Elementary function is a function generated by power function, exponential function, logarithmic function, trigonometric function, inverse trigonometric function and constant through finite rational operation (addition, subtraction, multiplication, division, rational power and rational root) and finite function synthesis, which can be expressed by an analytical expression. So far, I haven't heard of the concept of higher order function.

Polynomials with real coefficients are called integer rational functions. The simplest is the linear function y=α0+α 1x, and its image is a straight line with a slope of α 1, passing through the point y=α0 on the Y axis. The image of quadratic integral rational function y=α0+α 1x+α2x2 is a parabola.

Complex trigonometric functions:

For example, if the variable x in y=sinx and y=cosx is changed to the complex variable z, then the complex trigonometric functions w=sinz and w=cosz are obtained, which is the whole function. Tanz=sinz/cosz and cotz=cosz/sinz are meromorphic functions of z.

They have many properties similar to real trigonometric functions: periodicity, wechat quotient, trigonometric identity and so on. But |sinz|≤ 1, |cosz|≤ 1 does not hold for any z, trigonometric function is closely related to exponential function, so it is very convenient to apply. The univalent region of sinz maps Gk univalent to the region obtained by removing the line segments on the real axis [- 1, 1] and the negative imaginary axis on the whole plane.