what is Transcendental function?
Functions which are not algebraic are called transcendental. Some authors prefer to restrict this classification to “analytic” functions, which are locally representable by power series. The terminology is inherited from the shaky early times of calculus when people could not imagine functions other than analytic (except for singularities). This illusion was shattered initially by Fourier (his trigonometric series violated the belief in analytic continuation as the only way of extending a function), and later substantially by Weierstrass when he introduced continuous functions which are nowhere differentiable, and later yet by discoveries in functional analysis, when these supposed “monster functions” were shown to form an overwhelming majority, and analytic functions turned out to be extremely rare (but no less interesting or useful).
