NationMaster - Encyclopedia: Weierstrass's elliptic functions

In mathematics, Weierstrass's elliptic functions are a standard type of elliptic functions (the other is the Jacobi's elliptic functions). They are named for Karl Weierstrass. Euclid, detail from The School of Athens by Raphael. ... In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ... In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e. ... Karl WeierstraÃŸ Karl Theodor Wilhelm Weierstrass (WeierstraÃŸ) (October 31, 1815 â€“ February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Biography Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ...

1 Definitions
2 Invariants
- 2.1 Special cases
3 Differential equation
4 Integral equation
5 Modular discriminant
6 The constants e1, e2 and e3
7 Addition theorems
8 The case with 1 a basic half-period
9 General theory
10 References

Definitions

Weierstrass P function defined over a subset of the complex plane using a standard visualization technique in which white corresponds to a pole, black to a zero, and maximal saturation to $left|f(z)right|=left|f(x+iy)right|=1;.$ Note the regular lattice of poles, and two interleaving lattices of zeroes. Image File history File links Download high resolution version (1584x1224, 516 KB) I, the creator of this work, hereby release it into the public domain. ... Scale of saturation (0% at bottom). ...

The Weierstrass elliptic function can be defined in three closely related ways, each of which possesses certain advantages. One is as a function of a complex variable $z$ and a lattice $Λ$ in the complex plane. Another is in terms of $z$ and two complex numbers $ω 1$ and $ω 2$ defining a pair of generators, or periods, for the lattice. The third is in terms $z$ and of a modulus $τ$ in the upper half-plane. This is related to the previous definition by $τ = ω 2 / ω 1$ , which by the conventional choice on the pair of periods is in the upper half-plane. Using this approach, for fixed $z$ the Weierstrass functions become modular functions of $τ$ . See lattice for other meanings of this term, both within and without mathematics. ... In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ... In mathematics, modular functions are certain kinds of mathematical functions mapping complex numbers to complex numbers. ...

In terms of the two periods, Weierstrass's elliptic function is an elliptic function with periods $ω 1$ and $ω 2$ defined as

$wp(z;omega_1,omega_2)=frac{1}{z^2}+ sum_{m^2+n^2 ne 0} left{ frac{1}{(z-momega_1-nomega_2)^2}- frac{1}{left(momega_1+nomega_2right)^2} right}$

Then $Λ = m ω 1 + n ω 2$ are the points of the period lattice, so that In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. ...

$wp(z;Lambda)=wp(z;omega_1,omega_2)$

for any pair of generators of the lattice defines the Weierstrass function as a function of a complex variable and a lattice.

If $τ$ is a complex number in the upper half-plane, then

$wp(z;tau) = wp(z;1,tau) =frac{1}{z^2} + sum_{n^2+m^2 ne 0}{1 over (z-n-mtau)^2} - {1 over (n+mtau)^2}.$

The above sum is homogeneous of degree minus two, from which we may define the Weierstrass $wp$ function for any pair of periods, as

$wp(z;omega_1,omega_2) = wp(z/omega_1; omega_2/omega_1)/omega_1^2$ .

We may compute $wp$ very rapidly in terms of theta functions; because these converge so quickly, this is a more expeditious way of computing $wp$ than the series we used to define it. The formula here is In mathematics, theta functions are special functions of several complex variables. ...

$wp(z; tau) = pi^2 vartheta^2(0;tau) vartheta_{10}^2(0;tau){vartheta_{01}^2(z;tau) over vartheta_{11}^2(z;tau)} + e_2(tau)$

where

$e_2(tau) = -{pi^2 over {3}}(vartheta^4(0;tau) + vartheta_{10}^4(0;tau))$ .

There is a second order pole at each point of the period lattice (including the origin). With these definitions, $wp(z)$ is an even function and its derivative with respect to $z$ , $wp'$ , an odd function.

Further development of the theory of elliptic functions shows that the condition on Weierstrass's function (correctly called pe) is determined up to addition of a constant and multiplication by a non-zero constant by the condition on the poles alone, amongst all meromorphic functions with the given period lattice. In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ...

