Introductions to JacobiNC

Introduction to the Jacobi elliptic functions

General

Historical remarks

Jacobi functions are named for the famous mathematician C. G. J. Jacobi. In 
1827 he introduced the elliptic amplitude am(z|m) as the inverse function of the elliptic integral F(z|m) by the variable z and investigated the twelve functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m). 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 sn(z|m).

The modern notations for Jacobi functions were introduced later in the 
works of C. Gudermann (1838) (for functions cn, dn, and sn) and J. Glaisher (1882) (for functions cd, cs, dc, ds, nc, nd, ns, sc, and sd). V. A. Puiseux (1850) showed that amplitude am is a multivalued function.

Periodic functions

An analytic function f(z) is called periodic if there exists a complex constant ρ≠0, such that f(z+ρ)==f(z)/;z∈ℂ. The number ρ (with minimal possible value of |ρ|) is called the period of the function f(z).

Examples of well-known singly periodic functions are the 
exponential functions and all the trigonometric and hyperbolic functions: (e^z), 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 ρ=2 π i, ρ=2 π, ρ=π, ρ=2 π i, and ρ=π i. The study of such functions can be restricted to any period-strip 
{(Subscript[z, 0])+α ρ/;0≤α<1∧(Subscript[z, 0])∈ℂ}, 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 ρ, (Subscript[ρ, 1]), and (Subscript[ρ, 2]) are periods (basic primitive periods). The condition Im(((Subscript[ρ, 1])/(Subscript[ρ, 2])))≠0 for doubly periodic functions implies the existence of a 
period-parallelogram {(Subscript[z, 0])+(Subscript[α, 1]) (Subscript[ρ, 1])+(Subscript[α, 2])(Subscript[ρ, 2])/;0≤(Subscript[α, 1])<1∧0≤(Subscript[α, 1])<1∧(Subscript[z, 0])∈ℂ}, which is the analog of the period-strip {(Subscript[z, 0])+α ρ/;0≤α<1∧(Subscript[z, 0])∈ℂ} for the singly periodic function with period ρ.

In the case (Subscript[z, 0])==0∧Im(((Subscript[ρ, 1])/(Subscript[ρ, 2])))>0, this parallelogram is called the basic fundamental 
period-parallelogram: (Subscript[Ρ, 0,0])=={(Subscript[α, 1]) (Subscript[ρ, 1])+(Subscript[α, 2])(Subscript[ρ, 2])/;0≤(Subscript[α, 1])<1∧0≤(Subscript[α, 2])<1}. The two line segments {(Subscript[α, i]) (Subscript[ρ, i])/;0≤(Subscript[α, i])<1}/;i==1,2 lying on the boundary of the period-parallelogram and beginning from the 
origin 0 belong to (Subscript[Ρ, 0,0]). The region (Subscript[Ρ, 0,0]) includes only one corner point 0 from four points lying at the boundary of parallelogram with corners in {0, (Subscript[ρ, 1]), (Subscript[ρ, 1])+(Subscript[ρ, 2]),(Subscript[ρ, 2])}. Sometimes the convention (Subscript[Ρ, 0,0])=={(Subscript[α, 1]) (Subscript[ρ, 1])+(Subscript[α, 2])(Subscript[ρ, 2])/;-1/2≤(Subscript[α, 1])<1/2∧-1/2≤(Subscript[α, 2])<1/2} is used.

The set of all such period-parallelograms:

covers all complex planes: ℂ=={(Subscript[Ρ, m,n])|-∞<m,n<∞∧m∈ℤ∧n∈ℤ}.

Any doubly periodic function ℰ(z) is called an elliptic function. The set of numbers m (Subscript[ρ, 1])+n (Subscript[ρ, 2])/;{m,n}∈ℤ is called the period-lattice for the elliptic function ℰ(z).

An elliptic function ℰ(z), which does not have poles in the period-parallelogram, is equal 
to the constant Liouville's theorem.

Nonconstant elliptic (doubly periodic) functions ℰ(z) cannot be entire functions (this is not the case for singly periodic 
functions, for example, sin(z) is an entire function.

Any nonconstant elliptic function ℰ(z) 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 ℰ(z) in a fundamental period-parallelogram P are finite.

The number of zeros of ℰ(z)-A, where A is any complex number, in a fundamental period-parallelogram (Subscript[Ρ, 0,0]) does not depend on the value A and coincides with the number s of the poles (Subscript[b, 1]),(Subscript[b, 2]),…,(Subscript[b, s]) counted according to their multiplicity (s is called the order of the elliptic function ℰ(z)).

The simplest elliptic function has order two.

