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Entire Function (Integral Function): Simple Definition

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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:

  • ez
  • zn
  • sin(z)

Many of the simpler entire functions behave in a similar way, dynamically speaking, to polynomial functions. These include λ ez 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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Stephanie Glen. "Entire Function (Integral Function): Simple Definition" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/entire-function-integral/
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