Elliptic Rational Functions

In mathematics the elliptic rational functions are a sequence of rational functions with real coefficients. Elliptic rational functions are extensively used in the design of elliptic electronic filters. (These functions are sometimes called Chebyshev rational functions, not to be confused with certain other functions of the same name).

Rational elliptic functions are identified by a positive integer order n and include a parameter ξ ≥ 1 called the selectivity factor. A rational elliptic function of degree n in x with selectivity factor ξ is generally defined as:

  • cd is the Jacobi elliptic cosine function.
  • K is a complete elliptic integral of the first kind.
  • is the discrimination factor, equal to the minimum value of the magnitude of for .

For many cases, in particular for orders of the form n = 2a3b where a and b are integers, the elliptic rational functions can be expressed using algebraic functions alone. Elliptic rational functions are closely related to the Chebyshev polynomials: Just as the circular trigonometric functions are special cases of the Jacobi elliptic functions, so the Chebyshev polynomials are special cases of the elliptic rational functions.

Read more about Elliptic Rational Functions:  Expression As A Ratio of Polynomials, Particular Values

Famous quotes containing the words rational and/or functions:

    It is not to be forgotten that what we call rational grounds for our beliefs are often extremely irrational attempts to justify our instincts.
    Thomas Henry Huxley (1825–95)

    Adolescents, for all their self-involvement, are emerging from the self-centeredness of childhood. Their perception of other people has more depth. They are better equipped at appreciating others’ reasons for action, or the basis of others’ emotions. But this maturity functions in a piecemeal fashion. They show more understanding of their friends, but not of their teachers.
    Terri Apter (20th century)