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.

The logarithm and the exponential function are examples of transcendental functions.

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 a function is a prolific source of transcendental functions, in the way that the logarithm function arises from the reciprocal function. In differential algebra it is studied how integration frequently creates functions algebraically independent of some class taken as 'standard', such as it created by taking polynomials with trigonometric functions.