An **entire function** (also called an *integral function*) is analytic on the entire complex plane. It’s called an “entire” function because of this very fact.

This simple definition leads to a big problem when dealing with entire functions: **The space of (set of all) entire functions is huge; **So huge in fact, that it’s usually necessary to work with smaller families of maps to ensure strong results. A whole subset of complex analysis, called *entire function theory*, is devoted to the study of these useful functions.

## Examples of Entire Functions

Some of the simplest entire functions are the exponential functions, polynomial functions (as long as the functions are complex-valued), and any finite compositions, products or sums of those two types.

A few specific examples of entire functions:

- e
^{z} - z
^{n} - sin(z)

Many of the simpler entire functions behave in a similar way, dynamically speaking, to polynomial functions. These include λ e^{z} and acos z + b (Eremenko & Lyubich, 1992).

The natural logarithm function and the square root function are not analytic across the entire complex plane, so they are *not* entire functions.

## Special Classes of Entire Functions

**Speiser class**(*S*) only have a finite number of singular values.**Eremenko-Lyubich class functions of bounded type**(*B*) are where all singular values are contained in a bounded set in ℂ.

## References

Eremenk, A. & Lyubich, M. (1992). Dynamical properties of some classes of entire functions. Retrieved December 8, 2019 from: http://www.math.stonybrook.edu/~bishop/classes/math627.S13/EL-Fourier.pdf

Gardner, R. Zeros of an Analytic Function. Retrieved December 9, 2019 from: https://faculty.etsu.edu/gardnerr/5510/notes/IV-3.pdf

Knopp, K. (1996). Entire Transcendental Functions. Ch. 9 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 112-116, 1996.

Orloff, J. Analytic Functions. Retrieved December 8, 2019 from: https://math.mit.edu/~jorloff/18.04/notes/topic2.pdf

Schleicher,, D. Dynamics of Entire Functions. Retrieved December 8, 2019 from: http://www.math.stonybrook.edu/~bishop/classes/math627.S13/DynamicsEntireOverview.pdf

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