**1)**
**Definition of the rational function**

**2)**
** Möbius**
(or **homographic** or
**fractional linear**)
**transformations**

**3)**
**Fractional-linear function **

**4)**
**Zhukovskii
function **

**Definition
of the rational function**

In complex analysis, a
rational function
is the ratio of two polynomials
and
with complex coefficients:

,
**(1)**

where
0, and
and
have no common factor. The coefficients
of
and
polynomials are called the coefficients
of the rational function. The function
is called irreducible when
and
have no common zeros. Every rational
function can be written as an irreducible
fraction **(1)**.

If
has degree **m**
and
has degree **n**,
then the degree of
is the number **N = max{***m*,*
n*}. Some literature offers
the degree of
is the pair **(***m*,*
n*).

A rational function of
degree **(***m*,* n*)
with *n* = 0
is a polynomial. It is also called an
**entire rational function**.
Otherwise it is called a **fractional-rational
function**.

When *m *<*n*,
the fraction
is called proper. It is called improper
when **m ****>****n**.
An improper fraction can be uniquely
written as

,
**(2)**

where
is a polynomial, called the integral
part of the fraction ,
and
is a proper fraction.

**
Möbius**
(or **homographic** or
**fractional linear**)
**transformations**

Rational functions with
degree 1 are called **Möbius**
(or **homographic** or
**fractional linear**)
**transformations**.

,
**(3)**

satisfying *ad
- bc* **0**.

**Top**

**
Fractional-linear function **

Fractional-linear function
is a function of the type

,

where
are complex or real variables,

are complex or real coefficients, and

.

If ,

the fractional-linear
function is an integral-linear function.

*L*(*z*)
is a constant, if the rank of the matrix

is equal to one.

A proper fractional-linear
function **(3)** is obtained
if the rank of *A*
is equal to two, and if

.

**Top**

**
Zhukovskii function **

Zhukovskii function is
the rational function

,
where * z
*is complex.

It was discovered by Russian
scientist N. E. Zhukovskii (often spelled
"Joukowski"). Zhukovskii function
is important for applications in fluid
mechanics, particularly in constructing
and studying the Kutta ꨵkovskii
airfoils (aerofoils).

by
Tetyana Butler