Algebraic function

In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are: In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are: Some algebraic functions, however, cannot be expressed by such finite expressions (this is the Abel–Ruffini theorem). This is the case, for example, for the Bring radical, which is the function implicitly defined by In more precise terms, an algebraic function of degree n in one variable x is a function y = f ( x ) , {displaystyle y=f(x),} that is continuous in its domain and satisfies a polynomial equation where the coefficients ai(x) are polynomial functions of x, with integer coefficients. The value of an algebraic function at a rational number, and more generally, at an algebraic number is always an algebraic number .Sometimes, coefficients a i ( x ) {displaystyle a_{i}(x)} that are polynomial over a ring R are considered, and one then talks about 'functions algebraic over R'. A function which is not algebraic is called a transcendental function, as it is for example the case of exp ⁡ ( x ) , tan ⁡ ( x ) , ln ⁡ ( x ) , Γ ( x ) {displaystyle exp(x), an(x),ln(x),Gamma (x)} . A composition of transcendental functions can give an algebraic function: f ( x ) = cos ⁡ ( arcsin ⁡ ( x ) ) = 1 − x 2 {displaystyle f(x)=cos(arcsin(x))={sqrt {1-x^{2}}}} . As an equation of degree n has n roots, a polynomial equation does not implicitly define a single function, but nfunctions, sometimes also called branches. Consider for example the equation of the unit circle: y 2 + x 2 = 1. {displaystyle y^{2}+x^{2}=1.,} This determines y, except only up to an overall sign; accordingly, it has two branches: y = ± 1 − x 2 . {displaystyle y=pm {sqrt {1-x^{2}}}.,} An algebraic function in m variables is similarly defined as a function y = f ( x 1 , … , x m ) {displaystyle y=f(x_{1},dots ,x_{m})} which solves a polynomial equation in m + 1 variables: It is normally assumed that p should be an irreducible polynomial. The existence of an algebraic function is then guaranteed by the implicit function theorem. Formally, an algebraic function in m variables over the field K is an element of the algebraic closure of the field of rational functions K(x1,...,xm).