Fonction transcendante

In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed algebraically. Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions. Formally, an analytic function f (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). In describing Ptolemy's table of chords, an equivalent to a table of sines, Olaf Pedersen wrote: The mathematical notion of continuity as an explicit concept is unknown to Ptolemy. That he, in fact, treats these functions as continuous appears from his unspoken presumption that it is possible to determine a value of the dependent variable corresponding to any value of the independent variable by the simple process of linear interpolation. A revolutionary understanding of these circular functions occurred in the 17th century and was explicated by Leonhard 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 Grégoire de Saint-Vincent in 1647, two millennia after Archimedes had produced The Quadrature of the Parabola. The area under the hyperbola was shown to have the scaling property of constant area for a constant ratio of bounds. The hyperbolic logarithm function so described was of limited service until 1748 when Leonhard Euler related it to functions where a constant is raised to a variable exponent, such as the exponential function where the constant base is e.