nLab elementary function

Elementary functions are the functions of one complex variable obtained from the identity function $x$, constant functions, the (possibly fractional) power functions $x^\gamma$, trigonometric functions, the exponential functions and logarithmic functions by the algebraic operations (addition, subtraction, multiplication, division) and composition. Elementary functions form a well-defined class. Some of them are multivalued and have singular points (e.g. the square root $x^{1/2}$ has a branch point at the origin).

Notice that trigonometric functions (e.g., $\cos x = \frac1{2}(e^{i x} + e^{-i x})$) and inverse trigonometric trigometric functions (e.g., $\arctan (x) = \frac{i}{2}(\log (1 - i x) - \log (1 + i x))$) are elementary. Were we to consider functions of a real variable generated in like manner from polynomials, the exponential function, and the logarithmic function, this would no longer be the case.

Functions of a complex variable which are not elementary are sometimes called transcendental, although usually this term is used for the (strictly larger) class of functions that are not algebraic.

To be included: decidability results of Richardson, etc.