Definitions

Transcendental function

A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation. In other words a transcendental function is a function which "transcends" algebra in the sense that it cannot be expressed in terms 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.

Algebraic and transcendental functions

The logarithm and the exponential function are examples of transcendental functions. Transcendental function is a term often used to describe the trigonometric functions, i.e., sine, cosine, tangent, cotangent, secant, and cosecant, also.

A function that is not transcendental is said to be algebraic. Examples of algebraic functions are rational functions and the square root function.

The operation of taking the indefinite integral of an algebraic function is a source of transcendental functions. For example, the logarithm function arose from the reciprocal function in an effort to find the area of a hyperbolic sector. Thus the hyperbolic angle and the hyperbolic functions sinh, cosh, and tanh are all transcendental.

In differential algebra one studies how integration frequently creates functions algebraically independent of some class taken as 'standard', such as when one takes polynomials with trigonometric functions as variables.

Dimensional analysis

In dimensional analysis, transcendental functions are notable because they make sense only when their argument is dimensionless (possibly after algebraic reduction). Because of this, transcendental functions can be an easy-to-spot source of dimensional errors. For example, log(10 m) is a nonsensical expression (unlike, e.g. log(x meters/y meters) or log(10) m). One could attempt to apply a logarithmic identity to get log(10) + log(m), which highlights the problem: applying a non-algebraic operation to a dimension creates meaningless results.

Some Examples

$f\left(x\right) = c^x$,

$f\left(x\right)=x^x$,

$f\left(x\right)=x^\left\{x^\left\{-1\right\}\right\}.$