Weierstrass's elliptic functions

Short description: Class of mathematical functions

In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as p-functions and they are usually denoted by the symbol ℘. They play an important role in theory of elliptic functions. 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 [math]\displaystyle{ \wp }[/math]-function

Model of Weierstrass [math]\displaystyle{ \wp }[/math]-function

Definition

Visualization of the [math]\displaystyle{ \wp }[/math]-function with invariants [math]\displaystyle{ g_2=1+i }[/math] and [math]\displaystyle{ g_3=2-3i }[/math] in which white corresponds to a pole, black to a zero.

Let [math]\displaystyle{ \omega_1,\omega_2\in\mathbb{C} }[/math] be two complex numbers that are linear independent over [math]\displaystyle{ \mathbb{R} }[/math] and let [math]\displaystyle{ \Lambda:=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2:=\{m\omega_1+n\omega_2: m,n\in\mathbb{Z}\} }[/math] be the lattice generated by those numbers. Then the [math]\displaystyle{ \wp }[/math]-function is defined as follows:

[math]\displaystyle{ \weierp(z,\omega_1,\omega_2):=\weierp(z,\Lambda) := \frac{1}{z^2} + \sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac 1 {(z-\lambda)^2} - \frac 1 {\lambda^2}\right). }[/math]

This series converges locally uniformly absolutely in [math]\displaystyle{ \mathbb{C}\setminus\Lambda }[/math]. Oftentimes instead of [math]\displaystyle{ \wp(z,\omega_1,\omega_2) }[/math] only [math]\displaystyle{ \wp(z) }[/math] is written.

The Weierstrass [math]\displaystyle{ \wp }[/math]-function is constructed exactly in such a way that it has a pole of the order two at each lattice point.

Because the sum [math]\displaystyle{ \sum_{\lambda\in\Lambda}\frac 1{(z-\lambda)^2} }[/math] alone would not converge it is necessary to add the term [math]\displaystyle{ -\frac 1 {\lambda^2} }[/math].

It is common to use [math]\displaystyle{ 1 }[/math] and [math]\displaystyle{ \tau\in\mathbb{H}:=\{z\in\mathbb{C}:\operatorname{Im}(z)\gt 0\} }[/math] as generators of the lattice. Multiplying by [math]\displaystyle{ \frac 1{\omega_1} }[/math] maps the lattice [math]\displaystyle{ \mathbb{Z}\omega_1+\mathbb{Z}\omega_2 }[/math] isomorphically onto the lattice [math]\displaystyle{ \mathbb{Z}+\mathbb{Z}\tau }[/math] with [math]\displaystyle{ \tau=\frac{\omega_2}{\omega_1} }[/math]. By possibly substituting [math]\displaystyle{ \tau }[/math] by [math]\displaystyle{ -\tau }[/math] it can be assumed that [math]\displaystyle{ \tau\in\mathbb{H} }[/math]. One sets [math]\displaystyle{ \wp(z,\tau) := \wp(z, 1,\tau) }[/math].

Motivation

A cubic of the form [math]\displaystyle{ C_{g_2,g_3}^\mathbb{C}=\{(x,y)\in\mathbb{C}^2:y^2=4x^3-g_2x+g_3\} }[/math], where [math]\displaystyle{ g_2,g_3\in\mathbb{C} }[/math] are complex numbers with [math]\displaystyle{ g_2^3-27g_3^2\neq0 }[/math], can not be rationally parameterized. Yet one still wants to find a way to parameterize it.

For the quadric [math]\displaystyle{ K=\{(x,y)\in\mathbb{R}^2:x^2+y^2=1\} }[/math], the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function:

[math]\displaystyle{ \psi:\mathbb{R}/2\pi\mathbb{Z}\to K, \quad t\mapsto(\sin(t),\cos(t)) }[/math].

