** Introduction **

The circular functions arise from ratios of lengths in a circle. In a similar manner, the elliptic functions can be defined by means of ratios of lengths in an ellipse. Many of the key properties of the elliptic functions follow from simple geometric properties of the ellipse.

Originally, Carl Gustav Jacobi defined the elliptic functions , , using the integral

He called the *amplitude* and wrote . It can be difficult to understand what motivated his definitions. We will define the elliptic functions , , in a more intuitive way, as simple ratios associated with an ellipse.

** Circular Functions **

To set the scene, we recall some key properties of the circular functions. The familiar trigonometric functions , and are described effectively by ratios of quantities in a circle. The algebraic formula for a circle is , where is the radius of the circle. Then the trigonometric functions are defined as

The parametric description of the circle is

and the theorem of Pythagoras leads immediately to the identity

The derivatives of the trigonometric functions are related by

Moreover, the functions and both satisfy the nonlinear differential equation

with appropriate boundary conditions.

There are numerous trigonometric identities relating the circular functions, for example, the addition formula

These have counterparts for the elliptic functions. Introducing the reciprocols and ratios of and , we have in total six trigonometric functions:

For the elliptic case, we will find a family of *twelve* functions. Finally, we recall the characteristic forms of and as periodic functions. Similar behaviour for the elliptic functions and will easily be seen from the definitions as ratios of lengths in an ellipse.

** Elliptic Functions **

The ellipse with centre at the origin and horizontal major axis has the equation

The eccentricity is usually denoted by or , but here we will write it as . It is defined by

For the circle, and so . We assume below that . For a greatly flattened ellipse, approaches 1. In contrast to the circle, the radius vector from the origin to the ellipse varies in length, and we have the Pythagorean relationship

From these two relationships we will derive the main properties of the elliptic functions.

We introduce a variable defined by

We note that in the circular case it is just the arc-length . In the elliptic case, it allows for the variation of as the representative point progresses around the ellipse. The variable is intimately related to Jacobi’s amplitude function mentioned above. Indeed, using the equation for the ellipse, we can show that

If , we define the eccentricity by and the cosine is replaced by a sine, giving the form of Jacobi’s integral that appears at the top of this article.

** “Sun”, “cun” and “dun”**

By analogy with the circular functions, we define three elliptic functions:

These should be compared to the definitions of and . When reading formulas aloud, we may call the three functions “sun”, “cun” and “dun”. Note that the third function, , has no equivalent in the circular case, as it is identically equal to unity when .

From the equation of the ellipse, it follows immediately that

The Pythagorean relationship gives rise to another identity:

The definitions of the elliptic functions, together with the equation of the ellipse and the Pythagorean relationship, allow us to obtain expressions for the derivatives:

It follows from these that the elliptic functions satisfy nonlinear differential equations, for example

The elliptic functions satisfy a wealth of identities. For example, the addition formula for the function is

which clearly reduces to the addition formula for when .

We introduce the reciprocals of , and , denoting them by , and . We also define the ratio , and two other ratios in an obvious way. Finally, we denote the reciprocal , and so on. Thus, we have in total twelve distinct elliptic functions:

**What do the elliptic functions look like?**

The representative point on the ellipse may be represented as or, from the definitions of and , as . Thus, we have

Now let us consider how changes as the point moves around the ellipse, starting at . For descriptive purposes, we can approximate by the polar angle if is small. Clearly, varies periodically, starting from and oscillating in the interval as varies between and . The function starts from and varies periodically, oscillating in the interval as varies between and . Finally, the function starts from and varies periodically, but it remains positive, in the range with period half that of and .

There are numerous interrelationships between the twelve elliptic functions. For example, the function is just shifted by a quarter-period, reflecting the relationship between the sine and cosine (see Fig. below).

**Applications**

Nonlinear differential equations for of the form

arise in a wide range of problems in physics. By a judicious transformation of variables, such equations can be solved by one of the standard Jacobi elliptic functions.

Thus, the Jacobi elliptic functions occur in many applications, such as celestial mechanics, the dynamics of spinning tops, fluid dynamics, electromagnetic theory, evaluation of integrals and the solution of differential equations.

** Sources **

William A. Schwalm, 2018: Elliptic Functions and Elliptic Integrals. PDF. See also four 1-hour lectures on YouTube

Wikipedia article *Jacobi elliptic functions.*

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