"℘" redirects here; the symbol can also be used to denote a
power set.
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
scriptp. 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 -function
Model of Weierstrass -function
Motivation
A
cubic of the form , where are complex numbers with , cannot be
rationally parameterized.[1] Yet one still wants to find a way to parameterize it.
For the
quadric; the
unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function:
Because of the periodicity of the sine and cosine is chosen to be the domain, so the function is bijective.
In a similar way one can get a parameterization of by means of the doubly periodic -function (see in the section "Relation to elliptic curves"). This parameterization has the domain , which is topologically equivalent to a
torus.[2]
There is another analogy to the trigonometric functions. Consider the integral function
It can be simplified by substituting and :
That means . So the sine function is an inverse function of an integral function.[3]
It is common to use and in the
upper half-plane as
generators of the
lattice. Dividing by maps the lattice isomorphically onto the lattice with . Because can be substituted for , without loss of generality we can assume , and then define .
Set and . Then the -function satisfies the differential equation[6]
This relation can be verified by forming a linear combination of powers of and to eliminate the pole at . This yields an entire elliptic function that has to be constant by
Liouville's theorem.[6]
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 g2 and g3 are known as the invariants. Because they depend on the lattice they can be viewed as functions in and .
The series expansion suggests that g2 and g3 are
homogeneous functions of degree −4 and −6. That is[7] for .
If and are chosen in such a way that , g2 and g3 can be interpreted as functions on the
upper half-plane.
Let . One has:[8]
That means g2 and g3 are only scaled by doing this. Set
and
As functions of are so called
modular forms.
The real part of the discriminant as a function of the square of the nome q on the unit disk.
The modular discriminant Δ is defined as the
discriminant of the polynomial on the right-hand side of the above differential equation:
The discriminant is a modular form of weight 12. That is, under the action of the
modular group, it transforms as
where with ad − bc = 1.[10]
, and are usually used to denote the values of the -function at the half-periods.
They are pairwise distinct and only depend on the lattice and not on its generators.[12]
, and are the roots of the cubic polynomial and are related by the equation:
Because those roots are distinct the discriminant does not vanish on the upper half plane.[13] Now we can rewrite the differential equation:
That means the half-periods are zeros of .
The invariants and can be expressed in terms of these constants in the following way:[14], and are related to the
modular lambda function:
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:[15]
where and are the three roots described above and where the modulus k of the Jacobi functions equals
and their argument w equals
Relation to Jacobi's theta functions
The function can be represented by
Jacobi's theta functions:
where is the nome and is the period ratio .[16] This also provides a very rapid algorithm for computing .
For this cubic there exists no rational parameterization, if .[1] In this case it is also called an elliptic curve. Nevertheless there is a parameterization in
homogeneous coordinates that uses the -function and its derivative :[17]
Now the map is
bijective and parameterizes the elliptic curve .
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.[footnote 1]
In computing, the letter ℘ is available as \wp in
TeX. In
Unicode the code point is U+2118℘SCRIPT CAPITAL P (℘, ℘), with the more correct alias weierstrass elliptic function.[footnote 2] In
HTML, it can be escaped as ℘.
^
This symbol was also used in the version of Weierstrass's lectures published by Schwarz in the 1880s. The first edition of A Course of Modern Analysis by
E. T. Whittaker in 1902 also used it.[22]
^
The
Unicode Consortium has acknowledged two problems with the letter's name: the letter is in fact lowercase, and it is not a "script" class letter, like U+1D4C5𝓅MATHEMATICAL SCRIPT SMALL P, but the letter for Weierstrass's elliptic function.
Unicode added the alias as a correction.[23][24]
References
^
abHulek, Klaus. (2012), Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen (in German) (2., überarb. u. erw. Aufl. 2012 ed.), Wiesbaden: Vieweg+Teubner Verlag, p. 8,
ISBN978-3-8348-2348-9
^Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 259,
ISBN978-3-540-32058-6
^Jeremy Gray (2015), Real and the complex: a history of analysis in the 19th century (in German), Cham, p. 71,
ISBN978-3-319-23715-2{{
citation}}: CS1 maint: location missing publisher (
link)
^Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 294,
ISBN978-3-540-32058-6
^
abcdeApostol, Tom M. (1976), Modular functions and Dirichlet series in number theory (in German), New York: Springer-Verlag, p. 11,
ISBN0-387-90185-X
^Hulek, Klaus. (2012), Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen (in German) (2., überarb. u. erw. Aufl. 2012 ed.), Wiesbaden: Vieweg+Teubner Verlag, p. 12,
ISBN978-3-8348-2348-9
^Hulek, Klaus. (2012), Elementare Algebraische Geometrie : Grundlegende Begriffe und Techniken mit zahlreichen Beispielen und Anwendungen (in German) (2., überarb. u. erw. Aufl. 2012 ed.), Wiesbaden: Vieweg+Teubner Verlag, p. 111,
ISBN978-3-8348-2348-9
^
abRolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 286,
ISBN978-3-540-32058-6
^Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 287,
ISBN978-3-540-32058-6
^Rolf Busam (2006), Funktionentheorie 1 (in German) (4., korr. und erw. Aufl ed.), Berlin: Springer, p. 288,
ISBN978-3-540-32058-6
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
ISBN0-8218-4532-2
Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York
ISBN0-387-97127-0 (See chapter 1.)
K. Chandrasekharan, Elliptic functions (1980), Springer-Verlag
ISBN0-387-15295-4
Konrad Knopp, Funktionentheorie II (1947), Dover Publications; Republished in English translation as Theory of Functions (1996), Dover Publications
ISBN0-486-69219-1