Invariants

The real part of the invariant g_3 as a function of the nome q on the unit disk.

The imaginary part of the invariant g_3 as a function of the nome q on the unit disk.

If points close to the origin are considered the appropriate Laurent series is Weierstrass elliptic functions invarient g3, real part (600x600 pixels) Detailed description This image shows the real part of the Weierstrass elliptic functions invarient g3=140 G6 as a function of the square of the nome on the unit disk |q| &lt 1. ... Weierstrass elliptic functions invarient g3, real part (600x600 pixels) Detailed description This image shows the real part of the Weierstrass elliptic functions invarient g3=140 G6 as a function of the square of the nome on the unit disk |q| &lt 1. ... Weierstrass elliptic functions invarient g3, imaginary part (600x600 pixels) Detailed description This image shows the imaginary part of the Weierstrass elliptic functions invarient g3=140 G6 as a function of the square of the nome on the unit disk |q| &lt 1. ... Weierstrass elliptic functions invarient g3, imaginary part (600x600 pixels) Detailed description This image shows the imaginary part of the Weierstrass elliptic functions invarient g3=140 G6 as a function of the square of the nome on the unit disk |q| &lt 1. ... A Laurent series is defined with respect to a particular point c and a path of integration Î³. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ...

$wp(z;omega_1,omega_2)=z^{-2}+frac{1}{20}g_2z^2+frac{1}{28}g_3z^4+O(z^6)$

where

$g_2= 60sum{}' Omega_{m,n}^{-4},qquad g_3=140sum{}' Omega_{m,n}^{-6}.$

(here a dashed summation refers to summation over all pairs of integers except $m = n = 0$ and $Ω m, n = z - m ω 1 - n ω 2$ ). The numbers $g 2$ and $g 3$ are known as the invariants — they are two terms out of the Eisenstein series. (Abramowitz and Stegun restrict themselves to the case of real $g 2$ and $g 3$ , stating that this case "seems to cover most applications"; this may be true from the point of view of applied mathematics. If $ω 1$ is real and $ω 2$ pure imaginary, or if $omega_1=overline{omega_2}$ , the invariants are real). In mathematics, Eisenstein series are particular modular forms with infinite series expansions that may be written down directly. ... Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ... Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ...

Note that $g 2$ and $g 3$ are homogeneous functions of degree -4 and -6; that is, In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by some factor, then the result is multiplied by some power of this factor. ...

g 2 (λω 1,λω 2) = λ - 4 g 2 (ω 1,ω 2)

and

g 3 (λω 1,λω 2) = λ - 6 g 3 (ω 1,ω 2)

Thus, by convention, one frequently writes $g 2$ and $g 3$ in terms of the half-period ratio $τ = ω 2 / ω 1$ and take $τ$ to lie in the upper half-plane. Thus, $g 2 (τ) = g 2 (1,ω 2 / ω 1)$ and $g 3 (τ) = g 3 (1,ω 2 / ω 1)$ . In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by where K and iK are the quarter periods, and and are the fundamental pair of periods. ... In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ...

The Fourier series for $g 2$ and $g 3$ can be written in terms of the square of the nome $q = exp(i πτ)$ as Fourier series are a mathematical technique for analyzing an arbitrary periodic function by decomposing the function into a sum of much simpler sinusoidal component functions, which differ from each other only in amplitude and frequency. ... In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by where K and iK are the quarter periods, and and are the fundamental pair of periods. ...

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

and

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

where $σ a (k)$ is the divisor function. This formula may be re-written in terms of Lambert series. In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. ... A Lambert series, named after Johann Heinrich Lambert, is a series taking the form It can be resummed by expanding the denominator: where the coefficients of the new series are given by the Dirichlet convolution of with the constant function Since this last sum is a typical number-theortic sum...

The invariants may be expressed in terms of Jacobi's theta functions. This method is very convenient for numerical calculation: the theta functions converge very quickly. In the notation of Abramowitz and Stegun, but denoting the primitive half-periods by $ω 1,ω 2$ , the invariants satisfy In mathematics, theta functions are special functions of several complex variables. ... Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ...

