Weierstrass elliptic function explained

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 are 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

Motivation

A cubic of the form

C ^{C = \{(x,y)\in C} _{g _{2,g} _{3}} ^{2:y} ^{2=4x} ^{3- g} ^{2 x - g} _{3\}} , where g _{2,g} _{3\inC} are complex numbers with g ^{2 ≠ 0} _{3} , cannot be rationally parameterized. Yet one still wants to find a way to parameterize it.K = \left\{(x,y)\in 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:$$\backslash psi:\backslash mathbb/2\backslash pi\backslash mathbb\backslash to\; K,\; \backslash quad\; t\backslash mapsto(\backslash sin\; t,\backslash cos\; t).$$Because of the periodicity of the sine and cosine R / 2 \pi Z is chosen to be the domain, so the function is bijective.

In a similar way one can get a parameterization of

C ^{C} _{g _{2,g} _{3}} by means of the doubly periodic \wp -function (see in the section "Relation to elliptic curves"). This parameterization has the domain C / Λ , which is topologically equivalent to a torus.

There is another analogy to the trigonometric functions. Consider the integral function$$a(x)=\backslash int\_0^x\backslash frac\; .$$It can be simplified by substituting

y = \sin t and s = \arcsin x :$$a(x)=\backslash int\_0^s\; dt\; =\; s\; =\; \backslash arcsin\; x\; .$$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:$$u(z)=-\backslash int\_z^\backslash infin\backslash frac\; .$$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.^[1]

Definition

Let

\omega _{1,\omega} _{2\inC} be two complex numbers that are linearly independent over R and let Λ : = Z \omega _{1+Z\omega} _{2:=\{m\omega} _{1+n\omega} _{2:} m,n\in 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)} = (1)/(z ^{2}) + \sum _{λ\inΛ\setminus\{0\} }\left(\frac 1 - \frac 1 \right).This series converges locally uniformly absolutely in the complex torus C \setminus Λ .

It is common to use

1 and \tau in the upper half-plane H : = \{z\in C :\operatorname{Im}(z) > 0 \} as generators of the lattice. Dividing by $\backslash omega\_1$ maps the lattice Z \omega _{1+Z\omega} _{2} isomorphically onto the lattice Z + Z \tau with $\backslash tau=\backslash tfrac$. Because - \tau can be substituted for \tau , without loss of generality we can assume \tau\in H , and then define \wp(z,\tau) : = \wp(z, 1 ,\tau) .

Properties

\wp is a meromorphic function with a pole of order 2 at each period λ in Λ .\wp is an even function. That means \wp(z) = \wp( - z) for all z \in C \setminus Λ , which can be seen in the following way:\begin{align} \wp( - z) & = (1)/(( - z) ^{2}) + \sum _{λ\inΛ\setminus\{0\} }\left(\frac-\frac\right) \\& =\frac+\sum_\left(\frac-\frac\right) \\& =\frac+\sum_\left(\frac-\frac\right)=\wp(z).\end

The second last equality holds because

\{ -λ : λ \in Λ \} =Λ . Since the sum converges absolutely this rearrangement does not change the limit.

• The derivative of

\wp is given by: $$\backslash wp\text{'}(z)=-2\backslash sum\_\backslash frac1.$$\wp and \wp' are doubly periodic with the periods \omega _{1} and \omega _{2} . This means: It follows that \wp(z +λ ) = \wp(z) and \wp'(z +λ ) = \wp'(z) for all λ \in Λ .

Laurent expansion

Let

r: =min \{{| λ }|: 0 ≠ λ \in Λ \} . Then for 0 <|z|<r the \wp -function has the following Laurent expansion$$\backslash wp(z)=\backslash frac1+\backslash sum\_^\backslash infin\; (2n+1)G\_z^$$where$$G\_n=\backslash sum\_\backslash lambda^$$ 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$$\backslash wp\text{'}^2(z)\; =\; 4\backslash wp\; ^3(z)-g\_2\backslash wp(z)-g\_3.$$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 coefficients of the above differential equation g₂ and g₃ are known as the invariants. Because they depend on the lattice

Λ they can be viewed as functions in \omega _{1} and \omega _{2} .

