Definition:Entire Function
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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:
- $\displaystyle f: \C \to \C: f(z) = \sum_{j \mathop = 0}^{\infty} a_n z^n; \quad \lim_{j \to \infty} \sqrt[j]{|a_j|}=0$
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 if and only if $f$ is not a polynomial function.
