Entire rational function

(algebraic) polynomial

A function of the form

$$ w = P _ {n} ( z) = \ a _ {0} z ^ {n} + a _ {1} z ^ {n - 1 } + \dots + a _ {n} , $$

where $ n $ is a non-negative integer, the coefficients $ a _ {0} \dots a _ {n} $ are real or complex numbers, and $ z $ is a real or complex variable. If $ a _ {0} \neq 0 $, then $ n $ is called the degree of the polynomial; the polynomial $ P ( z) \equiv 0 $ does not have a degree. The simplest non-constant entire rational function is the linear function

$$ w = az + b,\ \ a \neq 0. $$

An entire rational function is analytic in the whole plane, that is, it is an entire function of the complex variable $ z $, and $ \infty $ is a pole of order $ n $ for $ P _ {n} ( z) $. ( $ P _ {n} ( z) \rightarrow \infty $ for $ n \geq 1 $, as $ z \rightarrow \infty $; conversely, if $ f ( z) $ is an entire function and $ f ( z) \rightarrow \infty $ as $ z \rightarrow \infty $, then $ f ( z) $ is an entire rational function.) Polynomials in several real or complex variables also play an important role in mathematical analysis. Entire rational functions are used for the approximate representation of more complicated functions because they are most convenient for computations.

See also Polynomial.

References

[1]	I.I. Privalov, "Introduction to the theory of functions of a complex variable" , Moscow (1977) (In Russian)

Comments

In non-Soviet literature the phrase "entire rational function" is not used.

References

[a1]	L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 24–26
[a2]	A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)