Rational elliptic functions

Representation of the rational elliptic functions between and for the orders 1, 2, 3 and 4 with the selective factor . ${\displaystyle x=-1}$ ${\displaystyle x=1}$ ${\displaystyle \xi =1{,}1}$

In mathematics, the rational elliptic functions represent a series of rational functions with real factors. They are used to design transfer functions for Cauer filters in electronic signal processing.

A certain rationally elliptic function is characterized by its order and a real selective factor. Formally, the rationally elliptic functions with the parameter are defined as: ${\displaystyle n}$ ${\displaystyle \xi \geq 1}$ ${\displaystyle x}$

${\displaystyle R_{n}(\xi ,x)\equiv \mathrm {cd} \left(n{\frac {K(1/L_{n})}{K(1/\xi )}}\,\mathrm {cd} ^{-1}(x,1/\xi ),1/L_{n}\right)}$,

where the function represents a derived Jacobian elliptic function , consisting of the cosinus amplitudinis and the delta amplitudinis . stands for the elliptic integral of the first kind and represents a discrimination factor which equals the smallest absolute value of . ${\displaystyle \operatorname {cd} (\cdot )}$ ${\displaystyle K(\cdot )}$ ${\displaystyle L_{n}(\xi )=R_{n}(\xi ,\xi )}$ ${\displaystyle |x|\geq \xi }$ ${\displaystyle R_{n}(\xi ,x)}$

Expression as a rational function

For orders in the form with and non-negative integers, the rationally elliptic functions can be expressed by analytic functions . ${\displaystyle n=2^{a}3^{b}}$ ${\displaystyle a}$ ${\displaystyle b}$

For even order , the rational elliptic functions in these cases can be expressed as the quotient of two polynomials , both with order , as: ${\displaystyle n}$ ${\displaystyle n}$

${\displaystyle R_{n}(\xi ,x)=r_{0}\,{\frac {\prod _{i=1}^{n}(x-x_{i})}{\prod _{i=1}^{n}(x-x_{pi})}}}$      ( straight) ${\displaystyle n}$

with the zeros and the poles . The factor is chosen so that applies. ${\displaystyle x_{i}}$ ${\displaystyle x_{pi}}$ ${\displaystyle r_{0}}$ ${\displaystyle R_{n}(\xi ,1)=1}$

For odd order there is a pole at and a zero at , which means rational elliptic functions for odd order in the form ${\displaystyle x=\infty }$ ${\displaystyle x=0}$

${\displaystyle R_{n}(\xi ,x)=r_{0}\,x\,{\frac {\prod _{i=1}^{n-1}(x-x_{i})}{\prod _{i=1}^{n-1}(x-x_{pi})}}}$      ( odd) ${\displaystyle n}$

can be expressed.

With this, the first orders of the rationally elliptic functions can be formulated:

${\displaystyle R_{1}(\xi ,x)=x}$

${\displaystyle R_{2}(\xi ,x)={\frac {(t+1)x^{2}-1}{(t-1)x^{2}+1}}}$, with . ${\displaystyle t\equiv {\sqrt {1-{\frac {1}{\xi ^{2}}}}}}$

${\displaystyle R_{3}(\xi ,x)=x\,{\frac {(1-x_{p}^{2})(x^{2}-x_{z}^{2})}{(1-x_{z}^{2})(x^{2}-x_{p}^{2})}}}$With , , ${\displaystyle G\equiv {\sqrt {4\xi ^{2}+(4\xi ^{2}(\xi ^{2}\!-\!1))^{2/3}}}}$ ${\displaystyle x_{p}^{2}\equiv {\frac {2\xi ^{2}{\sqrt {G}}}{{\sqrt {8\xi ^{2}(\xi ^{2}\!+\!1)+12G\xi ^{2}-G^{3}}}-{\sqrt {G^{3}}}}}}$ ${\displaystyle x_{z}^{2}=\xi ^{2}/x_{p}^{2}}$

Further orders can then be created using lower orders using the nesting property:

${\displaystyle R_{4}(\xi ,x)=R_{2}(R_{2}(\xi ,\xi ),R_{2}(\xi ,x))={\frac {(1+t)(1+{\sqrt {t}})^{2}x^{4}-2(1+t)(1+{\sqrt {t}})x^{2}+1}{(1+t)(1-{\sqrt {t}})^{2}x^{4}-2(1+t)(1-{\sqrt {t}})x^{2}+1}}}$

${\displaystyle R_{5}(\xi ,x)}$, not a rational function.

${\displaystyle R_{6}(\xi ,x)=R_{3}{\bigl (}R_{2}(\xi ,\xi ),R_{2}(\xi ,x){\bigr )}}$

properties

normalization

All rational elliptic functions are in on normalized: ${\displaystyle x=1}$ ${\displaystyle 1}$

${\displaystyle R_{n}(\xi ,1)=1}$.

Nesting

The following applies to the property of nesting:

${\displaystyle R_{m}(R_{n}(\xi ,\xi ),R_{n}(\xi ,x))=R_{m\cdot n}(\xi ,x)}$.

The above rule for specifying certain orders as a rational function follows directly from the nesting property, since and can be specified as a closed analytical expression. This means that all orders can be given in the form of analytical functions. ${\displaystyle R_{2}}$ ${\displaystyle R_{3}}$ ${\displaystyle n=2^{a}3^{b}}$

Limit values

The limit values ​​of the rational elliptic functions for can be expressed as Chebyshev polynomials of the first kind : ${\displaystyle \xi \to \infty }$ ${\displaystyle T_{n}}$

${\displaystyle \lim _{\xi =\rightarrow \,\infty }R_{n}(\xi ,x)=T_{n}(x)}$.

symmetry

The general rule is:

${\displaystyle R_{n}(\xi ,-x)=R_{n}(\xi ,x)}$for straight , ${\displaystyle n}$

${\displaystyle R_{n}(\xi ,-x)=-R_{n}(\xi ,x)\,}$for odd . ${\displaystyle n}$

Ripple

${\displaystyle R_{n}(\xi ,x)}$has a uniform ripple of in the interval . ${\displaystyle \pm 1}$ ${\displaystyle -1\leq x\leq 1}$

Reciprocal

It applies generally

${\displaystyle R_{n}(\xi ,\xi /x)={\frac {R_{n}(\xi ,\xi )}{R_{n}(\xi ,x)}}}$.

This means that the poles and zeros must occur in pairs and the relationship

${\displaystyle x_{pi}x_{zi}=\xi \,}$

have to suffice. Odd orders thus have a zero at and a pole at infinity. ${\displaystyle x=0}$

swell

• Elliptic rational functions on MathWorld (Engl.)

literature

• Max Koecher, Aloys Krieg: Elliptical functions and modular forms . 2nd Edition. Springer, 2007, ISBN 978-3-540-49324-2 .