Monday, 30 September 2013

Rational Functions and its geometric Notions

A function f(x) is called a rational function if and only if it can be written in the form
 f(x) = \frac{P(x)}{Q(x)}
where P\, and Q\, are polynomials in x\, and Q\, is not the zero polynomial. The domain of f\, is the set of all points x\, for which the denominator Q(x)\, is not zero, assuming \textstyle P and \textstyle Q have no common factors.

In abstract algebra the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any field. In this setting given a field F and some indeterminate X, a rational expression is any element of the field of fractions of the polynomial ring F[X]. Any rational expression can be written as the quotient of two polynomials P/Q with Q ≠ 0, although this representation isn't unique. P/Q is equivalent to R/S, for polynomials P, Q, R, and S, when PS = QR. However since F[X] is a unique factorization domain, there is a unique representation for any rational expression P/Q with P and Q polynomials of lowest degree and Q chosen to be monic. This is similar to how a fraction of integers can always be written uniquely in lowest terms by canceling out common factors.
The field of rational expressions is denoted F(X). This field is said to be generated (as a field) over F by (a transcendental element) X, because F(X) does not contain any proper subfield containing both F and the element X.

Complex rational functions

In complex analysis, a rational function
f(z) = \frac{P(z)}{Q(z)}
is the ratio of two polynomials with complex coefficients, where Q is not the zero polynomial and P and Q have no common factor (this avoids f taking the indeterminate value 0/0). The domain and range of f are usually taken to be the Riemann sphere, which avoids any need for special treatment at the poles of the function (where Q(z) is 0).
The degree of a rational function is the maximum of the degrees of its constituent polynomials P and Q. If the degree of f is d, then the equation
f(z) = w \,
has d distinct solutions in z except for certain values of w, called critical values, where two or more solutions coincide. The function f can therefore be thought of as a d-fold covering of the w-sphere by the z-sphere.
Rational functions with degree 1 are called Möbius transformations and form the automorphisms group of the Riemann sphere. Rational functions are representative examples of meromorphic functions.

Notion of a rational function on an algebraic variety

Like polynomials, rational expressions can also be generalized to n indeterminates X1,..., Xn, by taking the field of fractions of F[X1,..., Xn], which is denoted by F(X1,..., Xn).
An extended version of the abstract idea of rational function is used in algebraic geometry. There the function field of an algebraic variety V is formed as the field of fractions of the coordinate ring of V (more accurately said, of a Zariski-dense affine open set in V). Its elements f are considered as regular functions in the sense of algebraic geometry on non-empty open sets U, and also may be seen as morphisms to the projective line.

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