Jacobi elliptic functions

In mathematics, a Jacobi elliptic function is one of twelve special elliptic functions. The Jacobi elliptic functions have some analogies to the trigonometric functions and find numerous applications in mathematical physics, elliptic filters and in geometry, in particular for the pendulum equation and the arc length of an ellipse. Carl Gustav Jacob Jacobi, she introduced around 1830. Carl Friedrich Gauss had, however, already in 1796 examined two special Jacobi functions with the lemniscate sine and cosine, his notes about it but not published. Today less the Jacobian, but rather the Weierstrass elliptic functions, however, play a role in the general theory of elliptic functions.

The three basic Jacobi functions

There are twelve Jacobian elliptic functions, one of which can be nine form of three basic functions. Given a parameter k, the elliptical module, which satisfies the inequality. It is often also expressed as m, and, as a modular or angle,. In addition, often the so-called complementary parameters and used. The three basic Jacobi elliptic functions are then:

The sine amplitudinis,
The cosine amplitudinis,
The delta amplitudinis.

They are elliptic functions and accordingly have two periods. Overall, they are subject to the following properties:

Here hang the periods K and K ' with the parameter k together on the elliptic integrals

Thus sn example, has zeros at and, and poles at and.

Especially for result of the sine or cosine amplitudinis amplitudinis, up to a constant, introduced by Gauss lemniscate sine and cosine functions,

For the limit values and the Jacobi functions yield the ( non-elliptical ) trigonometric functions and hyperbolic functions:

Definitions

There are several equivalent definitions of the Jacobi functions.

Abstract definition as special meromorphic functions

Given as free parameters of the elliptic modular k and the above that depend on real numbers K and K ', with

Furthermore, it is a rectangle with sides of length K and K ' in the complex plane, with the vertices s, c, d, n is given, the area S lies at the origin and the side lengths K and K' have. The sides of length K are parallel to the real axis. Length K ' parallel to the imaginary axis The corner C is the point K, the point K d ik ' and the point n ik' on the imaginary axis. The twelve Jacobi elliptic functions are then formed from a combination of letters pq, where pq each one of the letters s, c, d, n.

A Jacobi elliptic function is then the unique double -periodic meromorphic function that satisfies the following three properties:

The function has a simple zero at q and a simple pole at p.
The function is periodic in the direction of P -Q, wherein the period is twice the distance from p to q. Similar periodically in the other two directions, but with a period corresponding to four times the distance from p to the other point.
If the function to the vertex p develops, the leading term is simply z ( by the coefficient 1 ), the leading term of the expansion around the point q is 1 / z, and the leading term of the expansion to the other two vertices respectively 1

Definition as inverse functions of elliptic integrals

The above definition as a unique meromorphic function is very abstract. Equivalent to a Jacobi elliptic function can be defined as a unique inverse function of the incomplete elliptic integral of the first kind. This is the usual and perhaps most understandable definition. Let k be a given parameter, and let

Therefore. Then the Jacobi elliptic functions are sn, cn and dn given by

And

The angle is the amplitude, for he is called delta amplitude. Furthermore, the free parameter k satisfies the inequality. For z is the quarter period K.

The other nine Jacobi elliptic functions are formed from these three basic, see the next section.

Definition using the theta functions

A further definition of the Jacobi functions uses the theta functions. Let k and k 'are two real constants and. Then are the three basic Jacobi functions

Where

And.

The derived Jacobi functions

Typically, the reciprocals of the three basic Jacobian functions are denoted by the inverse of the letter sequence, thus:

The ratios of the three basic Jacobian functions are denoted by the first letters of the numerator and the denominator, that is:

So Shortens we can write

Where p, q and r are each one of the letters s, c, d, n and ss = is cc = dd = nn = 1.

Addition theorems

The Jacobi functions satisfy the two algebraic relations

Thus parameterize (cn, sn, dn) be an elliptic curve, which represents the intersection of the two defined by the above equations quadrics. Furthermore, we can define a group with the addition theorems law for points on this curve:

Quadratic relations

With. More quadratic relationship can be established with and, where p, q and r are each one of the letters s, c, d, n and ss = is cc = dd = nn = 1.

Development as a Lambert series

The functions can be developed in a Lambert series with ( to English. Nome ) and the argument

And

The Jacobi elliptic functions as solutions of nonlinear differential equations

The derivatives of the three basic Jacobi elliptic functions are:

With the addition theorems above are therefore, for a given k with solutions of the following nonlinear differential equations:

Triggers and

Triggers and

Triggers and