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In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction, multiplication, and division (without the need of taking limits).
A transcendental equation need not be an equation between elementary functions, although most published examples are. In some cases, a transcendental equation can be solved by transforming it into an equivalent algebraic equation. Some such transformations are sketched below; computer algebra systems may provide more elaborated transformations. [a]
The name "transcendental" comes from Latin trānscendere ' to climb over or beyond, surmount ', [7] and was first used for the mathematical concept in Leibniz's 1682 paper in which he proved that sin x is not an algebraic function of x. [8] Euler, in the eighteenth century, was probably the first person to define transcendental numbers in the ...
In mathematics, the Lerch transcendent, is a special function that generalizes the Hurwitz zeta function and the polylogarithm.It is named after Czech mathematician Mathias Lerch, who published a paper about a similar function in 1887. [1]
Transcendental functions are functions that are not algebraic. Exponential function: raises a fixed number to a variable power. Hyperbolic functions: formally similar to the trigonometric functions. Inverse hyperbolic functions: inverses of the hyperbolic functions, analogous to the inverse circular functions.
The notation convention chosen here (with W 0 and W −1) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth. [3]The name "product logarithm" can be understood as follows: since the inverse function of f(w) = e w is termed the logarithm, it makes sense to call the inverse "function" of the product we w the "product logarithm".
Transcendental function, a function which does not satisfy a polynomial equation whose coefficients are themselves polynomials; Transcendental number theory, the branch of mathematics dealing with transcendental numbers and algebraic independence
A composition of transcendental functions can give an algebraic function: = =. As a polynomial equation of degree n has up to n roots (and exactly n roots over an algebraically closed field , such as the complex numbers ), a polynomial equation does not implicitly define a single function, but up to n functions, sometimes also called ...