. The points ${\displaystyle P=(\wp (u),\wp '(u)),Q=(\wp (v),\wp '(v)),}$ and ${\displaystyle R=(\wp (u+v),-\wp '(u+v))}$ are collinear and lie on the curve ${\displaystyle y^{2}=4x^{3}-g_{2}x-g_{3}}$. The slope of that line is $${\displaystyle m={\frac {y_{P}-y_{Q}}{x_{P}-x_{Q}}}={\frac {\wp '(u)-\wp '(v)}{\wp (u)-\wp (v)}}.}$$ So ${\displaystyle x=x_{P}=\wp (u)}$, ${\displaystyle x=x_{Q}=\wp (v)}$, and ${\displaystyle x=x_{R}=\wp (u+v)}$ all satisfy a cubic $${\displaystyle (mx+q)^{2}=4x^{3}-g_{2}x-g_{3},}$$ where ${\displaystyle q}$ is a constant. This becomes $${\displaystyle 4x^{3}-m^{2}x^{2}-(2mq+g_{2})x-g_{3}-q^{2}=0.}$$ Thus ${\displaystyle x_{P}+x_{Q}+x_{R}={\frac {m^{2}}{4}}}$ which provides the wanted formula ${\displaystyle \wp (u+v)+\wp (u)+\wp (v)={\frac {1}{4}}\left[{\frac {\wp '(u)-\wp '(v)}{\wp (u)-\wp (v)}}\right]^{2}.}$

A direct proof is as follows.^[22] Any elliptic function ${\displaystyle f}$ can be expressed as: $${\displaystyle f(u)=c\prod _{i=1}^{n}{\frac {\sigma (u-a_{i})}{\sigma (u-b_{i})}}\quad c\in \mathbb {C} }$$ where ${\displaystyle \sigma }$ is the Weierstrass sigma function and ${\displaystyle a_{i},b_{i}}$ are the respective zeros and poles in the period parallelogram. Considering the function ${\displaystyle \wp (u)-\wp (v)}$ as a function of ${\displaystyle u}$, we have $${\displaystyle \wp (u)-\wp (v)=c{\frac {\sigma (u+v)\sigma (u-v)}{\sigma (u)^{2}}}.}$$ Multiplying both sides by ${\displaystyle u^{2}}$ and letting ${\displaystyle u\to 0}$, we have ${\displaystyle 1=-c\sigma (v)^{2}}$, so ${\displaystyle c=-{\frac {1}{\sigma (v)^{2}}}\implies \wp (u)-\wp (v)=-{\frac {\sigma (u+v)\sigma (u-v)}{\sigma (u)^{2}\sigma (v)^{2}}}.}$

By definition the Weierstrass zeta function: ${\displaystyle {\frac {d}{dz}}\ln \sigma (z)=\zeta (z)}$ therefore we logarithmically differentiate both sides with respect to ${\displaystyle u}$ obtaining: $${\displaystyle {\frac {\wp '(u)}{\wp (u)-\wp (v)}}=\zeta (u+v)-2\zeta (u)-\zeta (u-v)}$$ Once again by definition ${\displaystyle \zeta '(z)=-\wp (z)}$ thus by differentiating once more on both sides and rearranging the terms we obtain $${\displaystyle -\wp (u+v)=-\wp (u)+{\frac {1}{2}}{\frac {\wp ''(v)[\wp (u)-\wp (v)]-\wp '(u)[\wp '(u)-\wp '(v)]}{[\wp (u)-\wp (v)]^{2}}}}$$ Knowing that ${\displaystyle \wp ''}$ has the following differential equation ${\displaystyle 2\wp ''=12\wp ^{2}-g_{2}}$ and rearranging the terms one gets the wanted formula $${\displaystyle \wp (u+v)={\frac {1}{4}}\left[{\frac {\wp '(u)-\wp '(v)}{\wp (u)-\wp (v)}}\right]^{2}-\wp (u)-\wp (v).}$$

The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘, which was Weierstrass's own notation introduced in his lectures of 1862–1863.^{[footnote 1]} It should not be confused with the normal mathematical script letters P: 𝒫 and 𝓅.

In computing, the letter ℘ is available as \wp in TeX. In Unicode the code point is U+2118 ℘ SCRIPT CAPITAL P (&weierp;, &wp;), with the more correct alias weierstrass elliptic function.^{[footnote 2]} In HTML, it can be escaped as ℘.

• Weierstrass functions
• Jacobi elliptic functions
• Lemniscate elliptic functions

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• Weierstrass's elliptic functions on Mathworld.
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• Weierstrass P function and its derivative implemented in C by David Dumas