A function of one variable is said to be algebraic if it satisfies a polynomial equation whose coefficients are polynomials in the same variable.
An algebraic function is a function $f(x)$ which satisfies $p(x,f(x))=0$, where $p(x,y)$ is a polynomial in $x$ and $y$ with integer coefficients.
The only difference (I think) is that allowing arbitrary real coefficients enables the use of transcendental constants. Since adding these constants doesn't add much complexity (it's the functions that are the interesting part), I'm inclined to allow them. What's the consensus here? Can an algebraic function contain transcendental constants?