Transcendental function

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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.

[edit] 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 function(s), 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 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.

[edit] 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.

[edit] Use in calculus

Logarithms can be used in aiding with differentiation and integration. The derivative of the expression ln(x) is 1/x. This definition can be used to find the derivative of the function y = e^x. Taking the natural log of both sides of the equation results in the equation ln(y) = ln(e^x), which can then be simplified to ln(y) = x ln(e), and since ln(e) = 1, the equation ln(y) = x is left. The derivative of this function is (dy/dx)/y = 1, simplified to dy/dx = y, therefore dy/dx = e^x, and it can be concluded that the derivative of e^x is e^x.