$g_2(omega_1,omega_2)= frac{pi^4}{12omega_1^4} left( theta_2(0,q)^8-theta_3(0,q)^4theta_2(0,q)^4+theta_3(0,q)^8 right)$

and

$g_3(omega_1,omega_2)= frac{pi^6}{(2omega_1)^6} left[ frac{8}{27}left(theta_2(0,q)^{12}+theta_3(0,q)^{12}right)right.$

$left. - frac{4}{9}left(theta_2(0,q)^4+theta_3(0,q)^4right)cdot theta_2(0,q)^4theta_3(0,q)^4 right]$

where $τ = ω 2 / ω 1$ is the half-period ratio and $q = e π i τ$ is the nome. In mathematics, the half-period ratio τ of an elliptic function j is the ratio of the two half-periods ω1 and ω2 of j, where j is defined in such a way that See also Modular form Categories: Math stubs ... In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by where K and iK are the quarter periods, and and are the fundamental pair of periods. ...

Special cases

If the invariants are $g 2 = 0$ , $g 3 = 1$ , then this is known as the equianharmonic case; $g 2 = 1$ , $g 3 = 0$ is the lemniscatic case. In mathematics, and in particular the study of Weierstrass elliptic functions, the Equianharmonic case occurs when the Weierstrass invariants satisfy and ; This page follows the terminology of Abramowitz and Stegun; see also the lemniscatic case. ... In mathematics, and in particular the study of Weierstrass elliptic functions, the lemniscatic case occurs when the Weierstrass invariants satisfy and . ...

Differential equation

With this notation, the $wp$ function satisfies the following differential equation: Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

$[wp'(z)]^2=4[wp(z)]^3-g_2wp(z)-g_3,$

where dependence on $ω 1$ and $ω 2$ is suppressed.

Integral equation

The Weierstrass elliptic function can be given as the inverse of an elliptic integral. Let In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ...

$u = int_y^infty frac {ds} {sqrt{4s^3 - g_2s -g_3}}$ .

Here, g₂ and g₃ are taken as constants. Then one has

$y=wp(u)$ .

The above follows directly by integrating the differential equation.

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 Modular discrimnant, real part, as function of nome. ... Modular discrimnant, real part, as function of nome. ...

$Delta=g_2^3-27g_3^2.$

This is studied in its own right, as a cusp form, in modular form theory (that is, as a function of the period lattice). In number theory, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion of the constant coefficient a0. ... Modular form - Wikipedia /**/ @import /skins-1. ...

Note that $Δ = (2π) 12 η 24$ where $η$ is the Dedekind eta function. The Dedekind eta function is a function defined on the upper half plane of complex numbers whose imaginary part is positive. ...

The discriminant is a modular form of weight 12. That is, under the action of the modular group, it transforms as In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...

$Delta left( frac {atau+b} {ctau+d}right) = left(ctau+dright)^{12} Delta(tau)$

with τ being the half-period ratio, and a,b,c and d being integers, with ad-bc=1.

The constants e₁, e₂ and e₃

Consider the algebraic equation $4 t 3 - g 2 t - g 3 = 0$ , and name its roots $e 1$ , $e 2$ , and $e 3$ . It can be shown from the non-vanishing of the discriminant that no two of these three are equal.

Algebraic considerations show that $e 1 + e 2 + e 3 = 0$ .

In the case of real invariants, the sign of $Δ$ determines the nature of the roots. If $Δ > 0$ , all three are real and it is conventional to name them so that $e 1 > e 2 > e 3$ . If $Δ < 0$ , it is conventional to write $e 1 = - α + β i$ (where $alphageq 0$ , $β > 0$ ), whence $e_3=overline{e_1}$ and $e 2$ is real and non-negative. We also have

$wp(omega_1/2)=e_1qquad wp(omega_2/2)=e_2qquad wp(omega_3/2)=e_3$

where $ω 3 = - (ω 1 + ω 2) / 2$ . Also, $wp'(omega_i/2)=0$ for $i = 1,2,3$ .

If $g 2$ and $g 3$ are real and $Δ > 0$ , the $e i$ are all real, and $wp()$ is real on the perimeter of the rectangle with corners $0$ , $ω 3$ , $ω 1 + ω 3$ , and $ω 1$ .

