Inmathematics, theWeierstrass elliptic functions areelliptic functions that take a particularly simple form. They are named forKarl Weierstrass. This class of functions is also referred to as℘-functions and they are usually denoted by the symbol ℘, a uniquely fancyscriptp. They play an important role in the theory of elliptic functions, i.e.,meromorphic functions that aredoubly periodic. A ℘-function together with its derivative can be used to parameterizeelliptic curves and they generate the field of elliptic functions with respect to a given period lattice.

Symbol for Weierstrass${\displaystyle \mathrm{\wp }}$-function

Acubic 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${\displaystyle g_2^3-27g_3^2\ne 0}$, cannot berationally parameterized.^[1] Yet one still wants to find a way to parameterize it.

For thequadric${\displaystyle K=\left\{(x,y)\in {\mathbb{R}}^2:x^2+y^2=1\right\}}$; theunit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function:$${\displaystyle \psi :\mathbb{R}/2\pi \mathbb{Z}\to K,t\mapsto (\sin t,\cos t).}$$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${\displaystyle \mathrm{\wp }}$-function and its derivative, namely via${\displaystyle (x,y)=(\mathrm{\wp }(z),{\mathrm{\wp }}^{\prime }(z))}$. This parameterization has the domain${\displaystyle \mathbb{C}/\mathrm{\Lambda }}$, which is topologically equivalent to atorus.^[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 ofelliptic integrals. In particular, let:$${\displaystyle u(z)={\int }_z^{\mathrm{\infty }}\frac{ds}{\sqrt{4s^3-g_{2}s-g_3}}.}$$Then the extension of${\displaystyle u^{-1}}$ to the complex plane equals the${\displaystyle \mathrm{\wp }}$-function.^[4] This invertibility is used incomplex analysis to provide a solution to certainnonlinear differential equations satisfying thePainlevé property, i.e., those equations that admitpoles as their onlymovable singularities.^[5]

Let${\displaystyle {\omega }_1,{\omega }_2\in \mathbb{C}}$ be twocomplex numbers that arelinearly independent over${\displaystyle \mathbb{R}}$ and let${\displaystyle \mathrm{\Lambda }:=\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2:=\{m{\omega }_1+n{\omega }_2:m,n\in \mathbb{Z}\}}$ be theperiod lattice generated by those numbers. Then the${\displaystyle \mathrm{\wp }}$-function is defined as follows:

${\displaystyle \mathrm{\wp }(z,{\omega }_1,{\omega }_2):=\mathrm{\wp }(z)=\frac{1}{z^2}+\sum _{\lambda \in \mathrm{\Lambda }\setminus \{0\}}\left(\frac{1}{(z-\lambda {)}^2}-\frac{1}{{\lambda }^2}\right).}$

This series converges locallyuniformly absolutely in thecomplex torus${\displaystyle \mathbb{C}/\mathrm{\Lambda }}$.

It is common to use${\displaystyle 1}$ and${\displaystyle \tau }$ in theupper half-plane${\displaystyle \mathbb{H}:=\{z\in \mathbb{C}:\mathrm{Im}(z)>0\}}$ asgenerators of thelattice. Dividing by${\omega }_1$ maps the lattice${\displaystyle \mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2}$ isomorphically onto the lattice${\displaystyle \mathbb{Z}+\mathbb{Z}\tau }$ with$\tau =\frac{{\omega }_2}{{\omega }_1}$. Because${\displaystyle -\tau }$ can be substituted for${\displaystyle \tau }$, without loss of generality we can assume${\displaystyle \tau \in \mathbb{H}}$, and then define${\displaystyle \mathrm{\wp }(z,\tau ):=\mathrm{\wp }(z,1,\tau )}$. With that definition, we have${\displaystyle \mathrm{\wp }(z,{\omega }_1,{\omega }_2)={\omega }_1^{-2}\mathrm{\wp }(z/{\omega }_1,{\omega }_2/{\omega }_1)}$.

