Elementary functions are the functions of one complex variable obtained from the identity function , constant functions, the (possibly fractional) power function?s , trigonometric function?s, the exponential functions and logarithmic function?s 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 has a branch point at the origin).
Notice that trigonometric functions (e.g., ) and inverse trigonometric trigometric functions (e.g., ) 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.