Algebraic functions

Algebraic functions are usually treated more generally as multivalued functions. The example of ${\displaystyle x^{2}+y^{2}-1=0}$ illustrates this, since it includes both the top semicircle ${\displaystyle y={\sqrt {1-x^{2}}}}$ and bottom semicircle ${\displaystyle y=-{\sqrt {1-x^{2}}}}$ 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 ${\displaystyle y}$ satisfying a polynomial equation $${\displaystyle P(x,y)=0}$$ where ${\displaystyle P(x,y)}$ is an irreducible polynomial of two variables, having positive degree in ${\displaystyle y}$ and complex coefficients. The example of ${\displaystyle x^{2}+y^{2}-1=0}$ can be expressed as having the two single-valued branches, ${\displaystyle y={\sqrt {1-x^{2}}}}$ and ${\displaystyle y=-{\sqrt {1-x^{2}}}}$, with branch points where the two branches come together, at ${\displaystyle x=\pm 1}$. This particular function can be written using finitely many algebraic operations and extraction of nth roots, but this is not generally the case such as with the Bring radical. Over the complex numbers, algebraic functions have local holomorphic branches away from finitely many branch points and poles, and are naturally studied as meromorphic functions on compact Riemann surfaces.