Let (Subscript[a, 1]),(Subscript[a, 2]),…,(Subscript[a, r]) (and (Subscript[b, 1]),(Subscript[b, 2]),…,(Subscript[b, s])) be the zeros (and poles) of a nonconstant elliptic function ℰ(z) in a fundamental period-parallelogram (Subscript[Ρ, 0,0]), 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 ℰ(z) in the fundamental period-parallelogram (Subscript[Ρ, 0,0]) is equal to the number of poles there and counted according to their 
multiplicity. The sum of zeros of a nonconstant elliptic function ℰ(z) in the fundamental period-parallelogram (Subscript[Ρ, 0,0]) differs from the sum of its poles by a period μ  (Subscript[ρ, 1])+ν (Subscript[ρ, 2]), where μ∈ℤ∧ν∈ℤ and the values of μ, ν depend on the function ℰ(z).

All elliptic functions ℰ(z) satisfy a common fundamental property, which generalizes addition, 
duplication, and multiple angle properties for the trigonometric and 
hyperbolic functions such as sin((Subscript[z, 1])+(Subscript[z, 2])), sin(n z)/;n∈(SuperPlus[ℕ]). It can be formulated as:

ℰ((∑_(k=1)^n)(Subscript[z, k])),

which can be expressed as an algebraic function of ℰ((Subscript[z, k]))/;1≤k≤n. In other words, there exists an irreducible polynomial C((Subscript[t, 1]),(Subscript[t, 2]),…,(Subscript[t, n+1])) in n+1 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 (Subscript[Ρ, 0,0]) have only a double pole with a residue zero (e.g., the Weierstrass 
elliptic functions   ℘(z;(Subscript[g, 2]),(Subscript[g, 3]))).

(2) Functions that at the period-parallelogram (Subscript[Ρ, 0,0]) have only two simple poles with residues, which are equal in absolute 
value but opposite in sign (e.g., the Jacobian elliptic functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m)).

Jacobian elliptic functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) arise as solutions to the differential equation:

with the following coefficients:

Definitions of Jacobi functions

The Jacobi elliptic amplitude am(z|m) and the twelve Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) are defined by the following formulas:

It is apparent that the amplitude function am(z|m) is the inverse function to elliptic integral F(w|m)==z, and the functions cn(z|m), sn(z|m), and dn(z|m) are the basic Jacobi functions that are built as the cosine, sine, and 
derivative of the amplitude function am(z|m). The other nine Jacobi functions are the ratios of these three basic 
Jacobi functions or their reciprocal functions.

A quick look at the Jacobi functions

Here is a quick look at the graphics for the twelve Jacobi elliptic 
functions along the real axis for m=1/2.

Connections within the group of Jacobi functions and with other elliptic 
functions

Representations through related equivalent functions

The twelve Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) can be represented through the Weierstrass sigma functions:

The twelve Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) can be represented through the Weierstrass ℘ function:

The twelve Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) can be represented through the elliptic theta functions:

The twelve Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) can be represented through the Neville theta functions:

Relations to inverse functions

The Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) are connected with the corresponding inverse functions by the following 
formulas:

Representations through other Jacobi functions

By definition, the three basic Jacobi functions have the following 
representations through the amplitude function am:

The other nine Jacobi functions can be easily expressed through the three 
basic Jacobi functions cn(z|m), cn(z|m), and dn(z|m) and, consequently, they can also be represented through the amplitude 
function am.

The twelve Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) are interconnected by formulas that include rational functions, simple 
powers, and arithmetical operations from other Jacobi functions. These 
formulas can be divided into the following eleven groups:

Representations of cd(z|m) through other Jacobi functions are:

Representations of cn(z|m) through other Jacobi functions are:

Representations of cs(z|m) through other Jacobi functions are:

Representations of dc(z|m) through other Jacobi functions are:

Representations of dn(z|m) through other Jacobi functions are:

Representations of ds(z|m) through other Jacobi functions are:

Representations of nc(z|m) through other Jacobi functions are:

Representations of nd(z|m) through other Jacobi functions are:

Representations of ns(z|m) through other Jacobi functions are:

Representations of sc(z|m) through other Jacobi functions are:

Representations of sd(z|m) through other Jacobi functions are:

Representations of sn(z|m) through other Jacobi functions are:

The best-known properties and formulas for Jacobi functions

Real values for real arguments

For real values of arguments z and m, the values of all Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) are real (or infinity).

