## What is an Entire Function?

An **entire 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.

The entire function is sometimes called the *integral function*. However, this can get a bit confusing because the term “integral function” also refers to an antiderivative for any function.

## 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.

## What is the Exponential Integral Function?

The **exponential integral function** is a special function used in astrophysics, quantum chemistry and many other applied sciences.

It can be defined in two different ways: as a real-valued function or as a complex-valued function.

## 1. Real-Valued Exponential Integral Function

The exponential integral function is defined as a definite integral, with a ratio of the exponential function and its argument:

The function is defined for x in the set of natural numbers. This set excludes zero.

Or, equivalently the function can be defined a little differently through a parity transformation. A *parity transformation* is where the signs are flipped: t→ -t and x→ -x. This gives what Enrico Masina calls a “more suitable” definition:

Note though, that despite the stated suitability of the above form, most authors use the notation E_{1} *only* for the complex-valued version of the function.

## 2. Exponential Integral Function Defined on the Complex Plane

On the complex plane, the function E_{1}is also a definite integral.

The notation is almost the same, with a couple of notable differences:

- The substitution of “z” (to denote a complex number) instead of “x”.
- While the “real” version of the function above is defined for the set of whole numbers, the complex-numbered version is defined for |arg(z) < π)|.*

*This comes from the exponential form of a complex number, z = |z|e^{iθ}, which is just z = x + i y rewritten with exponentials; θ = arg(z) = arctan(y/x).

The complex valued version is not valid when z = 0 or z = ∞, because of a specific kind of discontinuity called a branch point.

## 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.

Masina, E. (2017). A review on the Exponential-Integral special function and other strictly related special functions. Lectures from a seminar of Mathematical Physics. Retrieved December 10, 2019 from: https://www.researchgate.net/publication/323772322_A_review_on_the_Exponential-Integral_special_function_and_other_strictly_related_special_functions_Lectures_from_a_seminar_of_Mathematical_Physics

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