# Entire function

An entire function is a complex-valued function that is holomorphic, meaning complex differentiable in a neighborhood of every point, on the whole complex plane; the term is also called an integral function.<sup>[1](https://maths.free/complex-analysis/entire-function/)</sup> Polynomials, the exponential function, sine, and cosine are entire, while rational functions with finite poles and trigonometric functions such as tangent, cotangent, secant, and cosecant are not.<sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup><sup> • </sup><sup>[15](https://complexanalysis.org/web/sec_trig-hyperbolics.html)</sup> Entire functions are quite rare in the space of all complex functions, and strong rigidity results can be proved for them.<sup>[3](https://web.njit.edu/~matveev/Courses/M656_S11/Complex1.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | Holomorphic on all of \( \mathbb C \); equivalently, representable by a power series \( f(z)=\sum_{n=0}^{\infty} a_n z^n \) converging everywhere, uniformly on compact sets<sup>[1](https://maths.free/complex-analysis/entire-function/)</sup> |
| Examples | Polynomials, \( e^z \), \( \sin z \), \( \cos z \); rational functions with finite poles are not entire<sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup> |
| Liouville's theorem | Every bounded entire function is constant<sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup> |
| Picard's little theorem | A nonconstant entire function omits at most one complex value; its range is \( \mathbb C \) or \( \mathbb C \) minus a single point<sup>[3](https://web.njit.edu/~matveev/Courses/M656_S11/Complex1.pdf)</sup> |
| Polynomial growth | If \( |f(z)| \le C(1+|z|)^m \), then \( f \) is a polynomial of degree at most \( m \)<sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup> |
| Factorization | Entire functions of finite order admit Hadamard's canonical product over their zeros<sup>[1](https://maths.free/complex-analysis/entire-function/)</sup> |
| Zero counting | Jensen's formula bounds the number of zeros in a disc by the growth of \( \log|f| \) on a larger circle<sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup> |

## Power series representation

Since the complex plane is simply connected and has no boundary, any entire function \( f(z) \) admits an expansion \( f(z)=\sum_{n=0}^{\infty} a_n z^n \), one power series valid at every point of the plane and therefore uniformly convergent on compact subsets.<sup>[1](https://maths.free/complex-analysis/entire-function/)</sup>

## Liouville's theorem and its consequences

**Boundedness forces constancy.** Liouville's theorem states that every bounded entire function is constant.<sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup> The proof applies Cauchy's estimate \( |f'(a)| \le M/R \) on a circle of radius \( R \) centered at \( a \), where \( M \) bounds \( |f| \); since \( R \) is arbitrary, letting \( R \to \infty \) yields \( f'(a)=0 \) everywhere.<sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup><sup> • </sup><sup>[3](https://web.njit.edu/~matveev/Courses/M656_S11/Complex1.pdf)</sup> The result is a rigidity statement: a bounded, non-constant entire function simply cannot exist.<sup>[4](https://maxschottenfeld.github.io/complex-analysis/lessons/05-consequences-of-cauchys-formula/)</sup>

The theorem explains a contrast with real analysis. The real sine and cosine are bounded, but \( \sin z \) and \( \cos z \) are not bounded in the complex setting, since they are entire and not constant.<sup>[5](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_10.html)</sup>

**Fundamental theorem of algebra.** Liouville's theorem gives a short proof that every nonconstant complex polynomial has a root: if a polynomial \( p \) had no zeros, then \( 1/p \) would be a bounded entire function, hence constant, forcing \( p \) itself to be constant.<sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup><sup> • </sup><sup>[3](https://web.njit.edu/~matveev/Courses/M656_S11/Complex1.pdf)</sup>

