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These are functions which cannot be represented in the form:

Pn * yn + ... + P1 * y + P0 = 0

Where the P's are polynomials in x with rational coefficients. For example: The function

y = 1/(√(x+1))

is not transcendental because it can be split up into seperate polynomials:

P2 = x + 1, P1 = 0, and P0 = -1

Common tanscendental functions include the basic trigonometric functions such as sine and cosine, their inverses, exponential, and logarithmic functions.

Wolfram Research, the company which produces Mathematica, a popular and powerful tool for learning and "doing" mathematics on a Personal Computer (PC), lists the following mathematical functions as "Elementary Transcendental Functions":

The transcendental functions are functions that let you leave the algebraic world. That is, let K be a field (think of Q, the rational numbers) embedded inside a much larger field F such that F/K is not an algebraic field extension (think of R, the real numbers). A function f:KF (or even KnF) is called transcendental if there exist values of x for which f(x) is not algebraic over K.

In fact, it's probably more convenient to introduce a topology and consider only (piecewise) continuous functions; those are the only ones considered in practice. Even better is to consider the field of rational functions K(x), and to define a function as transcendental if it is not algebraic over K(x). This means that values f(x) of an algebraic (non-transcendental) function will satisfy the same algebraic equation (involving also x) for all values of x. The end result is the same for practical purposes, though.

For instance, the square root function sqrt is algebraic and not transcendental: the square root of any number is algebraic, as it satisfies (duh!)

sqrt(x)2 = x

But the exponential function exp is transcendental. For instance, e=exp(1) is not an algebraic number. Sine is also transcendental, a consequence of the same proof that we cannot trisect an angle with compass and straightedge.

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