Because of the periodicity of the sine and cosine [math]\displaystyle{ \mathbb{R}/2\pi\mathbb{Z} }[/math] is chosen to be the domain, so the function is bijective.

In a similar way one can get a parameterization of [math]\displaystyle{ C_{g_2,g_3}^\mathbb{C} }[/math] by means of the doubly periodic [math]\displaystyle{ \wp }[/math]-function (see in the section "Relation to ellitpic curves"). This parameterization has the domain [math]\displaystyle{ \mathbb{C}/\Lambda }[/math], which is topologically equivalent to a torus.

There is another analogy to the trigonometric functions. Consider the integral function

[math]\displaystyle{ a(x)=\int_0^x\frac{dy}{\sqrt{(1-y^2)}} }[/math].

It can be simplified by substituting [math]\displaystyle{ y=\sin(t) }[/math] and [math]\displaystyle{ s=\arcsin(x) }[/math]:

[math]\displaystyle{ a(x)=\int_0^sdt=s=\arcsin(x) }[/math].

That means [math]\displaystyle{ a^{-1}(x)=\sin(x) }[/math]. So the sine function is an inverse function of an integral function.

Elliptic functions are also inverse functions of integral functions, namely of elliptic integrals. In particular the [math]\displaystyle{ \wp }[/math]-function is obtained in the following way:

Let

[math]\displaystyle{ u(z)=-\int_z^\infin\frac{ds}{\sqrt{4s^3-g_2s-g_3}} }[/math].

Then [math]\displaystyle{ u^{-1} }[/math] can be extended to the complex plane and this extension equals the [math]\displaystyle{ \wp }[/math]-function.

Properties

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

[math]\displaystyle{ \weierp(-z)=\frac{1}{(-z)^2}+\sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac1{(-z-\lambda)^2}-\frac1{\lambda^2}\right)=\frac{1}{z^2}+\sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac1{(z+\lambda)^2}-\frac1{\lambda^2}\right)=\frac{1}{z^2}+\sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac1{(z-\lambda)^2}-\frac1{\lambda^2}\right)=\wp(z) }[/math]

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

• ℘ is meromorphic and its derivative is

[math]\displaystyle{ \wp'(z)=-2\sum_{\lambda \in \Lambda}\frac1{(z-\lambda)^3} }[/math].

• [math]\displaystyle{ \wp }[/math] and [math]\displaystyle{ \wp' }[/math] are doubly periodic with the periods [math]\displaystyle{ \omega_1 }[/math]und [math]\displaystyle{ \omega_2 }[/math]. This means:

[math]\displaystyle{ \wp(z+\omega_1)=\wp(z)=\wp(z+\omega_2) }[/math] and [math]\displaystyle{ \wp'(z+\omega_1)=\wp'(z)=\wp'(z+\omega_2) }[/math].

It follows that [math]\displaystyle{ \wp(z+\lambda)=\wp(z) }[/math] and [math]\displaystyle{ \wp'(z+\lambda)=\wp'(z) }[/math] for all [math]\displaystyle{ \lambda \in \Lambda }[/math]. Functions which are meromorphic and doubly periodic are also called elliptic functions.

Laurent expansion

Let [math]\displaystyle{ r:=\min\{{|\lambda}|:0\neq\lambda\in\Lambda\} }[/math]. Then for [math]\displaystyle{ 0\lt |z|\lt r }[/math] the [math]\displaystyle{ \wp }[/math]-function has the following Laurent expansion

[math]\displaystyle{ \wp(z)=\frac1{z^2}+\sum_{n=1}^\infin (2n+1)G_{2n+2}z^{2n} }[/math]

where

[math]\displaystyle{ G_n=\sum_{0\neq\lambda\in\Lambda}\lambda^{-n} }[/math] for [math]\displaystyle{ n\geq3 }[/math] are so called Eisenstein series.

