Elliptic rational functions

The Rational elliptic functions represent in mathematics a set of rational functions with real factors; they are for the design of transfer functions in Cauer filters in electronic signal processing.

A certain rational elliptic function is n by their order and a real selective factor ξ ≥ 1 characterized. Formally, the rational elliptic functions with the parameter x are defined as:

Where the function cd ( ·) is a derivative of Jacobi elliptic function, consisting of the cosine amplitudinis and the delta amplitudinis represents. K ( · ) is the elliptic integral of the first kind and represents a discrimination factor, which is equal to the least amount of value.

Expressed as a rational function

For orders in the form n = 2A3B, with a and b non-negative integers, the rational elliptic functions can be expressed by analytic functions.

The rational elliptic functions can be expressed as For just order n in these cases as the quotient of two polynomials, both n with order:

With the zeros xi and poles xpi. Factor R0 is selected so that the following applies.

For odd -order a pole at x = ∞ and a zero result at x = 0, which rationally elliptic functions with odd order in the form

Can be expressed.

This means that the first orders of the rational elliptic functions can be formulated:

Other orders can then form by means of the lower orders Verschachtelungseigenschaft:

Properties

Normalization

All rational elliptic functions are normalized at x = 1 to 1:

Nesting

When the nesting property holds:

Directly from the property to the nesting follows the above rule to specify certain orders as a rational function, as can be specified and as a closed analytical expression. Thus all orders let n = 2A3B in the form of analytic functions specify.

Limits

The limits of the rational elliptic functions for ξ → ∞ can be expressed as Chebyshev polynomials of the first kind Tn:

Symmetry

It is generally:

Ripple

Has a uniform ripple of ± 1 in the interval -1 ≤ x ≤ 1

Reciprocal

It is generally:

The means that the poles and zeros must occur in pairs and the relation:

Have to suffice. Odd orders thus have a place zeros at x = 0 and a pole at infinity on.

Swell

• Elliptic rational functions on MathWorld (English )