nLab elementary function

Elementary functions are the functions of one complex variable obtained from the identity function xx, constant functions, the (possibly fractional) power functions x γx^\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 1/2x^{1/2} has a branch point at the origin).

Notice that trigonometric functions (e.g., cosx=12(e ix+e ix)\cos x = \frac1{2}(e^{i x} + e^{-i x})) and inverse trigonometric trigometric functions (e.g., arctan(x)=i2(log(1ix)log(1+ix))\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.

Last revised on June 6, 2023 at 17:48:43. See the history of this page for a list of all contributions to it.