Weierstrass elliptic function

"P-function" redirects here. For the phase-space function representing a quantum state, see Glauber–Sudarshan P representation.

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.

Symbol for Weierstrass $\wp$-function

Model of Weierstrass $\wp$-function

Motivation

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

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

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

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

Elliptic functions are the inverse functions of elliptic integrals. In particular, let: $$u(z)=\int _{z}^{\infty }{\frac {ds}{\sqrt {4s^{3}-g_{2}s-g_{3}}}}.$$ Then the extension of $u^{-1}$ to the complex plane equals the $\wp$-function. 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.

Definition

Visualization of the $\wp$-function with invariants $g_{2}=1+i$ and $g_{3}=2-3i$ in which white corresponds to a pole, black to a zero.

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

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

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

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

Properties

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

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

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

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

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

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

Laurent expansion

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

Differential equation

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

Invariants

The real part of the invariant g₃ as a function of the square of the nome q on the unit disk.

The imaginary part of the invariant g₃ as a function of the square of the nome q on the unit disk.

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

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

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

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

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

Modular discriminant

The real part of the discriminant as a function of the square of the nome q on the unit disk.

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

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

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

The constants e₁, e₂ and e₃

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

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

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

Relation to Jacobi's elliptic functions

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

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

Relation to Jacobi's theta functions

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

Relation to elliptic curves

See also: Elliptic curve § Elliptic curves over the complex numbers

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

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

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

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

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

$\mathbb {C} /\Lambda$ 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 $g_{2},g_{3}\in \mathbb {C}$ with $\Delta =g_{2}^{3}-27g_{3}^{2}\neq 0$ there exists a lattice $\mathbb {Z} \omega _{1}+\mathbb {Z} \omega _{2}$, such that

$g_{2}=g_{2}(\omega _{1},\omega _{2})$ and $g_{3}=g_{3}(\omega _{1},\omega _{2})$.

The statement that elliptic curves over $\mathbb {Q}$ can be parameterized over $\mathbb {Q}$, 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.

Addition theorems

Let $z,w\in \mathbb {C}$, so that $z,w,z+w,z-w\notin \Lambda$. Then one has: $$\wp (z+w)={\frac {1}{4}}\left[{\frac {\wp '(z)-\wp '(w)}{\wp (z)-\wp (w)}}\right]^{2}-\wp (z)-\wp (w).$$

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

Proofs

1. These formulas can come with a geometric interpretation. If one looks at the elliptic curve $C_{g_{2},g_{3}}^{\mathbb {C} }$ a line $\lambda =\{(x,y)\in \mathbb {C} ^{2}:y=mx+q\}$ intersects it in three points:$C_{g_{2},g_{3}}^{\mathbb {C} }\cap \lambda =\{P,Q,R\}$. Since these points belong to the elliptic curve they can be labeled as $P=(\wp (u),\wp '(u))\quad Q=(\wp (v),\wp '(v))\quad$ $R=(\wp (u+v),\wp '(u+v))$ with $(u,v)\notin \Lambda$. From the formula of a secant line we have $m={\frac {y_{P}-y_{Q}}{x_{P}-x_{Q}}}={\frac {\wp '(u)-\wp '(v)}{\wp (u)-\wp (v)}}$ letting $C_{g_{2},g_{3}}^{\mathbb {C} }=\lambda$ we have the equation $(mx+q)^{2}=4x^{3}-g_{2}x-g_{3}$ which becomes $4x^{3}-m^{2}x^{2}-(2mq+g_{2})x-g_{3}-q^{2}=0$ using Vieta's formulas one obtains: $x_{P}+x_{Q}+x_{R}={\frac {m^{2}}{4}}$ which provides the wanted formula $\wp (u+v)+\wp (u)+\wp (v)={\frac {1}{4}}\left[{\frac {\wp '(u)-\wp '(v)}{\wp (u)-\wp (v)}}\right]^{2}$

2. A second proof from Akhiezer's book is the following:

if $f$ is arbitrary elliptic function then: $f(u)=c\prod _{i=1}^{n}{\frac {\sigma (u-a_{i})}{\sigma (u-b_{i})}}\quad c\in \mathbb {C}$

where $\sigma$ is one of the Weierstrass functions and $a_{i},b_{i}$ are the respective zeros and poles in the period parallelogram. We then let a function $k(u,v)=\wp (u)-\wp (v)$ From the previous lemma we have: $k(u,v)=\wp (u)-\wp (v)=c{\frac {\sigma (u+v)\sigma (u-v)}{\sigma (u)^{2}}}$

From some calculations one can find that $c={\frac {1}{\sigma (v)^{2}}}\implies \wp (u)-\wp (v)={\frac {\sigma (u+v)\sigma (u-v)}{\sigma (u)^{2}\sigma (v)^{2}}}$

By definition the Weierstrass Zeta function: ${\frac {d}{dz}}\ln \sigma (z)=\zeta (z)$ therefore we logarithmicly differentiate both sides obtaining: ${\frac {\wp '(u)}{\wp (u)-\wp (v)}}=\zeta (u+v)-2\zeta (u)-\zeta (u-v)$ Once again by definition $\zeta '(z)=-\wp (z)$ thus by differentiating once more on both sides and rearranging the terms we obtain

$-\wp (u+v)=-\wp (u)+{\frac {1}{2}}{\frac {\wp ''(v)[\wp (u)-\wp (v)]-\wp '(u)[\wp '(u)-\wp '(v)]}{[\wp (u)-\wp (v)]^{2}}}$

Knowing that $\wp ''$ has the following differential equation $2\wp ''=12\wp ^{2}-g_{2}$ and rearranging the terms one gets the wanted formula $$\wp (u+v)={\frac {1}{4}}\left[{\frac {\wp '(u)-\wp '(v)}{\wp (u)-\wp (v)}}\right]^{2}-\wp (u)-\wp (v).$$

Typography

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. 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. In HTML, it can be escaped as ℘.

Character information

Preview	℘
Unicode name	SCRIPT CAPITAL P / WEIERSTRASS ELLIPTIC FUNCTION
Encodings	decimal	hex
Unicode	8472	U+2118
UTF-8	226 132 152	E2 84 98
Numeric character reference	℘	℘
Named character reference	℘, ℘

See also

• Weierstrass functions
• Jacobi elliptic functions
• Lemniscate elliptic functions

References

• 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 as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
• Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York ISBN 0-387-97127-0 (See chapter 1.)
• K. Chandrasekharan, Elliptic functions (1980), Springer-Verlag ISBN 0-387-15295-4
• Konrad Knopp, Funktionentheorie II (1947), Dover Publications; Republished in English translation as Theory of Functions (1996), Dover Publications ISBN 0-486-69219-1
• Serge Lang, Elliptic Functions (1973), Addison-Wesley, ISBN 0-201-04162-6
• E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, 1952, chapters 20 and 21

External links

• "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