Editing Transcendental function (section)

==Algebraic and transcendental functions==
{{details|Elementary function (differential algebra)}}

The most familiar transcendental functions are the [[logarithm]], the [[exponential function|exponential]] (with any non-trivial base), the [[trigonometric function|trigonometric]], and the [[hyperbolic functions]], and the [[inverse function|inverses]] of all of these. Less familiar are the [[special functions]] of [[mathematical analysis|analysis]], such as the [[gamma function|gamma]], [[elliptic function|elliptic]], and [[zeta function]]s, all of which are transcendental. The [[generalized hypergeometric function|generalized hypergeometric]] and [[Bessel function|Bessel]] functions are transcendental in general, but algebraic for some special parameter values.

Transcendental functions cannot be defined using only the operations of addition, subtraction, multiplication, division, and <math>n</math>th roots (where <math>n</math> is any integer), without using some "limiting process".

A function that is not transcendental is '''algebraic'''. Simple examples of algebraic functions are the [[rational functions]] and the [[square root]] function, but in general, algebraic functions cannot be defined as finite formulas of the elementary functions, as shown by the example above with <math>f(x)^5+f(x)=x</math> (see [[Abel–Ruffini theorem]]).

The [[indefinite integral]] of many algebraic functions is transcendental. For example, the integral <math>\int_{t=1}^x\frac{1}{t}dt</math> turns out to equal the logarithm function <math>log_e(x)</math>. Similarly, the limit or the infinite sum of many algebraic function sequences is transcendental. For example, <math>\lim_{n\to \infty}(1+x/n)^n</math> converges to the exponential function <math>e^x</math>, and the infinite sum <math>\sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}</math> turns out to equal the hyperbolic cosine function <math>\cosh x</math>. In fact, it is ''impossible'' to define any transcendental function in terms of algebraic functions without using some such "limiting procedure" (integrals, sequential limits, and infinite sums are just a few).

[[Differential algebra]] examines how integration frequently creates functions that are algebraically independent of some class, such as when one takes polynomials with trigonometric functions as variables.