Addition theorems

The Weierstrass elliptic functions have several properties that may be proved:

$detbegin{bmatrix} wp(z) & wp'(z) & 1 wp(y) & wp'(y) & 1 wp(z+y) & -wp'(z+y) & 1 end{bmatrix}=0$

(a symmetrical version would be

$detbegin{bmatrix} wp(u) & wp'(u) & 1 wp(v) & wp'(v) & 1 wp(w) & wp'(w) & 1 end{bmatrix}=0$

where $u + v + w = 0$ ).

Also

$wp(z+y)=frac{1}{4} left{ frac{wp'(z)-wp'(y)}{wp(z)-wp(y)} right}^2 -wp(z)-wp(y).$

and the duplication formula

$wp(2z)= frac{1}{4}left{ frac{wp''(z)}{wp'(z)}right}^2-2wp(z),$

unless $2 z$ is a period.

The case with 1 a basic half-period

If $ω 1 = 1$ , much of the above theory becomes simpler; it is then conventional to write $τ$ for $ω 2$ . For a fixed τ in the upper half-plane, so that the imaginary part of τ is positive, we define the Weierstrass $wp$ function by: In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ...

$wp(z;tau) =frac{1}{z^2} + sum_{n^2+m^2 ne 0}{1 over (z-n-mtau)^2} - {1 over (n+mtau)^2}$

The sum extends over the lattice {n+mτ : n and m in Z} with the origin omitted. Here we regard τ as fixed and $wp$ as a function of $z$ ; fixing $z$ and letting τ vary leads into the area of elliptic modular functions. See lattice for other meanings of this term, both within and without mathematics. ... In mathematics, the j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half plane of complex numbers with positive imaginary part. ...

General theory

$wp$ is a meromorphic function in the complex plane with poles at the lattice points. It is doubly periodic with periods 1 and τ; this means that $wp$ satisfies A meromorphic function is a function that is holomorphic on an open subset of the complex number plane C (or on some other connected Riemann surface) except at points in a set of isolated poles, which are certain well-behaved singularities. ...

$wp(z+1) = wp(z+tau) = wp(z)$

The above sum is homogeneous of degree minus two, and if $c$ is any non-zero complex number,

$wp(cz;ctau) = wp(z;tau)/c^2$

from which we may define the Weierstrass $wp$ function for any pair of periods. We also may take the derivative (of course, with respect to z) and obtain a function algebraically related to $wp$ by There are several meanings of derivation: A derivation in abstract algebra is a linear map that satisfies Leibniz law. ...

$wp'^2 = wp^3 - g_2 wp - g_3$

where $g 2$ and $g 3$ depend only on τ, being modular forms. The equation A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. ...

Y 2 = X 3 - g 2 X - g 3

defines an elliptic curve, and we see that ( $wp$ , $wp'$ ) is a parametrization of that curve. In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ...

The totality of meromorphic doubly periodic functions with given periods defines an algebraic function field, associated to that curve. It can be shown that this field is A meromorphic function is a function that is holomorphic on an open subset of the complex number plane C (or on some other connected Riemann surface) except at points in a set of isolated poles, which are certain well-behaved singularities. ... In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ...

$Bbb{C}(wp, wp')$ ,

so that all such functions are rational functions in the Weierstrass function and its derivative. In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...

We can also wrap a single period parallelogram into a torus, or donut-shaped Riemann surface, and regard the elliptic functions associated to a given pair of periods to be functions defined on that Riemann surface. A torus. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...

The roots $e 1$ , $e 2$ , and $e 3$ of the equation $X 3 - g 2 X - g 3$ depend on τ and can be expressed in terms of theta functions; we have In mathematics, theta functions are special functions of several complex variables. ...

$e_1(tau) = {pi^2 over {3}}(vartheta^4(0;tau) + vartheta_{01}^4(0;tau))$

$e_2(tau) = -{pi^2 over {3}}(vartheta^4(0;tau) + vartheta_{10}^4(0;tau))$

$e_3(tau) = {pi^2 over {3}}(vartheta_{10}^4(0;tau) - vartheta_{01}^4(0;tau))$

Since $g 2 = - 4(e 1 e 2 + e 2 e 3 + e 3 e 1)$ and $g 3 = 4 e 1 e 2 e 3$ we have these in terms of theta functions also.