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

${\displaystyle \mathrm{\wp }(\lambda z,\lambda {\omega }_1,\lambda {\omega }_2)={\lambda }^{-2}\mathrm{\wp }(z,{\omega }_1,{\omega }_2).}$

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

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

• The derivative of${\displaystyle \mathrm{\wp }}$ is given by:^[6]$${\displaystyle {\mathrm{\wp }}^{\prime }(z)=-2\sum _{\lambda \in \mathrm{\Lambda }}\frac{1}{(z-\lambda {)}^3}.}$$
• ${\displaystyle \mathrm{\wp }}$ and${\displaystyle {\mathrm{\wp }}^{\prime }}$ aredoubly periodic with the periods${\displaystyle {\omega }_1}$ and${\displaystyle {\omega }_2}$.^[6] This means: It follows that${\displaystyle \mathrm{\wp }(z+\lambda )=\mathrm{\wp }(z)}$ and${\displaystyle {\mathrm{\wp }}^{\prime }(z+\lambda )={\mathrm{\wp }}^{\prime }(z)}$ for all${\displaystyle \lambda \in \mathrm{\Lambda }}$.

Let${\displaystyle r:=min\{|\lambda |:0\ne \lambda \in \mathrm{\Lambda }\}}$. Then for the${\displaystyle \mathrm{\wp }}$-function has the followingLaurent expansion$${\displaystyle \mathrm{\wp }(z)=\frac{1}{z^2}+\sum _{n=1}^{\mathrm{\infty }}(2n+1)G_{2n+2}{z}^{2n}}$$where$${\displaystyle G_n=\sum _{0\ne \lambda \in \mathrm{\Lambda }}{\lambda }^{-n}}$$ for${\displaystyle n\ge 3}$ are so calledEisenstein series.^[6]

Set${\displaystyle g_2=60G_4}$ and${\displaystyle g_3=140G_6}$. Then the${\displaystyle \mathrm{\wp }}$-function satisfies the differential equation^[6]$${\displaystyle {\mathrm{\wp }}^{\prime 2}(z)=4{\mathrm{\wp }}^3(z)-g_2\mathrm{\wp }(z)-g_3.}$$This relation can be verified by forming a linear combination of powers of${\displaystyle \mathrm{\wp }}$ and${\displaystyle {\mathrm{\wp }}^{\prime }}$ to eliminate the pole at${\displaystyle z=0}$. This yields an entire elliptic function that has to be constant byLiouville's theorem.^[6]

The coefficients of the above differential equation${\displaystyle g_2}$ and${\displaystyle g_3}$ are known as theinvariants. Because they depend on the lattice${\displaystyle \mathrm{\Lambda }}$ they can be viewed as functions in${\displaystyle {\omega }_1}$ and${\displaystyle {\omega }_2}$.

The series expansion suggests that${\displaystyle g_2}$ and${\displaystyle g_3}$ arehomogeneous 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${\displaystyle \lambda \ne 0}$.

If${\displaystyle {\omega }_1}$ and${\displaystyle {\omega }_2}$ are chosen in such a way that${\displaystyle \mathrm{Im}\left(\frac{{\omega }_2}{{\omega }_1}\right)>0}$,${\displaystyle g_2}$ and${\displaystyle g_3}$ can be interpreted as functions on theupper half-plane${\displaystyle \mathbb{H}:=\{z\in \mathbb{C}:\mathrm{Im}(z)>0\}}$.

Let${\displaystyle \tau =\frac{{\omega }_2}{{\omega }_1}}$. 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 meansg₂ andg₃ 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 calledmodular forms.

TheFourier series for${\displaystyle g_2}$ and${\displaystyle g_3}$ are given as follows:^[9]$${\displaystyle g_2(\tau )=\frac{4}{3}{\pi }^4\left[1+240\sum _{k=1}^{\mathrm{\infty }}{\sigma }_3(k)q^{2k}\right]}$$$${\displaystyle g_3(\tau )=\frac{8}{27}{\pi }^6\left[1-504\sum _{k=1}^{\mathrm{\infty }}{\sigma }_5(k)q^{2k}\right]}$$where$${\displaystyle {\sigma }_m(k):=\sum _{d\mid k}{d}^m}$$is thedivisor function and${\displaystyle q=e^{\pi i\tau }}$ is thenome.