The series expansion suggests that g₂ and g₃ are homogeneous functions of degree −4 and −6. That is^[2] $$g\_2(\backslash lambda\; \backslash omega\_1,\; \backslash lambda\; \backslash omega\_2)\; =\; \backslash lambda^\; g\_2(\backslash omega\_1,\; \backslash omega\_2)$$$$g\_3(\backslash lambda\; \backslash omega\_1,\; \backslash lambda\; \backslash omega\_2)\; =\; \backslash lambda^\; g\_3(\backslash omega\_1,\; \backslash omega\_2)$$ for

λ ≠ 0 .

If

\omega _{1} and \omega _{2} are chosen in such a way that \operatorname{Im}\left( \tfrac{\omega _{2}{\omega} _{1}} \right)> 0 , g₂ and g₃ can be interpreted as functions on the upper half-plane H : = \{z\in C :\operatorname{Im}(z)> 0 \} .

Let

\tau = \tfrac{\omega _{2}{\omega} _{1}} . One has:$$g\_2(1,\backslash tau)=\backslash omega\_1^4g\_2(\backslash omega\_1,\backslash omega\_2),$$$$g\_3(1,\backslash tau)=\backslash omega\_1^6\; g\_3(\backslash omega\_1,\backslash omega\_2).$$That means g₂ and g₃ are only scaled by doing this. Set$$g\_2(\backslash tau):=g\_2(1,\backslash tau)$$ and $$g\_3(\backslash tau):=g\_3(1,\backslash tau).$$As functions of \tau\in H g _{2,g} _{3} are so called modular forms.

The Fourier series for

g _{2} and g _{3} are given as follows:^[3] $$g\_2(\backslash tau)=\backslash frac43\backslash pi^4\; \backslash left[1+\; 240\backslash sum\_\{k=1\}^\backslash infty\; \backslash sigma\_3(k)\; q^\{2k\}\; \backslash right]$$$$g\_3(\backslash tau)=\backslash frac\backslash pi^6\; \backslash left[1-\; 504\backslash sum\_\{k=1\}^\backslash infty\; \backslash sigma\_5(k)\; q^\{2k\}\; \backslash right]$$where$$\backslash sigma\_a(k):=\backslash sum\_d^\backslash alpha$$is the divisor function and q = e ^{\pi} is the nome.

Modular discriminant

The modular discriminant Δ is defined as the discriminant of the characteristic polynomial of the differential equation $$\backslash wp\text{'}^2(z)\; =\; 4\backslash wp\; ^3(z)-g\_2\backslash wp(z)-g\_3$$ as follows:$$\backslash Delta=g\_2^3-27g\_3^2.$$The discriminant is a modular form of weight 12. That is, under the action of the modular group, it transforms as$$\backslash Delta\; \backslash left(\backslash frac\; \backslash right)\; =\; \backslash left(c\backslash tau+d\backslash right)^\; \backslash Delta(\backslash tau)$$where

a,b,d,c\in Z with ad − bc = 1.^[4]

Note that

\Delta = ( 2 \pi) ^{12} η ^{24} where η is the Dedekind eta function.^[5]

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.$$e\_1\backslash equiv\backslash wp\backslash left(\backslash frac\backslash right)$$$$e\_2\backslash equiv\backslash wp\backslash left(\backslash frac\backslash right)$$$$e\_3\backslash equiv\backslash wp\backslash left(\backslash frac\backslash right)$$They are pairwise distinct and only depend on the lattice Λ 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:$$\backslash wp\text{'}^2(z)=4(\backslash wp(z)-e\_1)(\backslash wp(z)-e\_2)(\backslash wp(z)-e\_3).$$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:$$\backslash lambda\; (\backslash tau)=\backslash frac,\backslash quad\; \backslash tau=\backslash frac.$$

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:^[6] $$\backslash wp(z)\; =\; e\_3\; +\; \backslash frac=\; e\_2\; +\; (e\_1\; -\; e\_3)\; \backslash frac=\; e\_1\; +\; (e\_1\; -\; e\_3)\; \backslash frac$$where

e _{1,e} _{2} and e _{3} are the three roots described above and where the modulus k of the Jacobi functions equals$$k\; =\; \backslash sqrt\backslash frac$$and their argument w equals$$w\; =\; z\; \backslash sqrt.$$

