Template:Short description Template:Redirect 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 <math>\wp</math>-function
MotivationEdit
A cubic of the form <math>C_{g_2,g_3}^\mathbb{C}=\{(x,y)\in\mathbb{C}^2:y^2=4x^3-g_2x-g_3\} </math>, where <math>g_2,g_3\in\mathbb{C}</math> are complex numbers with <math>g_2^3-27g_3^2\neq0</math>, cannot be rationally parameterized.<ref name=":5" /> Yet one still wants to find a way to parameterize it.
For the quadric <math>K=\left\{(x,y)\in\mathbb{R}^2:x^2+y^2=1\right\}</math>; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function: <math display="block">\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>\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>C_{g_2,g_3}^\mathbb{C} </math> by means of the doubly periodic <math>\wp </math>-function (see in the section "Relation to elliptic curves"). This parameterization has the domain <math>\mathbb{C}/\Lambda </math>, which is topologically equivalent to a torus.<ref>Template:Citation</ref>
There is another analogy to the trigonometric functions. Consider the integral function <math display="block">a(x)=\int_0^x\frac{dy}{\sqrt{1-y^2}} .</math> It can be simplified by substituting <math>y=\sin t </math> and <math>s=\arcsin x </math>: <math display="block">a(x)=\int_0^s dt = s = \arcsin x .</math> That means <math>a^{-1}(x) = \sin x </math>. So the sine function is an inverse function of an integral function.<ref>Template:Citation</ref>
Elliptic functions are the inverse functions of elliptic integrals. In particular, let: <math display="block">u(z)=\int_z^\infin\frac{ds}{\sqrt{4s^3-g_2s-g_3}} .</math> Then the extension of <math>u^{-1} </math> to the complex plane equals the <math>\wp </math>-function.<ref>Template:Citation</ref> 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.<ref>Template:Cite book</ref>
DefinitionEdit
Let <math>\omega_1,\omega_2\in\mathbb{C}</math> be two complex numbers that are linearly independent over <math>\mathbb{R}</math> and let <math>\Lambda:=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2:=\{m\omega_1+n\omega_2: m,n\in\mathbb{Z}\}</math> be the period lattice generated by those numbers. Then the <math>\wp</math>-function is defined as follows:
- <math>\weierp(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).</math>
This series converges locally uniformly absolutely in the complex torus <math>\mathbb{C} / \Lambda</math>.
It is common to use <math>1</math> and <math>\tau</math> in the upper half-plane <math>\mathbb{H}:=\{z\in\mathbb{C}:\operatorname{Im}(z) > 0\}</math> as generators of the lattice. Dividing by <math display="inline">\omega_1</math> maps the lattice <math>\mathbb{Z}\omega_1+\mathbb{Z}\omega_2</math> isomorphically onto the lattice <math>\mathbb{Z}+\mathbb{Z}\tau</math> with <math display="inline">\tau=\tfrac{\omega_2}{\omega_1}</math>. Because <math>-\tau</math> can be substituted for <math>\tau</math>, without loss of generality we can assume <math>\tau\in\mathbb{H}</math>, and then define <math>\wp(z,\tau) := \wp(z, 1,\tau)</math>. With that definition, we have <math>\wp(z,\omega_1,\omega_2) = \omega_1^{-2}\wp(z/\omega_1,\omega_2/\omega_1)</math>.
PropertiesEdit
- <math>\wp</math> is a meromorphic function with a pole of order 2 at each period <math>\lambda</math> in <math>\Lambda</math>.
- <math>\wp</math> is a homogeneous function in that:
- <math>\wp(\lambda z , \lambda\omega_{1}, \lambda\omega_{2}) = \lambda^{-2} \wp (z, \omega_{1},\omega_{2}).</math>
- <math>\wp</math> is an even function. That means <math>\wp(z)=\wp(-z)</math> for all <math>z \in \mathbb{C} \setminus \Lambda</math>, which can be seen in the following way:
- <math>\begin{align}
\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{align}</math>
- The second last equality holds because <math>\{-\lambda:\lambda \in \Lambda\}=\Lambda</math>. Since the sum converges absolutely this rearrangement does not change the limit.
- The derivative of <math>\wp</math> is given by:<ref name=":1">Template:Citation</ref> <math display="block">\wp'(z)=-2\sum_{\lambda \in \Lambda}\frac1{(z-\lambda)^3}.</math>
- <math>\wp</math> and <math>\wp'</math> are doubly periodic with the periods <math>\omega_1 </math> and <math>\omega_2</math>.<ref name=":1" /> This means: <math display="block">\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}</math> It follows that <math>\wp(z+\lambda)=\wp(z)</math> and <math>\wp'(z+\lambda)=\wp'(z)</math> for all <math>\lambda \in \Lambda</math>.
