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"℘" 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
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Transcription
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]
Elliptic functions are the inverse functions of elliptic integrals. In particular, let:
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 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.
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