In the mathematical field of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
, elliptic functions are a special kind of
meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from
elliptic integrals. Originally those integrals occurred at the calculation of the arc length of an
ellipse.
Important elliptic functions are
Jacobi elliptic functions and the
Weierstrass -function.
Further development of this theory led to
hyperelliptic functions and
modular forms.
Definition
A
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. ...
is called an elliptic function, if there are two
-
linear independent complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s
such that
:
and
.
So elliptic functions have two periods and are therefore also called ''doubly periodic''.
Period lattice and fundamental domain

If
is an elliptic function with periods
it also holds that
:
for every linear combination
with
.
The
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
:
is called the ''period lattice''.
The
parallelogram generated by
and
:
is called ''fundamental domain.''
Geometrically the complex plane is tiled with parallelograms. Everything that happens in the fundamental domain repeats in all the others. For that reason we can view elliptic function as functions with the
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
as their domain. This quotient group, called an
elliptic curve, can be visualised as a parallelogram where opposite sides are identified, which
topologically is a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not ...
.
Liouville's theorems
The following three theorems are known as ''
Liouville
Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer.
Life and work
He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérès ...
's theorems (1847).''
1st theorem
A holomorphic elliptic function is constant.
This is the original form of
Liouville's theorem and can be derived from it. A holomorphic elliptic function is bounded since it takes on all of its values on the fundamental domain which is compact. So it is constant by Liouville's theorem.
2nd theorem
Every elliptic function has finitely many poles in
and the sum of its
residues
Residue may refer to:
Chemistry and biology
* An amino acid, within a peptide chain
* Crop residue, materials left after agricultural processes
* Pesticide residue, refers to the pesticides that may remain on or in food after they are appli ...
is zero.
This theorem implies that there is no elliptic function not equal to zero with exactly one pole of order one or exactly one zero of order one in the fundamental domain.
3rd theorem
A non-constant elliptic function takes on every value the same number of times in
counted with multiplicity.
Weierstrass ℘-function
One of the most important elliptic functions is the Weierstrass
-function. For a given period lattice
it is defined by
:
It is constructed in such a way that it has a pole of order two at every lattice point. The term
is there to make the series convergent.
is an even elliptic function, that means
.
Its derivative
:
is an odd function, i.e.
[
One of the main results of the theory of elliptic functions is the following: Every elliptic function with respect to a given period lattice can be expressed as a rational function in terms of and .
The -function satisfies the differential equation
:
and are constants that depend on . More precisely and , where and are so called ]Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be general ...
.
In algebraic language: The field of elliptic functions is isomorphic to the field
: ,
where the isomorphism maps to and to .
File:Weierstrass-p-1.jpg, Weierstrass -function with period lattice
File:Weierstrass-dp-1.jpg, Derivative of the -function
Relation to elliptic integrals
The relation to elliptic integrals has mainly a historical background. Elliptic integrals had been studied by Legendre, whose work was taken on by Niels Henrik Abel and Carl Gustav Jacobi.
Abel discovered elliptic functions by taking the inverse function of the elliptic integral function
:
with .
Additionally he defined the functions
:
and
: .
After continuation to the complex plane they turned out to be doubly periodic and are known as Abel elliptic functions.
Jacobi elliptic functions are similarly obtained as inverse functions of elliptic integrals.
Jacobi considered the integral function
:
and inverted it: . stands for ''sinus amplitudinis'' and is the name of the new function. He then introduced the functions ''cosinus amplitudinis'' and ''delta amplitudinis'', which are defined as follows:
:
: .
Only by taking this step, Jacobi could prove his general transformation formula of elliptic integrals in 1827.
History
Shortly after the development of infinitesimal calculus the theory of elliptic functions was started by the Italian mathematician Giulio di Fagnano and the Swiss mathematician Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
. When they tried to calculate the arc length of a lemniscate they encountered problems involving integrals that contained the square root of polynomials of degree 3 and 4. It was clear that those so called elliptic integrals could not be solved using elementary functions. Fagnano observed an algebraic relation between elliptic integrals, what he published in 1750.[ Euler immediately generalized Fagnano's results and posed his algebraic addition theorem for elliptic integrals.][
Except for a comment by Landen his ideas were not pursued until 1786, when Legendre published his paper ''Mémoires sur les intégrations par arcs d’ellipse''. Legendre subsequently studied elliptic integrals and called them ''elliptic functions''. Legendre introduced a three-fold classification –three kinds– which was a crucial simplification of the rather complicated theory at that time. Other important works of Legendre are: ''Mémoire sur les transcendantes elliptiques'' (1792), ''Exercices de calcul intégral'' (1811–1817), ''Traité des fonctions elliptiques'' (1825–1832). Legendre's work was mostly left untouched by mathematicians until 1826.
Subsequently, Niels Henrik Abel and Carl Gustav Jacobi resumed the investigations and quickly discovered new results. At first they inverted the elliptic integral function. Following a suggestion of Jacobi in 1829 these inverse functions are now called ''elliptic functions''. One of Jacobi's most important works is ''Fundamenta nova theoriae functionum ellipticarum'' which was published 1829. The addition theorem Euler found was posed and proved in its general form by Abel in 1829. Note that in those days the theory of elliptic functions and the theory of doubly periodic functions were considered to be different theories. They were brought together by Briout and Bouquet in 1856. ]Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
discovered many of the properties of elliptic functions 30 years earlier but never published anything on the subject.
See also
* Elliptic integral
* Elliptic curve
* Modular group
* Theta function
References
Literature
* (only considers the case of real invariants).
* 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'', Springer-Verlag, New York, 1976. ''(See Chapter 1.)''
* E. T. Whittaker and G. N. Watson
George Neville Watson (31 January 1886 – 2 February 1965) was an English mathematician, who applied complex analysis to the theory of special functions. His collaboration on the 1915 second edition of E. T. Whittaker's ''A Course of Modern ...
. '' A course of modern analysis'', Cambridge University Press, 1952
External links
*
* MAA
Translation of Abel's paper on elliptic functions.
* , lecture by William A. Schwalm (4 hours)
*
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