Laurent expansionEdit
Let <math>r:=\min\{{|\lambda}|:0\neq\lambda\in\Lambda\}</math>. Then for <math>0<|z|<r</math> the <math>\wp</math>-function has the following Laurent expansion <math display="block">\wp(z)=\frac1{z^2}+\sum_{n=1}^\infin (2n+1)G_{2n+2}z^{2n} </math> where <math display="block">G_n=\sum_{0\neq\lambda\in\Lambda}\lambda^{-n}</math> for <math>n \geq 3</math> are so called Eisenstein series.<ref name=":1" />
Differential equation
Set <math>g_2=60G_4</math> and <math>g_3=140G_6</math>. Then the <math>\wp</math>-function satisfies the differential equation<ref name=":1" /> <math display="block"> \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>\wp</math> and <math>\wp'</math> to eliminate the pole at <math>z=0</math>. This yields an entire elliptic function that has to be constant by Liouville's theorem.<ref name=":1" />
Invariants
The coefficients of the above differential equation <math>g_2</math> and <math>g_3</math> are known as the invariants. Because they depend on the lattice <math>\Lambda</math> they can be viewed as functions in <math>\omega_1</math> and <math>\omega_2</math>.
The series expansion suggests that <math>g_2</math> and <math>g_3</math> are homogeneous functions of degree <math>-4</math> and <math>-6</math>. That is<ref name=":0">Template:Cite book</ref> <math display="block">g_2(\lambda \omega_1, \lambda \omega_2) = \lambda^{-4} g_2(\omega_1, \omega_2)</math> <math display="block">g_3(\lambda \omega_1, \lambda \omega_2) = \lambda^{-6} g_3(\omega_1, \omega_2)</math> for <math>\lambda \neq 0</math>.
If <math>\omega_1</math> and <math>\omega_2</math> are chosen in such a way that <math>\operatorname{Im}\left( \tfrac{\omega_2}{\omega_1} \right)>0 </math>, <math>g_2</math> and <math>g_3</math> can be interpreted as functions on the upper half-plane <math>\mathbb{H}:=\{z\in\mathbb{C}:\operatorname{Im}(z)>0\}</math>.
Let <math>\tau=\tfrac{\omega_2}{\omega_1}</math>. One has:<ref name=":2">Template:Citation</ref> <math display="block">g_2(1,\tau)=\omega_1^4g_2(\omega_1,\omega_2),</math> <math display="block">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 display="block">g_2(\tau):=g_2(1,\tau) </math> and <math display="block">g_3(\tau):=g_3(1,\tau).</math> As functions of <math>\tau\in\mathbb{H}</math>, <math>g_2</math> and <math>g_3</math> are so called modular forms.
The Fourier series for <math>g_2</math> and <math>g_3</math> are given as follows:<ref>Template:Cite book</ref> <math display="block">g_2(\tau)=\frac43\pi^4 \left[ 1+ 240\sum_{k=1}^\infty \sigma_3(k) q^{2k} \right] </math> <math display="block">g_3(\tau)=\frac{8}{27}\pi^6 \left[ 1- 504\sum_{k=1}^\infty \sigma_5(k) q^{2k} \right] </math> where <math display="block">\sigma_m(k):=\sum_{d\mid{k}}d^m</math> is the divisor function and <math>q=e^{\pi i\tau}</math> is the nome.
Modular discriminantEdit
The modular discriminant <math>\Delta</math> is defined as the discriminant of the characteristic polynomial of the differential equation <math display="block"> \wp'^2(z) = 4\wp ^3(z)-g_2\wp(z)-g_3</math> as follows: <math display="block"> \Delta=g_2^3-27g_3^2. </math> The discriminant is a modular form of weight <math>12</math>. That is, under the action of the modular group, it transforms as <math display="block">\Delta \left( \frac {a\tau+b} {c\tau+d}\right) = \left(c\tau+d\right)^{12} \Delta(\tau) </math> where <math>a,b,d,c\in\mathbb{Z}</math> with <math>ad-bc = 1</math>.<ref>Template:Cite book</ref>
Note that <math>\Delta=(2\pi)^{12}\eta^{24}</math> where <math>\eta</math> is the Dedekind eta function.<ref>Template:Cite book</ref>
For the Fourier coefficients of <math>\Delta</math>, see Ramanujan tau function.
