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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.

For every real number $a>0$ and nonnegative integer $m$, $x^m{e}^{-ax}\to 0$ as $x\to \infty$. Thus every fixed polynomial grows more slowly in magnitude than a positive-rate exponential function, regardless of the polynomial's coefficients.

A short proof comes from the power series of the exponential function. For $x>0$, $e^{ax}\ge (ax{)}^{m+1}/(m+1)!$, soApply this estimate term by term to a polynomial. For example, $x^{100}/e^x$ eventually tends to zero even though it can be extremely large at moderately sized $x$. This is an asymptotic statement, not a claim that the exponential function is larger at every positive input. It explains why exponential tails have finite moments of every fixed order and why the proof that the exponential function is a transcendental function works.

A Painlevé transcendent is a special function obtained as a solution of one of the six nonlinear second-order Painlevé equations. The family is organized by the differential equation, its parameters and its initial or asymptotic data. In general these solutions supply new transcendental functions, rather than combinations of classical special functions.^[ref]

The name of the family does not imply that every exceptional solution is a transcendental function. For the Painlevé II equation, $y=0$ at $\alpha =0$ and $y=-1/x$ at $\alpha =1$ are rational solutions. The Painlevé property concerns the absence of movable branch points, rather than the absence of movable poles; it describes how solutions behave under analytic continuation.

A transcendental function is an analytic function that is not an algebraic function. Explicitly, on a domain $U$, there is no nonzero polynomial $P(z,w)$ with complex coefficients for whichA rational function is algebraic, and a local branch of $\sqrt{z}$ is algebraic because it satisfies $w^2-z=0$. The exponential function is transcendental. This concerns a relation between functions, rather than the arithmetic nature of a single value: $e^0=1$ even though $z\mapsto e^z$ is transcendental.

Here is a direct proof for the exponential function. Suppose ${\sum }_{k=0}^m{p}_k(z)e^{kz}=0$ on $U$, with polynomials $p_k$ and $p_m\ne 0$. The expression is an entire function, so the identity theorem for holomorphic functions extends the identity to all complex $z$. For positive real $x$, divide by $e^{mx}$:By exponential domination of polynomials, the right side tends to zero as $x\to \infty$. A nonzero polynomial cannot tend to zero along the positive real axis, giving a contradiction. The case $m=0$ is already impossible because $p_0$ would vanish identically. Thus no such algebraic relation exists.