Differential equation

Set [math]\displaystyle{ g_2=60G_4 }[/math] and [math]\displaystyle{ g_3=140G_6 }[/math]. Then the [math]\displaystyle{ \wp }[/math]-function satisfies the differential equation

[math]\displaystyle{ \wp'^2(z) = 4\wp ^3(z)-g_2\wp(z)-g_3 }[/math].

This relation can be verified by forming a linear combination of powers of [math]\displaystyle{ \wp }[/math] and [math]\displaystyle{ \wp' }[/math] to eliminate the pole at [math]\displaystyle{ z=0 }[/math]. 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 nome q on the unit disk.

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

The coefficients of the above differential equation g₂ and g₃ are known as the invariants. Because they depent on the lattice [math]\displaystyle{ \Lambda }[/math] they can be viewed as functions in [math]\displaystyle{ \omega_1 }[/math]and [math]\displaystyle{ \omega_2 }[/math].

The series expansion suggests that g₂ and g₃ are homogeneous functions of degree −4 and −6. That is

[math]\displaystyle{ g_2(\lambda \omega_1, \lambda \omega_2) = \lambda^{-4} g_2(\omega_1, \omega_2) }[/math]

[math]\displaystyle{ g_3(\lambda \omega_1, \lambda \omega_2) = \lambda^{-6} g_3(\omega_1, \omega_2) }[/math] for [math]\displaystyle{ \lambda\neq0 }[/math].

If [math]\displaystyle{ \omega_1 }[/math]and [math]\displaystyle{ \omega_2 }[/math] are chosen in such a way that [math]\displaystyle{ \operatorname{Im}\left( \frac{\omega_2}{\omega_1} \right)\gt 0 }[/math] g₂ and g₃ can be interpreted as functions on the upper half-plane [math]\displaystyle{ \mathbb{H}:=\{z\in\mathbb{C}:\operatorname{Im}(z)\gt 0\} }[/math].

Let [math]\displaystyle{ \tau=\frac{\omega_2}{\omega_1} }[/math]. One has:

[math]\displaystyle{ g_2(1,\tau)=\omega_1^4g_2(\omega_1,\omega_2) }[/math],

[math]\displaystyle{ g_3(1,\tau)=\omega_1^6 g_3(\omega_1,\omega_2) }[/math].

That means g₂ and g₃ are only scaled by doing this. Set

[math]\displaystyle{ g_2(\tau):=g_2(1,\tau) }[/math], [math]\displaystyle{ g_3(\tau):=g_3(1,\tau) }[/math].

As functions of [math]\displaystyle{ \tau\in\mathbb{H} }[/math] [math]\displaystyle{ g_2,g_3 }[/math] are so called modular forms.

The Fourier series for [math]\displaystyle{ g_2 }[/math] and [math]\displaystyle{ g_3 }[/math] are given as follows:

[math]\displaystyle{ g_2(\tau)=\frac{4}{3}\pi^4 \left[ 1+ 240\sum_{k=1}^\infty \sigma_3(k) q^{2k} \right] }[/math]

[math]\displaystyle{ g_3(\tau)=\frac{8}{27}\pi^6 \left[ 1- 504\sum_{k=1}^\infty \sigma_5(k) q^{2k} \right] }[/math]

where [math]\displaystyle{ \sigma_a(k):=\sum_{d\mid{k}}d^\alpha }[/math] is the divisor function and [math]\displaystyle{ q:=\exp(i\pi\tau) }[/math].

Modular discriminant

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

The modular discriminant Δ is defined as the discriminant of the polynomial at right-hand side of the above differential equation:

[math]\displaystyle{ \Delta=g_2^3-27g_3^2. \, }[/math]

The discriminant is a modular form of weight 12. That is, under the action of the modular group, it transforms as

[math]\displaystyle{ \Delta \left( \frac {a\tau+b} {c\tau+d}\right) = \left(c\tau+d\right)^{12} \Delta(\tau) }[/math]

where [math]\displaystyle{ a,b,d,c\in\mathbb{Z} }[/math] with ad − bc = 1.

