Weierstrass elliptic function

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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 P function

Symbol for Weierstrass <math>\wp</math>-function

Model of Weierstrass <math>\wp</math>-function

Motivation

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 and its derivative, namely via <math>(x,y)=(\wp(z),\wp'(z))</math>. 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>

Definition

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

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>.

Properties

<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 expansion

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 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 <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 discriminant

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

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 e3

<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 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:<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 functions

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 curves

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 theorem

The addition theorem states<ref>Template:Citation</ref> that if <math>z,w,</math> and <math>z+w</math> do not belong to <math>\Lambda</math>, then <math display="block">\det\begin{bmatrix}1 & \wp(z) & \wp'(z) \\ 1 & \wp(w) & \wp'(w) \\ 1 & \wp(z+w) & -\wp'(z+w)\end{bmatrix}=0.</math> This states that the points <math>P=(\wp(z),\wp'(z)),</math> <math>Q=(\wp(w),\wp'(w)),</math> and <math>R=(\wp(z+w),-\wp'(z+w))</math> are collinear, the geometric form of the group law of an elliptic curve.

This can be proven<ref>Template:Citation</ref> by considering constants <math>A,B</math> such that <math display="block">\wp'(z) = A\wp(z) + B, \quad \wp'(w) = A\wp(w) + B.</math> Then the elliptic function <math display="block">\wp'(\zeta) - A\wp(\zeta) - B</math> has a pole of order three at zero, and therefore three zeros whose sum belongs to <math>\Lambda</math>. Two of the zeros are <math>z</math> and <math>w</math>, and thus the third is congruent to <math>-z-w</math>.

Alternative form

The addition theorem can be put into the alternative form, for <math>z,w,z-w,z+w\not\in\Lambda</math>:<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>

Proofs

This can be proven from the addition theorem shown above. The points <math>P=(\wp(u),\wp'(u)), Q=(\wp(v),\wp'(v)),</math> and <math> R=(\wp(u+v),-\wp'(u+v))</math> are collinear and lie on the curve <math>y^2=4x^3-g_2x-g_3</math>. The slope of that line is <math display="block">m=\frac{y_P-y_Q}{x_P-x_Q}=\frac{\wp'(u)-\wp'(v)}{\wp(u)-\wp(v)}.</math> So <math>x=x_P=\wp(u)</math>, <math>x=x_Q=\wp(v)</math>, and <math>x=x_R=\wp(u+v)</math> all satisfy a cubic <math display="block"> (mx+q)^2=4x^3-g_2x-g_3,</math> where <math>q</math> is a constant. This becomes <math display="block"> 4x^3-m^2x^2-(2mq+g_2)x-g_3-q^2=0.</math> Thus <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>

A direct proof is as follows.<ref>Template:Citation</ref> Any elliptic function <math> f </math> can be expressed as: <math display="block">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 the Weierstrass sigma function and <math> a_i , b_i</math> are the respective zeros and poles in the period parallelogram. Considering the function <math>\wp(u)-\wp(v)</math> as a function of <math>u</math>, we have <math display="block">\wp(u)-\wp(v)=c\frac{\sigma(u+v)\sigma(u-v)}{\sigma(u)^2}. 

</math>

Multiplying both sides by <math>u^2</math> and letting <math>u\to 0</math>, we have <math> 1 = -c\sigma(v)^2</math>, so <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 logarithmically differentiate both sides with respect to <math>u</math> obtaining: <math display="block">\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 display="block">-\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>

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.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 &weierp; or &wp;. Template:Charmap

See also

Footnotes

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References

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