Definition:Entire Function

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

Transcendental Entire Function

Let $f$ be an entire function that has an essential singularity at $\infty$.

Then $f$ is a transcendental entire function.

Also see

Historical Note

The concept of an entire function was of particular interest to Karl Weierstrass during his work on rebuilding the discipline of complex analysis.