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:K→F (or
even Kn→F) 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
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