The constants e1, e2 and e3Edit
<math>e_1</math>, <math>e_2</math> and <math>e_3</math> are usually used to denote the values of the <math>\wp</math>-function at the half-periods. <math display="block">e_1\equiv\wp\left(\frac{\omega_1}{2}\right)</math> <math display="block">e_2\equiv\wp\left(\frac{\omega_2}{2}\right)</math> <math display="block">e_3\equiv\wp\left(\frac{\omega_1+\omega_2}{2}\right)</math> They are pairwise distinct and only depend on the lattice <math>\Lambda</math> and not on its generators.<ref>Template:Citation</ref>
<math>e_1</math>, <math>e_2</math> and <math>e_3</math> are the roots of the cubic polynomial <math>4\wp(z)^3-g_2\wp(z)-g_3</math> and are related by the equation: <math display="block">e_1+e_2+e_3=0.</math> Because those roots are distinct the discriminant <math>\Delta</math> does not vanish on the upper half plane.<ref>Template:Citation</ref> Now we can rewrite the differential equation: <math display="block">\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>\wp'</math>.
The invariants <math>g_2</math> and <math>g_3</math> can be expressed in terms of these constants in the following way:<ref>Template:Citation</ref> <math display="block">g_2 = -4 (e_1 e_2 + e_1 e_3 + e_2 e_3)</math> <math display="block">g_3 = 4 e_1 e_2 e_3</math> <math>e_1</math>, <math>e_2</math> and <math>e_3</math> are related to the modular lambda function: <math display="block">\lambda (\tau)=\frac{e_3-e_2}{e_1-e_2},\quad \tau=\frac{\omega_2}{\omega_1}.</math>
Relation to Jacobi's elliptic functionsEdit
For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of Jacobi's elliptic functions.
The basic relations are:<ref>Template:Cite book</ref> <math display="block"> \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>e_1,e_2</math> and <math>e_3</math> are the three roots described above and where the modulus k of the Jacobi functions equals <math display="block">k = \sqrt\frac{e_2 - e_3}{e_1 - e_3}</math> and their argument w equals <math display="block">w = z \sqrt{e_1 - e_3}.</math>
Relation to Jacobi's theta functionsEdit
The function <math>\wp (z,\tau)=\wp (z,1,\omega_2/\omega_1)</math> can be represented by Jacobi's theta functions: <math display="block">\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)</math> where <math>q=e^{\pi i\tau}</math> is the nome and <math>\tau</math> is the period ratio <math>(\tau\in\mathbb{H})</math>.<ref>Template:Dlmf</ref> This also provides a very rapid algorithm for computing <math>\wp (z,\tau)</math>.
Relation to elliptic curvesEdit
Template:See also Consider the embedding of the cubic curve in the complex projective plane
- <math>\bar C_{g_2,g_3}^\mathbb{C} = \{(x,y)\in\mathbb{C}^2:y^2=4x^3-g_2x-g_3\}\cup\{O\}\subset \mathbb{C}^{2} \cup \mathbb{P}_1(\mathbb{C}) = \mathbb{P}_2(\mathbb{C}).</math>
where <math>O</math> is a point lying on the line at infinity <math>\mathbb{P}_1(\mathbb{C})</math>. For this cubic there exists no rational parameterization, if <math>\Delta \neq 0</math>.<ref name=":5">Template:Citation</ref> In this case it is also called an elliptic curve. Nevertheless there is a parameterization in homogeneous coordinates that uses the <math>\wp</math>-function and its derivative <math>\wp'</math>:<ref>Template:Citation</ref>
- <math> \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} </math>
Now the map <math>\varphi</math> is bijective and parameterizes the elliptic curve <math>\bar C_{g_2,g_3}^\mathbb{C}</math>.