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

For the Fourier coefficients of [math]\displaystyle{ \Delta }[/math], see Ramanujan tau function.

The constants e₁, e₂ and e₃

[math]\displaystyle{ e_1 }[/math], [math]\displaystyle{ e_2 }[/math] and [math]\displaystyle{ e_3 }[/math] are usually used to denote the values of the [math]\displaystyle{ \wp }[/math]-function at the half-periods.

[math]\displaystyle{ e_1\equiv\wp\left(\frac{\omega_1}{2}\right) }[/math]

[math]\displaystyle{ e_2\equiv\wp\left(\frac{\omega_2}{2}\right) }[/math]

[math]\displaystyle{ e_3\equiv\wp\left(\frac{\omega_1+\omega_2}{2}\right) }[/math]

They are pairwise distinct and only depend on the lattice [math]\displaystyle{ \Lambda }[/math] and not on its generators.

[math]\displaystyle{ e_1 }[/math], [math]\displaystyle{ e_2 }[/math] and [math]\displaystyle{ e_3 }[/math] are the roots of the cubic polynomial [math]\displaystyle{ 4\wp(z)^3-g_2\wp(z)-g_3 }[/math] and are related by the equation:

[math]\displaystyle{ e_1+e_2+e_3=0 }[/math].

Because those roots are distinct the discriminant [math]\displaystyle{ \Delta }[/math] does not vanish on the upper half plane. Now we can rewrite the differential equation:

[math]\displaystyle{ \wp'^2(z)=4(\wp(z)-e_1)(\wp(z)-e_2)(\wp(z)-e_3) }[/math].

That means the half-periods are zeros of [math]\displaystyle{ \wp' }[/math].

The invariants [math]\displaystyle{ g_2 }[/math] and [math]\displaystyle{ g_3 }[/math] can be expressed in terms of these constants in the following way:

[math]\displaystyle{ g_2=-4(e_1e_2+e_1e_3+e_2e_3) }[/math]

[math]\displaystyle{ g_3=4e_1e_2e_3 }[/math]

Relation to elliptic curves

Consider the projective cubic curve

[math]\displaystyle{ \bar C_{g_2,g_3}^\mathbb{C}=\{(x,y)\in\mathbb{C}^2:y^2=4x^3-g_2x+g_3\}\cup\{\infin\}\subset\mathbb{P}_\mathbb{C}^2 }[/math].

For this cubic, also called Weierstrass cubic, there exists no rational parameterization, if [math]\displaystyle{ \Delta\neq0 }[/math]. In this case it is also called an elliptic curve. Nevertheless there is a parameterization that uses the [math]\displaystyle{ \wp }[/math]-function and its derivative [math]\displaystyle{ \wp' }[/math]:

[math]\displaystyle{ \varphi: \mathbb{C}/\Lambda\to\bar C_{g_2,g_3}^\mathbb{C}, \quad \bar{z}\mapsto \begin{cases} (\wp(z),\wp'(z),1) & \bar{z}\neq0\\ \infin \quad &\bar{z}=0 \end{cases} }[/math]

Now the map [math]\displaystyle{ \varphi }[/math] is bijective and parameterizes the elliptic curve [math]\displaystyle{ \bar C_{g_2,g_3}^\mathbb{C} }[/math].

[math]\displaystyle{ \mathbb{C}/\Lambda }[/math] 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 [math]\displaystyle{ g_2,g_3\in\mathbb{C} }[/math] with [math]\displaystyle{ \Delta=g_2^3-27g_3^2\neq0 }[/math] there exists a lattice [math]\displaystyle{ \mathbb{Z}\omega_1+\mathbb{Z}\omega_2 }[/math], such that

[math]\displaystyle{ g_2=g_2(\omega_1,\omega_2) }[/math] and [math]\displaystyle{ g_3=g_3(\omega_1,\omega_2) }[/math].