Themodular discriminant${\displaystyle \mathrm{\Delta }}$ is defined as thediscriminant of the characteristic polynomial of the differential equation$${\displaystyle {\mathrm{\wp }}^{\prime 2}(z)=4{\mathrm{\wp }}^3(z)-g_2\mathrm{\wp }(z)-g_3}$$ as follows:$${\displaystyle \mathrm{\Delta }=g_2^3-27g_3^2.}$$The discriminant is a modular form of weight${\displaystyle 12}$. That is, under the action of themodular group, it transforms as$${\displaystyle \mathrm{\Delta }\left(\frac{a\tau +b}{c\tau +d}\right)={\left(c\tau +d\right)}^{12}\mathrm{\Delta }(\tau )}$$where${\displaystyle a,b,d,c\in \mathbb{Z}}$ with${\displaystyle ad-bc=1}$.^[10]

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

For the Fourier coefficients of${\displaystyle \mathrm{\Delta }}$, seeRamanujan tau function.

${\displaystyle e_1}$,${\displaystyle e_2}$ and${\displaystyle e_3}$ are usually used to denote the values of the${\displaystyle \mathrm{\wp }}$-function at the half-periods.$${\displaystyle e_1\equiv \mathrm{\wp }\left(\frac{{\omega }_1}{2}\right)}$$$${\displaystyle e_2\equiv \mathrm{\wp }\left(\frac{{\omega }_2}{2}\right)}$$$${\displaystyle e_3\equiv \mathrm{\wp }\left(\frac{{\omega }_1+{\omega }_2}{2}\right)}$$They are pairwise distinct and only depend on the lattice${\displaystyle \mathrm{\Lambda }}$ and not on its generators.^[12]

${\displaystyle e_1}$,${\displaystyle e_2}$ and${\displaystyle e_3}$ are the roots of the cubic polynomial${\displaystyle 4\mathrm{\wp }(z{)}^3-g_2\mathrm{\wp }(z)-g_3}$ and are related by the equation:$${\displaystyle e_1+e_2+e_3=0.}$$Because those roots are distinct the discriminant${\displaystyle \mathrm{\Delta }}$ does not vanish on the upper half plane.^[13] Now we can rewrite the differential equation:$${\displaystyle {\mathrm{\wp }}^{\prime 2}(z)=4(\mathrm{\wp }(z)-e_1)(\mathrm{\wp }(z)-e_2)(\mathrm{\wp }(z)-e_3).}$$That means the half-periods are zeros of${\displaystyle {\mathrm{\wp }}^{\prime }}$.

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 themodular lambda function:$${\displaystyle \lambda (\tau )=\frac{e_3-e_2}{e_1-e_2},\tau =\frac{{\omega }_2}{{\omega }_1}.}$$

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

The basic relations are:^[15]$${\displaystyle \mathrm{\wp }(z)=e_3+\frac{e_1-e_3}{{\mathrm{sn}}^{2}w}=e_2+(e_1-e_3)\frac{{\mathrm{dn}}^{2}w}{{\mathrm{sn}}^{2}w}=e_1+(e_1-e_3)\frac{{\mathrm{cn}}^{2}w}{{\mathrm{sn}}^{2}w}}$$where${\displaystyle e_1,e_2}$ and${\displaystyle e_3}$ are the three roots described above and where the modulusk of the Jacobi functions equals$${\displaystyle k=\sqrt{\frac{e_2-e_3}{e_1-e_3}}}$$and their argumentw equals$${\displaystyle w=z\sqrt{e_1-e_3}.}$$

The function${\displaystyle \mathrm{\wp }(z,\tau )=\mathrm{\wp }(z,1,{\omega }_2/{\omega }_1)}$ can be represented byJacobi's theta functions:$${\displaystyle \mathrm{\wp }(z,\tau )={\left(\pi {\theta }_2(0,q){\theta }_3(0,q)\frac{{\theta }_4(\pi z,q)}{{\theta }_1(\pi z,q)}\right)}^2-\frac{{\pi }^2}{3}\left({\theta }_2^4(0,q)+{\theta }_3^4(0,q)\right)}$$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${\displaystyle \mathrm{\wp }(z,\tau )}$.