We may also express $wp$ in terms of theta functions; because these converge very rapidly, this is a more expeditious way of computing $wp$ than the series we used to define it.

$wp(z; tau) = pi^2 vartheta^2(0;tau) vartheta_{10}^2(0;tau){vartheta_{01}^2(z;tau) over vartheta_{11}^2(z;tau)} + e_2(tau)$

The function $wp$ has two zeroes (modulo periods) and the function $wp'$ has three. The zeroes of $wp'$ are easy to find: since $wp'$ is an odd function they must be at the half-period points. On the other hand it is very difficult to express the zeroes of $wp$ by closed formula, except for special values of the modulus (e.g. when the period lattice is the Gaussian integers). An expression was found, by Zagier and Eichler. The word modulo is the Latin ablative of modulus. ... In mathematics, an equation or system of equations is said to have a closed-form solution just in case a solution can be expressed analytically in terms of a bounded number of well_known operations. ... A Gaussian integer is a complex number whose real and imaginary part are both integers. ... Don Bernhard Zagier (1951 - ) is an American mathematician. ... Martin Eichler was a number theorist of the twentieth century who stated that there were five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms. ...

The Weierstrass theory also includes the Weierstrass zeta function, which is an indefinite integral of $wp$ and not doubly-periodic, and a theta function called the Weierstrass sigma function, of which his zeta-function is the log-derivative. The sigma-function has zeroes at all the period points (only), and can be expressed in terms of Jacobi's functions. This gives one way to convert between Weierstrass and Jacobi notations. In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function called pe. Weierstrass sigma function The Weierstrass sigma function is defined as the product where denotes and is the two-dimensional lattice. ... In mathematics, theta functions are special functions of several complex variables. ... In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function called pe. Weierstrass sigma-function The Weierstrass sigma-function is defined as the product where denotes and is the two-dimensional lattice. ... In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula f′/f where f′ is the derivative of f. ... In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e. ...

The Weierstrass sigma-function is an entire function; it played the role of 'typical' function in a theory of random entire functions of J. E. Littlewood. In complex analysis, an entire function is a function that is holomorphic everywhere on the whole complex plane. ... John Edensor Littlewood (June 9, 1885 - September 6, 1977) was a British mathematician. ...

References

Naum Illyich 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
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
Abramowitz and Stegun, chapter 18

Categories: Modular forms | Algebraic curves | Elliptic functions Edmund Taylor Whittaker (24 October 1873 - 24 March 1956) was an English mathematician, who contributed widely to applied mathematics, mathematical physics and the theory of special functions. ... (George) Neville Watson (31 January 1886 - 2 February 1965) was an English mathematician, a noted master in the application of complex analysis to the theory of special functions. ... The headquarters of the Cambridge University Press, in Trumpington Street, Cambridge. ... 1952 (MCMLII) was a Leap year starting on Tuesday (link will take you to calendar). ... Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ...

Results from FactBites:

Weierstrass's elliptic functions - Wikipedia, the free encyclopedia (1115 words)
	Further development of the theory of elliptic functions shows that the condition on Weierstrass's function (correctly called pe) is determined up to addition of a constant and multiplication by a non-zero constant by the condition on the poles alone, amongst all meromorphic functions with the given period lattice.
	are homogeneous functions of degree -4 and -6; that is,
	is a meromorphic function in the complex plane with poles at the lattice points.

List of mathematical functions - Wikipedia, the free encyclopedia (861 words)
	A modern, abstract point of view contrasts large function spaces, which are infinite-dimensional and within which most functions are 'anonymous', with special functions picked out by properties such as symmetry, or relationship to harmonic analysis and group representations.
	Superadditive function : The value of a sum is greater than or equal to the sum of the values of the summands.
	Elliptic functions : The inverses of elliptic integrals; used to model double-periodic phenomena.

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Want to know more?
Search encyclopedia, statistics and forums:

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.

Contents