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:
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).
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.
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
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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