Algebraic function ($P(z,f(z))=0$)

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 $q\ne 0$. In contrast, the exponential function is a transcendental function.