Consider the embedding of the cubic curve in thecomplex projective plane

${\displaystyle {\overline{C}}_{g_2,g_3}^{\mathbb{C}}=\{(x,y)\in {\mathbb{C}}^2:y^2=4x^3-g_{2}x-g_3\}\cup \{O\}\subset {\mathbb{C}}^2\cup {\mathbb{P}}_1(\mathbb{C})={\mathbb{P}}_2(\mathbb{C}).}$

where${\displaystyle O}$ is a point lying on theline at infinity${\displaystyle {\mathbb{P}}_1(\mathbb{C})}$. For this cubic there exists no rational parameterization, if${\displaystyle \mathrm{\Delta }\ne 0}$.^[1] In this case it is also called an elliptic curve. Nevertheless there is a parameterization inhomogeneous coordinates that uses the${\displaystyle \mathrm{\wp }}$-function and its derivative${\displaystyle {\mathrm{\wp }}^{\prime }}$:^[17]

Now the map${\displaystyle \phi }$ isbijective and parameterizes the elliptic curve${\displaystyle {\overline{C}}_{g_2,g_3}^{\mathbb{C}}}$.

${\displaystyle \mathbb{C}/\mathrm{\Lambda }}$ is anabelian group and atopological space, equipped with thequotient 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${\displaystyle \mathrm{\Delta }=g_2^3-27g_3^2\ne 0}$ 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 themodularity theorem. This is an important theorem innumber theory. It was part ofAndrew Wiles' proof (1995) ofFermat's Last Theorem.

The addition theorem states^[19] that if${\displaystyle z,w,}$ and${\displaystyle z+w}$ do not belong to${\displaystyle \mathrm{\Lambda }}$, thenThis states that the points${\displaystyle P=(\mathrm{\wp }(z),{\mathrm{\wp }}^{\prime }(z)),}$${\displaystyle Q=(\mathrm{\wp }(w),{\mathrm{\wp }}^{\prime }(w)),}$ and${\displaystyle R=(\mathrm{\wp }(z+w),-{\mathrm{\wp }}^{\prime }(z+w))}$ are collinear, the geometric form of thegroup law of an elliptic curve.

This can be proven^[20] by considering constants${\displaystyle A,B}$ such that$${\displaystyle {\mathrm{\wp }}^{\prime }(z)=A\mathrm{\wp }(z)+B,{\mathrm{\wp }}^{\prime }(w)=A\mathrm{\wp }(w)+B.}$$Then the elliptic function$${\displaystyle {\mathrm{\wp }}^{\prime }(\zeta )-A\mathrm{\wp }(\zeta )-B}$$has a pole of order three at zero, and therefore three zeros whose sum belongs to${\displaystyle \mathrm{\Lambda }}$. 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${\displaystyle z,w,z-w,z+w\notin \mathrm{\Lambda }}$:^[21]$${\displaystyle \mathrm{\wp }(z+w)=\frac{1}{4}{\left[\frac{{\mathrm{\wp }}^{\prime }(z)-{\mathrm{\wp }}^{\prime }(w)}{\mathrm{\wp }(z)-\mathrm{\wp }(w)}\right]}^2-\mathrm{\wp }(z)-\mathrm{\wp }(w).}$$

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

This can be proven from the addition theorem shown above. The points${\displaystyle P=(\mathrm{\wp }(u),{\mathrm{\wp }}^{\prime }(u)),Q=(\mathrm{\wp }(v),{\mathrm{\wp }}^{\prime }(v)),}$ and${\displaystyle R=(\mathrm{\wp }(u+v),-{\mathrm{\wp }}^{\prime }(u+v))}$ are collinear and lie on the curve${\displaystyle y^2=4x^3-g_{2}x-g_3}$. The slope of that line is$${\displaystyle m=\frac{y_P-y_Q}{x_P-x_Q}=\frac{{\mathrm{\wp }}^{\prime }(u)-{\mathrm{\wp }}^{\prime }(v)}{\mathrm{\wp }(u)-\mathrm{\wp }(v)}.}$$So${\displaystyle x=x_P=\mathrm{\wp }(u)}$,${\displaystyle x=x_Q=\mathrm{\wp }(v)}$, and${\displaystyle x=x_R=\mathrm{\wp }(u+v)}$ 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${\displaystyle \mathrm{\wp }(u+v)+\mathrm{\wp }(u)+\mathrm{\wp }(v)=\frac{1}{4}{\left[\frac{{\mathrm{\wp }}^{\prime }(u)-{\mathrm{\wp }}^{\prime }(v)}{\mathrm{\wp }(u)-\mathrm{\wp }(v)}\right]}^2.}$

