HOME

TheInfoList



OR:

In the mathematical field of complex analysis, elliptic functions are a special kind of
meromorphic 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 pole (complex analysis), pol ...
functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from
elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...
s. Originally those integrals occurred at the calculation of the arc length of an
ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse i ...
. Important elliptic functions are
Jacobi elliptic functions In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While tr ...
and the Weierstrass \wp-function. Further development of this theory led to hyperelliptic functions and
modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory ...
s.


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 pole (complex analysis), pole ...
is called an elliptic function, if there are two \mathbb- linear independent complex numbers \omega_1,\omega_2\in\mathbb such that : f(z + \omega_1) = f(z) and f(z + \omega_2) = f(z), \quad \forall z\in\mathbb. So elliptic functions have two periods and are therefore also called ''doubly periodic''.


Period lattice and fundamental domain

Iff is an elliptic function with periods \omega_1,\omega_2 it also holds that : f(z+\gamma)=f(z) for every linear combination \gamma=m\omega_1+n\omega_2 with m,n\in\mathbb . 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 comm ...
: \Lambda:=\langle \omega_1,\omega_2\rangle_:=\mathbb Z\omega_1+\mathbb Z\omega_2:=\ is called the ''period lattice''. The
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equ ...
generated by \omega_1and \omega_2 : \ 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 exampl ...
\mathbb/\Lambda as their domain. This quotient group, called an
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
, can be visualised as a parallelogram where opposite sides are identified, which
topologically In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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 tou ...
.


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 Thérèse ...
'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 \mathbb/\Lambda and the sum of its residues 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 \mathbb/\Lambda counted with multiplicity.


Weierstrass ℘-function

One of the most important elliptic functions is the Weierstrass \wp-function. For a given period lattice \Lambda it is defined by : \wp(z)=\frac1+\sum_\left(\frac1-\frac1\right). It is constructed in such a way that it has a pole of order two at every lattice point. The term -\frac1 is there to make the series convergent. \wp is an even elliptic function, that means \wp(-z)=\wp(z). Its derivative : \wp'(z)=-2\sum_\frac1 is an odd function, i.e. \wp'(-z)=-\wp'(z). One of the main results of the theory of elliptic functions is the following: Every elliptic function with respect to a given period lattice \Lambda can be expressed as a rational function in terms of \wp and \wp'. The \wp-function satisfies the
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
: \wp'^2(z)=4\wp(z)^3-g_2\wp(z)-g_3. g_2 and g_3 are constants that depend on \Lambda. More precisely g_2(\omega_1,\omega_2)=60G_4(\omega_1,\omega_2) and g_3(\omega_1,\omega_2)=140G_6(\omega_1,\omega_2), where G_4 and G_6 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 generaliz ...
. In algebraic language: The field of elliptic functions is isomorphic to the field : \mathbb C(X) (Y^2-4X^3+g_2X+g_3), where the isomorphism maps \wp to X and \wp' to Y. File:Weierstrass-p-1.jpg, Weierstrass \wp-function with period lattice \Lambda=\mathbb+e^\mathbb File:Weierstrass-dp-1.jpg, Derivative of the \wp-function


Relation to elliptic integrals

The relation to
elliptic integrals In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising ...
has mainly a historical background. Elliptic integrals had been studied by Legendre, whose work was taken on by
Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
and
Carl Gustav Jacobi Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, Dynamics (mechanics), dynamics, differential equations, determinants, and number theory. H ...
. Abel discovered elliptic functions by taking the inverse function \varphi of the elliptic integral function : \alpha(x)=\int_0^x \frac with x=\varphi(\alpha). Additionally he defined the functions : f(\alpha)=\sqrt and : F(\alpha)=\sqrt. After continuation to the complex plane they turned out to be doubly periodic and are known as
Abel elliptic functions In mathematics Abel elliptic functions are a special kind of elliptic functions, that were established by the Norwegian mathematician Niels Henrik Abel. He published his paper "Recherches sur les Fonctions elliptiques" in Crelle's Journal in 1827. I ...
.
Jacobi elliptic functions In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While tr ...
are similarly obtained as inverse functions of elliptic integrals. Jacobi considered the integral function : \xi(x)=\int_0^x \frac and inverted it: x=\operatorname(\xi). \operatorname 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: : \operatorname(\xi):=\sqrt : \operatorname(\xi):=\sqrt . 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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arit ...
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 ...
. 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 Landen () is a city and municipality located in the Belgian province of Flemish Brabant. The municipality comprises the city of Landen proper and the villages of Attenhoven, Eliksem, Ezemaal, Laar, Neerlanden, Neerwinden, Overwinden, Rumsdorp, ...
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 Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
and
Carl Gustav Jacobi Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, Dynamics (mechanics), dynamics, differential equations, determinants, and number theory. H ...
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 In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...
*
Elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
*
Modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
*
Theta function In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field the ...


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 Tom Mike Apostol (August 20, 1923 – May 8, 2016) was an American analytic number theorist and professor at the California Institute of Technology, best known as the author of widely used mathematical textbooks. Life and career Apostol was bor ...
, ''Modular Functions and Dirichlet Series in Number Theory'', Springer-Verlag, New York, 1976. ''(See Chapter 1.)'' *
E. T. Whittaker Sir Edmund Taylor Whittaker (24 October 1873 – 24 March 1956) was a British mathematician, physicist, and historian of science. Whittaker was a leading mathematical scholar of the early 20th-century who contributed widely to applied mathema ...
and G. N. Watson. ''
A course of modern analysis ''A Course of Modern Analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions'' (colloquially known as Whittaker and Watson) is a landmark textb ...
'', Cambridge University Press, 1952


External links

* * MAA
Translation of Abel's paper on elliptic functions.
* , lecture by William A. Schwalm (4 hours) * {{Authority control