Definition:Entire Function

Definition
A holomorphic self-map $f$ of the complex plane is called an entire function.

Since holomorphic functions are analytic, this is the same as saying that $f$ is given by an everywhere convergent power series:
 * $\displaystyle f: \C \to \C: f \left({z}\right) = \sum_{j \mathop = 0}^\infty a_n z^n; \quad \lim_{j \to \infty} \sqrt [j] {\left|{a_j}\right|} = 0$

Transcendental Entire Function
If $f$ is an entire function that has an essential singularity at $\infty$, then $f$ is called a transcendental entire function.

In terms of the power series expansion of $f$, this is equivalent to infinitely many of the power series coefficients $a_j$ being nonzero.

That is, an entire function $f$ is transcendental iff $f$ is not a polynomial function.