Algebraic function
= Algebraic function
{title2=$P(z,f(z))=0$}
An <algebraic function> is a <function> whose values satisfy a nonzero <polynomial> relation $P(z,f(z))=0$. In <complex analysis> one usually studies a local <holomorphic> branch of this relation; the relation itself need not specify one globally single-valued <function>.
For example, $w^2-z=0$ gives two <square root> branches away from zero. On a slit <domain>, one can choose a <holomorphic> branch, but following it around zero exchanges the two values. This is why <Riemann surfaces> arise naturally when studying an <algebraic function>. Every <rational function> $f=p/q$ is an <algebraic function>, since $q(z)f(z)-p(z)=0$ wherever $q\ne0$. In contrast, the <exponential function> is a <transcendental function>.