The statement that elliptic curves over [math]\displaystyle{ \mathbb{Q} }[/math] can be parameterized over [math]\displaystyle{ \mathbb{Q} }[/math], 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 [math]\displaystyle{ z,w\in\mathbb{C} }[/math], so that [math]\displaystyle{ z,w,z+w,z-w\notin\Lambda }[/math]. Then one has:

[math]\displaystyle{ \wp(z+w)=\frac1{4} \left[\frac{\wp'(z)-\wp'(w)}{\wp(z)-\wp(w)}\right]^2-\wp(z)-\wp(w) }[/math].

As well as the duplication formula:

[math]\displaystyle{ \wp(2z)=\frac1{4}\left[\frac{\wp''(z)}{\wp'(z)}\right]^2-2\wp(z) }[/math].

These formulas also have a geometric interpretation, if one looks at the elliptic curve [math]\displaystyle{ \bar C_{g_2,g_3}^\mathbb{C} }[/math] together with the mapping [math]\displaystyle{ {\varphi}:\mathbb{C}/\Lambda\to\bar C_{g_2,g_3}^\mathbb{C} }[/math] as in the previous section.

The group structure of [math]\displaystyle{ (\mathbb{C}/\Lambda,+) }[/math] translates to the curve [math]\displaystyle{ \bar C_{g_2,g_3}^\mathbb{C} }[/math]and can be geometrically interpreted there:

The sum of three pairwise different points [math]\displaystyle{ a,b,c\in\bar C_{g_2,g_3}^\mathbb{C} }[/math]is zero if and only if they lie on the same line in [math]\displaystyle{ \mathbb{P}_\mathbb{C}^2 }[/math].

This is equivalent to:

[math]\displaystyle{ \det\left(\begin{array}{rrr} 1&\wp(u+v)&-\wp'(u+v)\\ 1&\wp(v)&\wp'(v)\\ 1&\wp(u)&\wp'(u)\\ \end{array}\right) =0 }[/math],

where [math]\displaystyle{ \wp(u)=a }[/math], [math]\displaystyle{ \wp(v)=b }[/math] and [math]\displaystyle{ u,v\notin\Lambda }[/math].

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:

[math]\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} }[/math]

where [math]\displaystyle{ e_1,e_2 }[/math]and [math]\displaystyle{ e_3 }[/math] are the three roots described above and where the modulus k of the Jacobi functions equals

[math]\displaystyle{ k =\sqrt{\frac{e_2 - e_3}{e_1 - e_3}} }[/math]

and their argument w equals

[math]\displaystyle{ w =z \sqrt{e_1 - e_3}. }[/math]

Typography

The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘.

In computing, the letter ℘ is available as \wp in TeX. In Unicode the code point is U+2118 ℘ SCRIPT CAPITAL P (HTML ℘ · ℘), with the more correct alias weierstrass elliptic function. In HTML, it can be escaped as ℘.

Character information

Preview	Template:Charmap/showcharTemplate:Charmap/showcharTemplate:Charmap/showcharTemplate:Charmap/showchar
Unicode name	SCRIPT CAPITAL P / WEIERSTRASS ELLIPTIC FUNCTION
Encodings	decimal	hex
Unicode	8472 0 0 0	U+2118
UTF-8	226 132 152 0 0 0	E2 84 98 00 00 00
Numeric character reference	℘���	℘���
Named character reference	℘

See also

• Weierstrass functions

References

• Abramowitz, Milton; Stegun, Irene Ann, eds (1983). "Chapter 18". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 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. pp. 627. LCCN 65-12253. ISBN 978-0-486-61272-0. http://www.math.sfu.ca/~cbm/aands/page_627.htm. 
• 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

• Hazewinkel, Michiel, ed. (2001), "Weierstrass elliptic functions", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/w097450 
• 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.