General
Historical remarks
The Weierstrass elliptic functions are identified with the famous mathematicians N. H. Abel (1827) and K. Weierstrass (1855, 1862). In the year 1849, C. Hermite first used the notation ℘123 for the basic Weierstrass doubly periodic function with only one double pole. The sigma and zeta Weierstrass functions were introduced in the works of F. G. Eisenstein (1847) and K. Weierstrass (1855, 1862, 1895).
The Weierstrass elliptic and related functions can be defined as inversions of elliptic integrals like and . Such integrals were investigated in the works of L. Euler (1761) and J.‐L. Lagrange (1769), who basically introduced the functions that are known today as the inverse Weierstrass functions.
Periodic functions
An analytic function is called periodic if there exists a complex constant such that . The number (with a minimal possible value of ) is called the period of the function .
Examples of well‐known singly periodic functions are the exponential functions, all the trigonometric and hyperbolic functions: , sin(z), cos(z), csc(z), sec(z), tan(z), cot(z), sinh(z), cosh(z), csch(z), sech(z), tanh(z), and coth(z), which have periods , , , , and . The study of such functions can be restricted to any period‐strip , because outside this strip, the values of these functions coincide with their corresponding values inside the strip.
Nonconstant analytic functions over the field of complex numbers cannot have more than two independent periods. So, generically, periodic functions can satisfy the following relations:
where , , and are periods (basic primitive periods). The condition for doubly periodic functions implies the existence of a period‐parallelogram , which is the analog of the period‐strip for singly periodic functions with period .
In the case , this parallelogram is called the basic fundamental period‐parallelogram: . The two line segments lying on the boundary of the periodparallelogram and beginning from the origin belong to . The region includes only one corner point from four points lying at the boundary of the parallelogram with corners in .
Sometimes the convention is used.
The set of all such period‐parallelograms:
covers all complex planes: .
Any doubly periodic function is called an elliptic function. The set of numbers is called the period‐lattice for elliptic function .
An elliptic function , which does not have poles in the period‐parallelogram, is equal to a constant (Liouville's theorem).
Nonconstant elliptic (doubly periodic) functions cannot be entire functions. This is not the case for singly periodic functions, for example, is entire function.
Any nonconstant elliptic function has at least two simple poles or at least one double pole in any period‐parallelogram. The sum of all its residues at the poles inside a period‐parallelogram is zero.
The numbers of the zeros and poles of a nonconstant elliptic function in a fundamental period‐parallelogram P are finite.
The number of the zeros of , where is any complex number, in a fundamental period‐parallelogram does not depend on the value and coincides with number of the poles counted according to their multiplicity ( is called the order of the elliptic function ).
The simplest elliptic function has order 2.
Let (and ) be the zeros (and poles) of a nonconstant elliptic function in a fundamental period‐parallelogram , both listed one or more times according to their multiplicity. Then the following hold:
So, the number of zeros of a nonconstant elliptic function in the fundamental period‐parallelogram is equal to the number of poles there and counted according to their multiplicity. The sum of zeros of a nonconstant elliptic function in the fundamental period‐parallelogram differs from the sum of its poles by a period , where and the values of , depend on the function .
All elliptic functions satisfy a common fundamental property, which generalizes addition, duplication, and multiple angle properties for trigonometric and hyperbolic functions (like , ). It can be formulated as the following:
.
It can also be expressed as an algebraic function of .
In other words, there exists an irreducible polynomial in variables with constant coefficients, for which the following relation holds:
And conversely, among all smooth functions, only elliptic functions and their degenerations have algebraic addition theorems.
The simplest elliptic functions (with order 2) can be divided into two classes:
(1) Functions that at the period‐parallelogram have only a double pole with residue zero (e.g., the Weierstrass elliptic functions ).
(2) Functions that in the period‐parallelogram have only two simple poles with residues, which are equal in absolute value but opposite in sign (e.g., Jacobian elliptic functions etc.).
Any elliptic function with periods and can be expressed as a rational function of the Weierstrassian elliptic functions and their derivative with the same periods .
The Weierstrass elliptic function arises as a solution to the following ordinary nonlinear differential equation:
