General

Historical remarks

Jacobi functions are named for the famous mathematician C. G. J. Jacobi. In 1827 he introduced the elliptic amplitude as the inverse function of the elliptic integral by the variable and investigated the twelve functions , , , , , , , , , , , and . In the same year, N. H. Abel independently studied properties of these functions. But earlier K. F. Gauss (1799) gave some attention to one of these function, namely .

The modern notations for Jacobi functions were introduced later in the works of C. Gudermann (1838) (for functions , , and ) and J. Glaisher (1882) (for functions , , , , , , , , and ). V. A. Puiseux (1850) showed that amplitude is a multivalued function.

Periodic functions

An analytic function is called periodic if there exists a complex constant , such that . The number (with minimal possible value of ) is called the period of the function .

Examples of well‐known singly periodic functions are the exponential functions and 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 the singly periodic function with period .

In the case this parallelogram is called the basic fundamental period‐parallelogram: . The two line segments lying on the boundary of the period-parallelogram and beginning from the origin belong to . The region includes only one corner point from four points lying at the boundary of 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 the elliptic function .

An elliptic function , which does not have poles in the period‐parallelogram, is equal to the 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 an 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 number of zeros and poles of a nonconstant elliptic function in a fundamental period‐parallelogram P are finite.

The number of zeros of , where is any complex number, in a fundamental period‐parallelogram does not depend on the value and coincides with the number of the poles counted according to their multiplicity ( is called the order of the elliptic function ).

The simplest elliptic function has order two.

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. This results in the following:

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 the trigonometric and hyperbolic functions such as , . It can be formulated as:

,

which can 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:

Conversely, among all smooth functions, only elliptic functions and their degenerations have algebraic addition theorems.

The simplest elliptic functions (of order two) can be divide into the following two classes:

(1) Functions that at the period‐parallelogram have only a double pole with a residue zero (e.g., the Weierstrass elliptic functions ).

(2) Functions that at the period‐parallelogram have only two simple poles with residues, which are equal in absolute value but opposite in sign (e.g., the Jacobian elliptic functions , , , , , , , , , , , and ).

Jacobian elliptic functions , , , , , , , , , , , and arise as solutions to the differential equation:

with the following coefficients: