Weierstrass elliptic function

A cubic of the form C_{g_2,g_3}^\mathbb{C}=\{(x,y)\in\mathbb{C}^2:y^2=4x^3-g_2x-g_3\} , where g_2,g_3\in\mathbb{C} are complex numbers with g_2^3-27g_3^2\neq0, cannot be rationally parameterized. 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^\infin\frac{ds}{\sqrt{4s^3-g_2s-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==

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: :\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). 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 \omega_1 maps the lattice \mathbb{Z}\omega_1+\mathbb{Z}\omega_2 isomorphically onto the lattice \mathbb{Z}+\mathbb{Z}\tau with \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{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} :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}\frac1{(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 the \wp-function has the following Laurent expansion \wp(z)=\frac1{z^2}+\sum_{n=1}^\infin (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 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_1^4g_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)=\frac43\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 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 = 4 e_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 ==

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_2x-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 theorem==

The addition theorem states that if z,w, and z+w do not belong to \Lambda, then \det\begin{bmatrix}1 & \wp(z) & \wp'(z) \\ 1 & \wp(w) & \wp'(w) \\ 1 & \wp(z+w) & -\wp'(z+w)\end{bmatrix}=0. This states that the points P=(\wp(z),\wp'(z)), Q=(\wp(w),\wp'(w)), and R=(\wp(z+w),-\wp'(z+w)) are collinear, the geometric form of the group law of an elliptic curve. This can be proven by considering constants A,B such that \wp'(z) = A\wp(z) + B, \quad \wp'(w) = A\wp(w) + B. Then the elliptic function \wp'(\zeta) - A\wp(\zeta) - B has a pole of order three at zero, and therefore three zeros whose sum belongs to \Lambda. Two of the zeros are z and w, and thus the third is congruent to -z-w. Alternative form The addition theorem can be put into the alternative form, for z,w,z-w,z+w\not\in\Lambda: \wp(z+w)=\frac14 \left[\frac{\wp'(z)-\wp'(w)}{\wp(z)-\wp(w)}\right]^2-\wp(z)-\wp(w). As well as the duplication formula: Any elliptic function f can be expressed as: f(u)=c\prod_{i=1}^n \frac{\sigma(u-a_i)}{\sigma(u-b_i)} \quad c \in \mathbb{C} where \sigma is the Weierstrass sigma function and a_i , b_i are the respective zeros and poles in the period parallelogram. Considering the function \wp(u)-\wp(v) as a function of u, we have \wp(u)-\wp(v)=c\frac{\sigma(u+v)\sigma(u-v)}{\sigma(u)^2}. Multiplying both sides by u^2 and letting u\to 0, we have 1 = -c\sigma(v)^2, so c=-\frac1{\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 logarithmically differentiate both sides with respect to u 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)+\frac12 \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)=\frac14 \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 , with the more correct alias . In HTML, it can be escaped as ℘ or ℘. == See also ==