Elliptic function

In the mathematical branch of function theory elliptic functions are doubly periodic meromorphic functions. " Doubly periodic " means that there is in the two linearly independent real vector space periods in the form of two complex numbers, so that the two periodicity conditions in terms of the functional equations

Are satisfied for all. " Meromorphic " means that the function up isolated on poles everywhere regular ( holomorphic = analytic ), that there is infinitely differentiable and can be developed locally in a power series.

The elliptic functions are inverse functions of elliptic integrals.

Generalizations of elliptic functions are the hyperelliptic functions.

Relationship with ellipses

The name of the elliptic functions indicates that they were first used in the calculation of the circumference of ellipses. A further application is the calculation of the period of oscillation of a pendulum.

Period gratings and basic stitch

If and as above, so also applies

For any linear combination of integers. The abelian group

Is called the period lattice. It is in a complete lattice.

The parallelogram spanned by and

Ie basic mesh of the grid.

Simple properties

• A holomorphic elliptic function is constant: It's all, and they are limited because they already accept on the ground stitch all its values ​​and the basic stitch is compact. By the theorem of Liouville it is constant.
• An elliptic function can not have exactly one simple pole in the base mesh, since the integral vanishes over the edge of the base mesh due to the periodicity; this includes after the Cauchy integral formula from a simple pole.
• The corresponding statement is also true for exactly one simple zero, since also is an elliptic function.

The Weierstrass ℘ - function

Always exists for a period gratings a non-constant elliptic function, the Weierstrass ℘ - function (the symbol is called Weierstrass p):

In essence, therefore made ​​by translations to a - invariant function; the summands are only used to make the series convergent.

Is an even elliptic function, ie. its derivative

Is an odd elliptic function, that is,

The central result of the theory of elliptic functions is the following statement: Every elliptic function of the period lattice can be described as rational function in and write. Every relation between and follows from the differential equation of the function

Here are constants that are more accurate and depend of Eisenstein series to the grid. In algebraic language of this sentence means: The body of elliptic functions to the periodic lattice is isomorphic to the body

Under this isomorphism is on and ready.

Derivation of this ℘ - function in the same area and with the same color.

℘ - function for the lattice in the same area and with the same color.

History of the elliptic functions

Soon after the development of calculus mathematicians came on problems where integrals occurring in which occurred the square roots of polynomials Grade 3 and 4. It was recognized that they could not be expressed in closed form using the hitherto customary functions. Euler (1707-1783) brought in 1766 in a relationship with each other with the help of a sentence, after which he could be represented as an integral of the same type, the sum certain of such integrals again. He pointed out that one can introduce as symbols in mathematics these integrals as well as the zyklometrischen functions and the logarithm function.

His ideas were - apart from a remark Landing - only in 1786 by Legendre ( 1752-1833 ) in his Mémoires sur les integration par two arcs d' ellipse ( treatises on the integration of elliptical arcs ) pursued. Legendre has repeatedly employed from then on with this kind of integrals and named it " elliptic functions ". From Legendre's works are to be mentioned: Mémoire sur les transcendantes elliptiques (1792 ), Exercices de calcul intégral ( 1811-1817 ), Traité des fonctions elliptiques ( 1825-1832 ). Legendre introduced the elliptic functions on three solid forms - Genres - back, making it the then very difficult access to their investigation much relieved himself. His work remained completely unnoticed until 1826.

Only from that time these studies took the two mathematicians Abel (1802-1829) and Jacobi (1804-1851) again, and came quickly to unexpected new findings. First, they returned to the problem by understood it the variable imaginary upper limit of the integral as a function of the integral value, ie considered the inverse of the elliptic integral functions. These inverse functions are called after a suggestion from 1829 Jacobi elliptic functions now. The work of Jacobi and Abel can be found in the Journal of Crelle in 1826. Moreover, Jacobi's Fundamenta nova theoriae functionum ellipticarum (1829 ) should be mentioned. Jacobi proved in 1835 that the unique functions of one variable have at most two independent periods. The elliptic functions have exactly two. The addition theorem discovered by Euler in a very special form was in its general form in 1829 and proved extremely Abel. Gauss had, as he himself remarked, as has also be detected, thirty years ago found many of the properties of elliptic functions, but published nothing about it.

Further development has led to the hyperelliptic functions, the Abelian functions and modular functions.