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

By definition theWeierstrass zeta function:${\displaystyle \frac{d}{dz}\ln \sigma (z)=\zeta (z)}$ therefore we logarithmically differentiate both sides with respect to${\displaystyle u}$ obtaining:$${\displaystyle \frac{{\mathrm{\wp }}^{\prime }(u)}{\mathrm{\wp }(u)-\mathrm{\wp }(v)}=\zeta (u+v)-2\zeta (u)-\zeta (u-v)}$$ Once again by definition${\displaystyle {\zeta }^{\prime }(z)=-\mathrm{\wp }(z)}$ thus by differentiating once more on both sides and rearranging the terms we obtain$${\displaystyle -\mathrm{\wp }(u+v)=-\mathrm{\wp }(u)+\frac{1}{2}\frac{{\mathrm{\wp }}^{″}(v)[\mathrm{\wp }(u)-\mathrm{\wp }(v)]-{\mathrm{\wp }}^{\prime }(u)[{\mathrm{\wp }}^{\prime }(u)-{\mathrm{\wp }}^{\prime }(v)]}{[\mathrm{\wp }(u)-\mathrm{\wp }(v){]}^2}}$$ Knowing that${\displaystyle {\mathrm{\wp }}^{″}}$ has the following differential equation${\displaystyle 2{\mathrm{\wp }}^{″}=12{\mathrm{\wp }}^2-g_2}$ and rearranging the terms one gets the wanted formula$${\displaystyle \mathrm{\wp }(u+v)=\frac{1}{4}{\left[\frac{{\mathrm{\wp }}^{\prime }(u)-{\mathrm{\wp }}^{\prime }(v)}{\mathrm{\wp }(u)-\mathrm{\wp }(v)}\right]}^2-\mathrm{\wp }(u)-\mathrm{\wp }(v).}$$

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 inTeX. InUnicode the code point isU+2118 ℘SCRIPT CAPITAL P, with the more correct aliasweierstrass elliptic function.^{[footnote 2]} InHTML, it can be escaped as℘ or℘.

• Weierstrass functions
• Jacobi elliptic functions
• Lemniscate elliptic functions

• Abramowitz, Milton;Stegun, Irene Ann, eds. (1983) [June 1964]."Chapter 18".Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 627.ISBN 978-0-486-61272-0.LCCN 64-60036.MR 0167642.LCCN 65-12253.
• N. I. Akhiezer,Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English asAMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode IslandISBN 0-8218-4532-2
• Tom M. Apostol,Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New YorkISBN 0-387-97127-0 (See chapter 1.)
• K. Chandrasekharan,Elliptic functions (1980), Springer-VerlagISBN 0-387-15295-4
• Konrad Knopp,Funktionentheorie II (1947), Dover Publications; Republished in English translation asTheory of Functions (1996), Dover PublicationsISBN 0-486-69219-1
• Serge Lang,Elliptic Functions (1973), Addison-Wesley,ISBN 0-201-04162-6
• E. T. Whittaker andG. N. Watson,A Course of Modern Analysis,Cambridge University Press, 1952, chapters 20 and 21

Wikimedia Commons has media related toWeierstrass's elliptic functions.

• "Weierstrass elliptic functions",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Weierstrass's elliptic functions on Mathworld.
• Chapter 23,Weierstrass Elliptic and Modular Functions in DLMF (Digital Library of Mathematical Functions) by W. P. Reinhardt and P. L. Walker.
• Weierstrass P function and its derivative implemented in C by David Dumas

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