Transcendental function

A transcendental function is a function which does not satisfy a polynomial equation whose coefficients are themselves polynomials. Saying it more technically, a function of one variable is transcendental if it is algebraically independent of that variable.

Algebraic and transcendental functions

Further information: elementary function (differential algebra)

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 which 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. Because of this, transcendental functions can be an easy-to-spot source of dimensional errors. For example, log(10 m) is a nonsensical expression. One could attempt to apply a logarithm 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(x)=c^x,

f(x)=x^x,

f(x)=x^x^-1,