Weierstrass's elliptic functions

Weierstrass's elliptic functions

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Weierstrass's elliptic functions are elliptic function
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...

s that take a particularly simple form; they are named for Karl Weierstrass
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....

. This class of functions are also referred to as p-functions and generally written using the symbol ℘ (or ) (a stylised letter p called Weierstrass p
Weierstrass p
In mathematics, the Weierstrass p , also called pe, is used for the Weierstrass's elliptic function. It is occasionally used for the power set, although for that purpose a cursive capital, rather than lower-case, p is more widespread...

).

Definitions

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
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...

Λ in the complex plane. Another is in terms of z and two complex numbers ω1 and ω2 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 = ω21, 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.

In terms of the two periods, Weierstrass's elliptic function is an elliptic function with periods ω1 and ω2 defined as

Then are the points of the period lattice, so that

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

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

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

We may compute ℘ very rapidly in terms of theta functions; because these converge so quickly, this is a more expeditious way of computing
℘ than the series we used to define it. The formula here is

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

s with the given period lattice.

Invariants

If points close to the origin are considered the appropriate Laurent series
Laurent series
In mathematics, the Laurent series of a complex function f is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where...

is

where

The numbers g2 and g3 are known as the invariants—they are two terms out of the Eisenstein series
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly...

.

Note that g2 and g3 are homogeneous function
Homogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...

s of degree −4 and −6; that is,

and

Thus, by convention, one frequently writes and in terms of the half-period ratio  and take to lie in the upper half-plane. Thus, and .

The Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

for and can be written in terms of the square of the nome as

and

where is the divisor function
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...

. This formula may be rewritten in terms of Lambert series.

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 , the invariants satisfy

and

where is the half-period ratio and is the nome.

Special cases

If the invariants are g2 = 0, g3 = 1, then this is known as the equianharmonic
Equianharmonic
In mathematics, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy g_2=0 and g_3=1;This page follows the terminology of Abramowitz and Stegun; see also the lemniscatic case...

case; g2 = 1, g3 = 0 is the lemniscatic
Lemniscatic elliptic function
In mathematics, and in particular the study of Weierstrass elliptic functions, the lemniscatic case occurs when the Weierstrass invariants satisfy g2=1 and g3=0...

case.

Differential equation

With this notation, the ℘ function satisfies the following differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

:

where dependence on and is suppressed.

This relation can be quickly verified by comparing the poles of both sides, for example, the pole at z = 0 of lhs is

while the pole at z = 0 of

Comparing these two yields the relation above.

Integral equation

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

. Let

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

The above follows directly by integrating the differential equation.

Modular discriminant

The modular discriminant Δ is defined as

This is studied in its own right, as a cusp form
Cusp form
In number theory, a branch of mathematics, 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 \Sigma a_n q^n...

, in modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...

theory (that is, as a function of the period lattice).

Note that where is the Dedekind eta function
Dedekind eta function
The Dedekind eta function, named after Richard Dedekind, is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive...

.

The presence of 24
24 (number)
24 is the natural number following 23 and preceding 25.The SI prefix for 1024 is yotta , and for 10−24 yocto...

can be understood by connection with other occurrences, as in the eta function and the Leech lattice.

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

, it transforms as

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 cubic polynomial equation with roots , , and . If the discriminant is not zero, no two of these roots are equal. Since the quadratic term of this cubic polynomial is zero, the roots are related by the equation

The linear and constant coefficients (g2 and g3, respectively) are related to the roots by the equations

In the case of real invariants, the sign of determines the nature of the roots. If , all three are real and it is conventional to name them so that . If , it is conventional to write (where , ), whence and is real and non-negative.

The half-periods ω1 and ω2 of Weierstrass' elliptic function are related to the roots
where . Since the derivative of Weierstrass' elliptic function equals the above cubic polynomial of the function's value, for ; if the function's value equals a root of the polynomial, the derivative is zero.

If and are real and , the are all real, and is real on the perimeter of the rectangle with corners , , , and . If the roots are ordered as above (), then the first half-period is completely real

whereas the third half-period is completely imaginary

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

(a symmetrical version would be
where ).

Also

and the duplication formula
unless is a period.

The case with 1 a basic half-period

If , much of the above theory becomes simpler; it is then conventional to
write for . For a fixed τ in the upper half-plane, so that the imaginary part of τ is positive, we define the
Weierstrass ℘ function by

The sum extends over the lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...

{n+mτ : n and m in Z} with the origin omitted.
Here we regard τ as fixed and ℘ as a function of z; fixing z and letting τ vary leads into the area of elliptic modular functions.

General theory

℘ is a meromorphic function in the complex plane with a double pole at each lattice points. It is doubly periodic with periods 1 and τ; this means that
℘ satisfies

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

from which we may define the Weierstrass ℘ function for any pair of periods. We also may take the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

(of course, with respect to z) and obtain a function algebraically related to ℘ by

where and depend only on τ, being modular forms. The equation

defines an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

, and we see that is a parametrization of that curve.

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

so that all such functions are rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

s in the Weierstrass function and its derivative.

We can also wrap a single period parallelogram into a torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

, or donut-shaped Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

, and regard the elliptic functions associated to a given pair of periods to be functions defined on that Riemann surface.

The roots e1, e2, and e3 of the equation depend on τ and can be expressed in terms of theta functions; we have

Since and we have these in terms of theta functions also.

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

The function ℘ has two zeros (modulo
Modulo (jargon)
The word modulo is the Latin ablative of modulus which itself means "a small measure."It was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801...

periods) and the function ℘′ has three. The zeros of ℘′ are easy to find: since ℘′ is an odd function they must be at the half-period points. On the other hand it is very difficult to express the zeros of ℘ by closed formula, except for special values of the modulus (e.g. when the period lattice is the Gaussian integer
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The Gaussian integers are a special case of the quadratic...

s). An expression was found, by Zagier
Don Zagier
Don Bernard Zagier is an American mathematician whose main area of work is number theory. He is currently one of the directors of the Max Planck Institute for Mathematics in Bonn, Germany, and a professor at the Collège de France in Paris, France.He was born in Heidelberg, Germany...

and Eichler
Martin Eichler
Martin Eichler was a German number theorist.Eichler received his Ph.D. from the Martin Luther University of Halle-Wittenberg in 1936....

.

The Weierstrass theory also includes the Weierstrass zeta function, which is an indefinite integral of ℘ and not doubly periodic, and a theta function called the Weierstrass sigma function
Weierstrass sigma function
In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass.-Weierstrass sigma-function:...

, of which his zeta-function is the log-derivative. The sigma-function has zeros at all the period points (only), and can be expressed in terms of Jacobi's functions
Jacobi's elliptic functions
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...

. This gives one way to convert between Weierstrass and Jacobi notations.

The Weierstrass sigma-function is an entire function
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...

; it played the role of 'typical' function in a theory of random entire functions of J. E. Littlewood.

Relation to Jacobi elliptic functions

For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of the Jacobi's elliptic functions
Jacobi's elliptic functions
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...

. The basic relations are

where e1-3 are the three roots described above and where the modulus k of the Jacobi functions equals

and their argument w equals