Definition:Entire Function

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:
 * $$f:\C\to\C; f(z) = \sum_{j=0}^{\infty} a_n z^n; \quad \lim_{j\to\infty} \sqrt[j]{|a_j|}=0.$$

Transcendental entire functions
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 if and only if $$f$$ is not a polynomial.