<math>\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>g_2,g_3\in\mathbb{C}</math> with <math>\Delta = g_2^3 - 27g_3^2 \neq 0 </math> there exists a lattice <math>\mathbb{Z}\omega_1+\mathbb{Z}\omega_2</math>, such that
<math>g_2=g_2(\omega_1,\omega_2) </math> and <math>g_3=g_3(\omega_1,\omega_2) </math>.<ref>Template:Citation</ref>
The statement that elliptic curves over <math>\mathbb{Q}</math> can be parameterized over <math>\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 theoremsEdit
Let <math>z,w\in\mathbb{C}</math>, so that <math>z,w,z+w,z-w\notin\Lambda </math>. Then one has:<ref name=":3">Template:Citation</ref> <math display="block">\wp(z+w)=\frac14 \left[\frac{\wp'(z)-\wp'(w)}{\wp(z)-\wp(w)}\right]^2-\wp(z)-\wp(w).</math>
As well as the duplication formula:<ref name=":3" /> <math display="block">\wp(2z)=\frac14\left[\frac{\wp(z)}{\wp'(z)}\right]^2-2\wp(z).</math>
ProofsEdit
1. These formulas can come with a geometric interpretation. If one looks at the elliptic curve <math>C_{g_2,g_3}^{\mathbb{C}} </math> a line <math>\lambda= \{(x,y)\in\mathbb{C}^2:y=mx+q\}</math> intersects it in three points:<math>C_{g_2,g_3}^{\mathbb{C}} \cap \lambda=\{P,Q,R\} </math>. Since these points belong to the elliptic curve they can be labeled as <math>P=(\wp(u),\wp'(u)) \quad Q=(\wp(v),\wp'(v)) \quad</math> <math> R=(\wp(u+v),\wp'(u+v))</math> with <math>(u,v)\notin \Lambda </math>. From the formula of a secant line we have <math>m=\frac{y_P-y_Q}{x_P-x_Q}=\frac{\wp'(u)-\wp'(v)}{\wp(u)-\wp(v)}</math> letting <math>C_{g_2,g_3}^{\mathbb{C}} = \lambda </math> we have the equation <math> (mx+q)^2=4x^3-g_2x-g_3</math> which becomes <math> 4x^3-m^2x^2-(2mq+g_2)x-g_3-q^2=0</math> using Vieta's formulas one obtains: <math> x_P+x_Q+x_R=\frac{m^2}4 </math> which provides the wanted formula <math>\wp(u+v)+\wp(u)+\wp(v)=\frac14 \left[ \frac{\wp'(u)-\wp'(v)}{\wp(u)-\wp(v)} \right]^2 </math>
2. A second proof from Akhiezer's book<ref>Akhiezer's book Elements of the theory of elliptic functions https://www.ams.org/books/mmono/079/mmono079-endmatter.pdf</ref> is the following:
if <math> f </math> is arbitrary elliptic function then: <math>f(u)=c\prod_{i=1}^n \frac{\sigma(u-a_i)}{\sigma(u-b_i)} \quad c \in \mathbb{C}</math>
where <math> \sigma </math> is one of the Weierstrass functions and <math> a_i , b_i</math> are the respective zeros and poles in the period parallelogram. We then let a function <math>k(u,v)=\wp(u)-\wp(v)</math> From the previous lemma we have: <math>k(u,v)= \wp(u)-\wp(v)=c\frac{\sigma(u+v)\sigma(u-v)}{\sigma(u)^2}
</math>
From some calculations one can find that <math>c=\frac1{\sigma(v)^2} \implies\wp(u)-\wp(v)=\frac{\sigma(u+v)\sigma(u-v)}{\sigma(u)^2\sigma(v)^2}</math>
By definition the Weierstrass Zeta function: <math> \frac{d}{dz}\ln \sigma(z)=\zeta(z)</math> therefore we logarithmicly differentiate both sides obtaining: <math>\frac{\wp'(u)}{\wp(u)-\wp(v)}=\zeta(u+v)-2\zeta(u)-\zeta(u-v)</math> Once again by definition <math> \zeta'(z)=-\wp(z)</math> thus by differentiating once more on both sides and rearranging the terms we obtain
<math>-\wp(u+v)=-\wp(u)+\frac12 \frac{ \wp(v)[\wp(u)-\wp(v) ]-\wp'(u)[\wp'(u)-\wp'(v)] }{ [\wp(u)-\wp(v) ]^2 } </math>
Knowing that <math>\wp </math> has the following differential equation <math>2\wp=12\wp^2-g_2</math> and rearranging the terms one gets the wanted formula <math display="block">\wp(u+v)=\frac14 \left[\frac{\wp'(u)-\wp'(v)}{\wp(u)-\wp(v)}\right]^2-\wp(u)-\wp(v).</math>
TypographyEdit
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.Template:Refn 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 Template:Unichar, with the more correct alias Template:Smallcaps.Template:Refn In HTML, it can be escaped as ℘.
Template:Charmap
See alsoEdit
FootnotesEdit
ReferencesEdit
- Template:AS ref
- 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 Template:Isbn
- Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York Template:Isbn (See chapter 1.)
- K. Chandrasekharan, Elliptic functions (1980), Springer-Verlag Template:Isbn
- Konrad Knopp, Funktionentheorie II (1947), Dover Publications; Republished in English translation as Theory of Functions (1996), Dover Publications Template:Isbn
- Serge Lang, Elliptic Functions (1973), Addison-Wesley, Template:Isbn
- E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, 1952, chapters 20 and 21
External linksEdit
- Template:Springer
- 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