Rational functions

Rational functions are mathematical expressions formed by the ratio of two polynomials. In more formal terms, a rational function \( R(x) \) can be expressed as: \[ R(x) = \frac{P(x)}{Q(x)} \] where \( P(x) \) and \( Q(x) \) are polynomial functions, and \( Q(x) \neq 0 \) (the denominator cannot be zero).

Partial fractions is a mathematical technique used to decompose a rational function into a sum of simpler fractions, called partial fractions. This method is particularly useful in algebra, calculus, and differential equations, as it simplifies the process of integrating rational functions. A rational function is typically expressed as the ratio of two polynomials, say \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.

Chebyshev rational functions are specific types of rational functions that are associated with Chebyshev polynomials, which are a sequence of orthogonal polynomials that arise in various areas of numerical analysis, approximation theory, and many applications in engineering and mathematics.

Elliptic rational functions are mathematical functions that arise in the study of elliptic curves and, more generally, in the theory of elliptic functions. They can be thought of as generalizations of rational functions that incorporate properties of elliptic functions. To understand elliptic rational functions, it's helpful to break down the components of the term: 1. **Elliptic Functions:** These are meromorphic functions that are periodic in two directions (often associated with the complex plane's lattice structure).

The Hartogs–Rosenthal theorem is a result in the field of functional analysis, particularly dealing with Banach spaces. It describes a certain property of bounded linear operators between infinite-dimensional Banach spaces.

Legendre rational functions are a family of rational functions constructed from Legendre polynomials, which are orthogonal polynomials defined on the interval \([-1, 1]\). These functions are used in various areas of mathematics, including numerical analysis and approximation theory.

A linear fractional transformation (LFT), also known as a Möbius transformation, is a function that maps the complex plane to itself. It is defined by the formula: \[ f(z) = \frac{az + b}{cz + d} \] where \(a\), \(b\), \(c\), and \(d\) are complex numbers, and \(ad - bc \neq 0\) to ensure that the transformation is well-defined and non-degenerate.

A Padé approximant is a type of rational function used to approximate a given function, typically a power series. It is defined as the ratio of two polynomials, \( P(x) \) and \( Q(x) \), where \( P(x) \) is of degree \( m \) and \( Q(x) \) is of degree \( n \).

The polylogarithm is a special mathematical function denoted as \(\text{Li}_s(z)\), which generalizes the concept of logarithms to allow for the exponentiation of complex variables.

A rational function is a type of mathematical function that can be expressed as the ratio of two polynomial functions. Specifically, a rational function can be written in the form: \[ R(x) = \frac{P(x)}{Q(x)} \] where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not equal to zero.