Simple values at zero

All thirteen Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) have the following simple values at the origin:

Specific values for specialized parameter values

All Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) can be represented through elementary functions when m==0 or m==1. The twelve elliptic functions degenerate into trigonometric and 
hyperbolic functions:

All Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) have very simple values at z==0:

The twelve Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) have the following values at the half-quarter-period 
points:

The partial derivatives of all Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) at the points z==0, m==0, or m==1 can be represented through trigonometric functions, for example:

Analyticity

All Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) are analytical meromorphic functions of z and m that are defined over (ℂ^2).

Poles and essential singularities

The amplitude function am(z|m) does not have poles and essential singularities with respect to m and z.

For fixed m, all Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) have an infinite set of singular points, including simple poles in finite 
points and an essential singular point z== (OverTilde[∞]).

The following formulas describe the sets of the simple poles for the 
corresponding Jacobi functions:

The values of the residues of the Jacobi functions at the simple poles are 
given by the following formulas:

Branch points and branch cuts

For fixed z, all Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) are meromorphic functions in m that have no branch points and branch cuts.

For fixed m, all Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) do not have branch points and branch cuts.

Periodicity

The Jacobi amplitude am(z|m) is a pseudo-periodic function with respect to z with period 2 i K(1-m) and pseudo-period 2 K(m):

The Jacobi functions cd(z|m), dc(z|m), ns(z|m), and sn(z|m) are doubly periodic functions with respect to z with periods 2 i K(1-m) and 4 K(m). The Jacobi functions cs(z|m), dn(z|m), nd(z|m), and sc(z|m) are doubly periodic functions with respect to z with periods 4 i K(1-m) and 2K(m). The Jacobi functions cn(z|m), ds(z|m), nc(z|m), and sd(z|m) are doubly periodic functions with respect to z with periods 4 i K(1-m) and 4K(m). That periodicity can be described by the following formulas:

The periodicity of Jacobi functions follow from more general formulas that 
also describe quasi-periodicity situations such as sn(z+2K(m)|m)==-sn(z|m):

Parity and symmetry

All Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) have mirror symmetry:

The Jacobi functions cd(z|m), cn(z|m), dc(z|m), dn(z|m), nc(z|m), and nd(-z|m)  are even functions with respect to z:

The Jacobi functions am(z|m), cs(z|m), ds(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) are odd functions with respect to z:

Series representations

The Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) have the following series expansions at the point z==0:

The Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) have the following series expansions at the point m==0:

The Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) have the following series expansions at the point m==1:

q-series representations

The Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) have the following so-called q-series representations:

where q(m) is the elliptic nome and K(m) is the complete elliptic integral.

Product representations

The twelve Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) have the following product representations:

where q(m) is the elliptic nome and K(m) is the complete elliptic integral.

Transformations

The amplitude function am(z|m) satisfies numerous relations that allow for transformations of its 
arguments, for example:

The twelve Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) with specific arguments can sometimes be represented through elliptic 
functions with other mostly simpler arguments, for example:

The twelve Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) with the argument complex z==u+v can be represented through elliptic functions with arguments z==u and z==v, for example:

The twelve Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) satisfy the following half-angle formulas:

The twelve Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) satisfy the following double-angle (or multiplication) formulas:

Identities

The twelve Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) satisfy the following nonlinear functional equations:

Simple representations of derivatives

The derivatives of all Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) with respect to variable z have rather simple and symmetrical representations that can be expressed 
through other Jacobi functions:

The derivatives of all Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) with respect to variable m have more complicated representations that include other Jacobi functions 
and the elliptic integral E(am(z|m)|m):

Integration

The indefinite integrals of the twelve Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) with respect to variable z can be expressed through Jacobi and elementary functions by the following 
formulas:

Differential equations

The Jacobi amplitude am(z|m) satisfies the following differential equations:

All Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) are special solutions of ordinary second-order nonlinear differential 
equations:

The twelve Jacobi functions cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) satisfy very complicated ordinary differential equations with respect to 
variable m, for example:

Zeros

All Jacobi functions am(z|m), cd(z|m), cn(z|m), cs(z|m), dc(z|m), dn(z|m), ds(z|m), nc(z|m), nd(z|m), ns(z|m), sc(z|m), sd(z|m), and sn(z|m) are equal to zero in the points z==k K(m)+n i K(1-m), where K(m) is the complete elliptic integral of the first kind and k, n are even or odd integers:

Applications of Jacobi elliptic functions

Applications of Jacobi elliptic functions include conformal mappings, 
electrostatics and magnetostatics, fluid dynamics, mechanics of tops, 
nonlinear integrable equations, the Ising model of statistical mechanics, 
celestial mechanics, closed-form solutions for nonlinear Schr
ödinger equations, analysis of exactly solvable 
chaos-exhibiting sequences, solution of equations of motion for 
quartic potentials, and elliptic function theory.

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