Algebraic function

In mathematics, an algebraic function is a function that satisfies a polynomial equation. Thus an equation of the following form holds: a n ( x ) f ( x ) n + a n − 1 ( x ) f ( x ) n − 1 + ⋯ + a 1 ( x ) f ( x ) + a 0 ( x ) = 0 {\displaystyle a_{n}(x)f(x)^{n}+a_{n-1}(x)f(x)^{n-1}+\cdots +a_{1}(x)f(x)+a_{0}(x)=0} where the a k ( x ) {\displaystyle a_{k}(x)} are polynomials (not all zero). Basic examples of algebraic functions are polynomial functions, rational functions, the nth root function, and functions obtained from these by composition and algebraic operations (addition, multiplication, subtraction, and division). Thus an example of an algebraic function is the function f ( x ) = 1 − x 2 {\displaystyle f(x)={\sqrt {1-x^{2}}}} (for − 1 < x < 1 {\displaystyle -1<x<1} ), whose graph is the top half of the standard unit circle. This function satisfies x 2 + f ( x ) 2 − 1 = 0 {\displaystyle x^{2}+f(x)^{2}-1=0} . Algebraic functions are contrasted with transcendental functions, such as the exponential function, logarithm, and the trigonometric functions. Algebraic functions are usually treated more generally as multivalued functions. The example of x 2 + y 2 − 1 = 0 {\displaystyle x^{2}+y^{2}-1=0} illustrates this, since it includes both the top semicircle y = 1 − x 2 {\displaystyle y={\sqrt {1-x^{2}}}} and bottom semicircle y = − 1 − x 2 {\displaystyle y=-{\sqrt {1-x^{2}}}} in the same package. Algebraic functions are most often studied over the complex numbers. Formally, an algebraic function over the complex numbers is defined to be a multivalued function y {\displaystyle y} satisfying a polynomial equation P ( x , y ) = 0 {\displaystyle P(x,y)=0} where P ( x , y ) {\displaystyle P(x,y)}...