**Polynomial growth.** Repeating the circle-of-radius-\( R \) reasoning for the higher derivatives yields a classification by growth: whenever an entire function obeys the bound \( |f(z)| \le C(1+|z|)^m \), it must be a polynomial whose degree does not exceed \( m \), since Cauchy's estimates make every derivative of order beyond \( m \) equal to zero.<sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup><sup> • </sup><sup>[6](https://ondara.net/course/complex-analysis/subunit/u4-6-liouville-consequences)</sup> A related uniqueness fact follows directly from Liouville: if \( f(z)=O(1) \) as \( z \to \infty \) and \( f(z_n) \to 0 \) for some sequence \( z_n \to \infty \), then \( f \) is identically zero.<sup>[7](https://lib.ysu.am/open_books/417657.pdf)</sup>

## Picard's theorem

Picard's little theorem sharpens Liouville dramatically. It states that a nonconstant entire function omits at most one complex value; equivalently, the range of a nonconstant entire function is either the entire complex plane or the plane minus a single point.<sup>[3](https://web.njit.edu/~matveev/Courses/M656_S11/Complex1.pdf)</sup><sup> • </sup><sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup> Where Liouville requires the entire image to be bounded to force constancy, Picard shows that an entire function omitting two values is already constant.<sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup> A proof follows from the modular covering of \( \mathbb C \setminus \{0,1\} \) or from normal-family theory.<sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup> The exponential function illustrates the sharpness: \( e^z \) is nonconstant and omits exactly one value, 0.

## Growth and order

Growth at infinity classifies entire functions beyond polynomials. The order of growth measures how fast \( |f| \) increases with \( |z| \) and controls the complexity of zero factorizations.<sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup> For example, the order of growth of \( e^{z^2} \) is 2.<sup>[8](https://www.fing.edu.uy/~cerminar/Complex_Analysis.pdf)</sup> A function of order 1 and type \( \sigma \) is said to be of exponential type \( \sigma \).<sup>[1](https://maths.free/complex-analysis/entire-function/)</sup>

Growth bounds and Taylor coefficients control each other in both directions. Optimizing over the radius \( r \) translates a bound \( M(r) \le C e^{A r^{\rho}} \) into coefficient decay: choosing \( r=(n/A\rho)^{1/\rho} \) gives a factorial-type bound on \( a_n \), and conversely coefficient decay reconstructs growth through the power series.<sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup>

## Zeros and factorization

How quickly an entire function grows and where its zeros lie are the central concerns studied in the theory of entire functions.<sup>[7](https://lib.ysu.am/open_books/417657.pdf)</sup> Karl Weierstrass, whose Berlin lectures developed the power-series approach to the function concept, showed that uniform functions developable into convergent power series can be decomposed into a product of an infinite number of primary factors.<sup>[9](https://en.wikisource.org/wiki/A_History_of_Mathematics_(1893)/Recent_Times/Theory_of_Functions)</sup><sup> • </sup><sup>[10](https://mathshistory.st-andrews.ac.uk/Biographies/Weierstrass/)</sup>

**Hadamard factorization.** Entire functions of finite order have Hadamard's canonical representation

\[ f(z)=z^m e^{P(z)} \prod_{n=1}^{\infty}\left(1-\frac{z}{z_n}\right)\exp\left(\frac{z}{z_n}+\cdots+\frac{1}{p}\left(\frac{z}{z_n}\right)^p\right), \]

where the \( z_n \) are the nonzero roots of \( f \), \( m \) is the order of the zero at \( z=0 \) (with \( m=0 \) meaning \( f(0) \neq 0 \)), \( P \) is a polynomial, and \( p \) is the smallest non-negative integer such that \( \sum_{n=1}^{\infty} 1/|z_n|^{p+1} \) converges.<sup>[1](https://maths.free/complex-analysis/entire-function/)</sup> The exponential prefactor and the elementary factors correct for zeros so that the product converges while matching the function's growth.