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:$$\backslash wp\; (z,\backslash tau)=\backslash left(\backslash pi\; \backslash theta\_2(0,q)\backslash theta\_3(0,q)\backslash frac\backslash right)^2-\backslash frac\backslash left(\backslash theta\_2^4(0,q)+\backslash theta\_3^4(0,q)\backslash right)$$where q = e ^{\pi} is the nome and \tau is the period ratio (\tau\in 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 ^{C} _{g _{2,g} _{3}} = \{(x,y)\in C ^{2:y} ^{2=4x} ^{3- g} ^{2 x - g} _{3\}\cup\{infin\}\subset} C ^{2} \cup \{ inf ty\} = P _{2(C).}

For this cubic there exists no rational parameterization, if

\Delta ≠ 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'): C / Λ \to\bar C ^{C ,} _{g _{2,g} _{3}} z \mapsto \begin{cases} \left[\wp(z):\wp'(z): 1 \right] & z \notin Λ \\ \left[ 0 : 1 : 0 \right] & z \in Λ \end{cases}

Now the map

\varphi is bijective and parameterizes the elliptic curve \bar C ^{C} _{g _{2,g} _{3}} .C / Λ 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\inC} with \Delta = g ^{3} _{2} - 27 g ^{2} _{3} ≠ 0 there exists a lattice Z \omega _{1+Z\omega} _{2} , such thatg _{2=g} _{2(\omega} _{1,\omega} _{2)} and g _{3=g} _{3(\omega} _{1,\omega} _{2)} .

The statement that elliptic curves over

Q can be parameterized over 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 C , so that z,w,z + w,z - w\notin Λ . Then one has:$$\backslash wp(z+w)=\backslash frac14\; \backslash left[\backslash frac\{\backslash wp\text{'}(z)-\backslash wp\text{'}(w)\}\{\backslash wp(z)-\backslash wp(w)\}\backslash right]^2-\backslash wp(z)-\backslash wp(w).$$

As well as the duplication formula:$$\backslash wp(2z)=\backslash frac14\backslash left[\backslash frac\{\backslash wp\text{'}\text{'}(z)\}\{\backslash wp\text{'}(z)\}\backslash right]^2-2\backslash wp(z).$$

These formulas also have a geometric interpretation, if one looks at the elliptic curve

\bar C ^{C} _{g _{2,g} _{3}} together with the mapping {\varphi}: C / Λ \to\bar C ^{C} _{g _{2,g} _{3}} as in the previous section.

The group structure of

( C / Λ , + ) translates to the curve \bar C ^{C} _{g _{2,g} _{3}} and can be geometrically interpreted there:

The sum of three pairwise different points

a,b,c\in\bar C ^{C} _{g _{2,g} _{3}} is zero if and only if they lie on the same line in P ^{2} _{C} .

This is equivalent to:where

\wp(u) = a , \wp(v) = b and u,v\notin Λ .

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.

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 ℘.

See also

• Weierstrass functions
• Jacobi elliptic functions
• Lemniscate elliptic functions

References

• • 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
• Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York (See chapter 1.)
• K. Chandrasekharan, Elliptic functions (1980), Springer-Verlag
• Konrad Knopp, Funktionentheorie II (1947), Dover Publications; Republished in English translation as Theory of Functions (1996), Dover Publications
• Serge Lang, Elliptic Functions (1973), Addison-Wesley,
• E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, 1952, chapters 20 and 21

External links

• • 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

Notes and References

1. Book: Ablowitz, Mark J. . Fokas . Athanassios S. . Complex Variables: Introduction and Applications . Cambridge University Press . 2003 . 978-0-521-53429-1 . 10.1017/cbo9780511791246. 185.
2. Book: Apostol, Tom M.. Modular functions and Dirichlet series in number theory. 1976. Springer-Verlag. 0-387-90185-X. New York. 14. 2121639.
3. Book: Apostol, Tom M.. Modular functions and Dirichlet series in number theory. 1990. Springer-Verlag. 0-387-97127-0. 2nd. New York. 20. 20262861.
4. Book: Apostol, Tom M.. Modular functions and Dirichlet series in number theory. 1976. Springer-Verlag. 0-387-90185-X. New York. 50. 2121639.
5. Book: Chandrasekharan, K. (Komaravolu), 1920-. Elliptic functions. 1985. Springer-Verlag. 0-387-15295-4. Berlin. 122. 12053023.
6. Book: . 1961 . Mathematical Handbook for Scientists and Engineers . McGraw–Hill . New York . 721 . 59014456.