Transcendental function

A transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. (The polynomials sometimes must have rational coefficients.) In other words, a transcendental function "transcends" algebra in that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, and root extraction.

Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions.

Formally, an analytic function ƒ(z) of one real or complex variable z is transcendental if it is algebraically independent of that variable. This can be extended to functions of several variables.

The transcendental functions sine and cosine were tabulated from physical measurements in antiquity, as evidenced in Greece (Hipparchus) and India (jya and koti-jya). A revolutionary understanding of these circular functions occurred in the 17th century and was explicated by Leonard Euler in 1748 in his Introduction to the Analysis of the Infinite. These ancient transcendental functions became known as continuous functions through quadrature of the rectangular hyperbola xy = 1 by Gregoire de Saint Vincent in 1647, two millennia after Archimedes had produced The Quadrature of the Parabola.

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