**Jensen's formula.** For \( f \) holomorphic near \( \overline{D(0,R)} \) with \( f(0) \neq 0 \) and zeros \( a_k \) in the disc counted with multiplicity,

\[ \log|f(0)| + \sum_{|a_k|<R} \log\frac{R}{|a_k|} = \frac{1}{2\pi}\int_0^{2\pi} \log|f(Re^{it})|\,\mathrm{d}t. \]

In the proof, f is divided by finite Blaschke-type factors which cancel its zeros yet leave the modulus on the boundary intact up to R/|a_k|; for the zero-free holomorphic function thus obtained, log|f| is a harmonic function, and applying the mean-value property at the origin yields the identity.<sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup> Jensen's formula bounds the number of zeros in a smaller disc by growth on a larger circle, and it is the elementary precursor of [Nevanlinna theory](https://www.edgepedia.org/nevanlinna-theory).<sup>[2](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)</sup>

## Related classes and applications

Entire functions of exponential type appear in [Fourier analysis](https://www.edgepedia.org/fourier-analysis). The Paley-Wiener class \( W^p \) consists of entire functions of exponential type in \( \mathbb C^n \) belonging to \( L^p(\mathbb R^n) \) for \( 1 < p < \infty \), and analogs of the Paley-Wiener theorem characterize the structure of the [Fourier transform](https://www.edgepedia.org/fourier-transform) for such functions.<sup>[11](https://matem.anrb.ru/en/article?art_id=27)</sup> Characterizations also exist for when an entire function is the Fourier transform of some \( f \in L^2(\mu) \) for a singular measure \( \mu \).<sup>[12](https://arxiv.org/abs/1709.07522)</sup>

Liouville-type rigidity extends beyond the classical statement: for entire and subharmonic functions of finite order, boundedness outside a set of zero planar density already forces the Liouville conclusion.<sup>[13](https://arxiv.org/abs/2009.01019)</sup> Standard graduate treatments of the subject collect the central tools as infinite products, Jensen's formula, the Weierstrass product theorem, and the Hadamard factorization theorem.<sup>[14](https://www.sas.rochester.edu/mth/graduate/prelim-topics/prelim-topics-analysis2.html)</sup>

## References

1. [Entire function, step-by-step with animation, maths.free](https://maths.free/complex-analysis/entire-function/)
2. [Entire functions and Liouville theorem, lecture notes of Hanqing Li](https://lihanqing1997.github.io/notes/complex-analysis/entire-functions-and-liouville-theorem/)
3. [Math 656 Complex Variables I, NJIT (V. Matveev)](https://web.njit.edu/~matveev/Courses/M656_S11/Complex1.pdf)
4. [Lesson 5, Consequences of Cauchy's Formula, Complex Analysis notes](https://maxschottenfeld.github.io/complex-analysis/lessons/05-consequences-of-cauchys-formula/)
5. [Lecture 10: using Cauchy's formula, UC Berkeley](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_10.html)
6. [Liouville's theorem and consequences, Complex Analysis course notes](https://ondara.net/course/complex-analysis/subunit/u4-6-liouville-consequences)
7. [Part I. Entire Functions of Finite Order](https://lib.ysu.am/open_books/417657.pdf)
8. [Complex Analysis, Princeton Lectures in Analysis, Volume II (Stein & Shakarchi)](https://www.fing.edu.uy/~cerminar/Complex_Analysis.pdf)
9. [A History of Mathematics (1893), Recent Times: Theory of Functions, Wikisource](https://en.wikisource.org/wiki/A_History_of_Mathematics_(1893)/Recent_Times/Theory_of_Functions)
10. [Karl Weierstrass (1815–1897), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Weierstrass/)
11. [Wiener classes of entire functions of exponential type, Institute of Mathematics UFRC RAS](https://matem.anrb.ru/en/article?art_id=27)
12. [A Paley-Wiener Type Theorem for Singular Measures on the Torus, arXiv](https://arxiv.org/abs/1709.07522)
13. [To the Liouville theorem for entire functions of finite order, arXiv](https://arxiv.org/abs/2009.01019)
14. [Analysis II Prelim topics, University of Rochester Department of Mathematics](https://www.sas.rochester.edu/mth/graduate/prelim-topics/prelim-topics-analysis2.html)
15. [Complex Analysis](https://complexanalysis.org/web/sec_trig-hyperbolics.html)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Complex analysis*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license