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 can be expressed using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power: are typical examples. However, some algebraic functions cannot be expressed by such finite expressions, as it is for example the case of the function defined by . In more precise terms, an algebraic function of degree
n in one variable
x is a function that satisfies a polynomial equation where the coefficients
ai are polynomial functions of
x, with coefficients belonging to a set
S. Quite often,, and one then talks about "function algebraic over ", and the evaluation at a given rational value of such an algebraic function gives an algebraic number. A function which is not algebraic is called a transcendental function, as it is for example the case of. A composition of transcendental functions can give an algebraic function:.