Editing Entire function (section)

==Properties==
Every entire function <math> f(z) </math> can be represented as a single [[power series]]:
<math display="block">\ f(z) = \sum_{n=0}^\infty a_n z^n\ </math>
that [[convergence (mathematics)|converges]] everywhere in the complex plane, hence [[Compact convergence|uniformly on compact sets]]. The [[radius of convergence]] is infinite, which implies that
<math display="block">\ \lim_{n\to\infty} |a_n|^{\frac{1}{n}} = 0\ </math>
or, equivalently,{{efn|If necessary, the logarithm of zero is taken to be equal to minus infinity.}}
<math display="block">\ \lim_{n\to\infty} \frac{\ln|a_n|}n = -\infty ~.</math>
Any power series satisfying this criterion will represent an entire function.

If (and only if) the coefficients of the power series are all real then the function evidently takes real values for real arguments, and the value of the function at the [[complex conjugate]] of <math>z</math> will be the complex conjugate of the value at <math>z ~.</math> Such functions are sometimes called self-conjugate (the conjugate function, <math>F^*(z),</math> being given by {{nowrap|<math>\bar F(\bar z)</math>).}}{{sfn|Boas|1954|p=1}}

If the real part of an entire function is known in a neighborhood of a point then both the real and imaginary parts are known for the whole complex plane, [[up to]] an imaginary constant. For instance, if the real part is known in a [[Neighbourhood (mathematics)|neighborhood]] of zero, then we can find the coefficients for <math>n>0</math> from the following derivatives with respect to a real variable <math>r</math>:

<math display="block">\begin{align}
\operatorname\mathcal{Re} \left\{\ a_n\ \right\} &= \frac{1}{n!} \frac{d^n}{dr^n}\ \operatorname\mathcal{Re} \left\{\ f(r)\ \right\} && \quad \mathrm{ at } \quad r = 0 \\
\operatorname\mathcal{Im}\left\{\ a_n\ \right\} &= \frac{1}{n!} \frac{d^n}{dr^n}\ \operatorname\mathcal{Re} \left\{\ f\left( r\ e^{-\frac{i\pi}{2n}} \right)\ \right\} && \quad \mathrm{ at } \quad r = 0
\end{align}</math>

(Likewise, if the imaginary part is known in a neighborhood then the function is determined up to a real constant.) In fact, if the real part is known just on an arc of a circle, then the function is determined up to an imaginary constant.{{efn|
For instance, if the real part is known on part of the unit circle, then it is known on the whole unit circle by [[analytic extension]], and then the coefficients of the infinite series are determined from the coefficients of the [[Fourier series]] for the real part on the unit circle.
}}}
Note however that an entire function is '''''not''''' determined by its real part on all curves. In particular, if the real part is given on any curve in the complex plane where the real part of some other entire function is zero, then any multiple of that function can be added to the function we are trying to determine. For example, if the curve where the real part is known is the real line, then we can add <math>i</math> times any self-conjugate function. If the curve forms a loop, then it is determined by the real part of the function on the loop since the only functions whose real part is zero on the curve are those that are everywhere equal to some [[imaginary number]].

The [[Weierstrass factorization theorem]] asserts that any entire function can be represented by a product involving its [[zero of a function|zeroes]] (or "roots").

The entire functions on the complex plane form an [[integral domain]] (in fact a [[Prüfer domain]]). They also form a [[commutative]] [[unital algebra|unital]] [[associative algebra]] over the [[complex number]]s.

[[Liouville's theorem (complex analysis)|Liouville's theorem]] states that any [[bounded function|bounded]] entire function must be constant.{{efn|
Liouville's theorem may be used to elegantly prove the [[fundamental theorem of algebra]].
}}

As a consequence of Liouville's theorem, any function that is entire on the whole [[Riemann sphere]]{{efn|
The [[Riemann sphere]] is the whole complex plane augmented with a single point at infinity.
}}
is constant. Thus any non-constant entire function must have a [[mathematical singularity|singularity]] at the complex [[point at infinity]], either a [[pole (complex analysis)|pole]] for a polynomial or an [[essential singularity]] for a [[transcendental function|transcendental]] entire function. Specifically, by the [[Casorati–Weierstrass theorem]], for any transcendental entire function <math>f</math> and any complex <math>w</math> there is a [[sequence]] <math>(z_m)_{m\in\N}</math> such that

:<math>\ \lim_{m\to\infty} |z_m| = \infty, \qquad \text{and} \qquad \lim_{m\to\infty} f(z_m) = w ~.</math>

[[Picard theorem|Picard's little theorem]] is a much stronger result: Any non-constant entire function takes on every complex number as value, possibly with a single exception. When an exception exists, it is called a [[lacunary value]] of the function. The possibility of a lacunary value is illustrated by the [[exponential function]], which never takes on the value <math>0</math>. One can take a suitable branch of the logarithm of an entire function that never hits <math>0</math>, so that this will also be an entire function (according to the [[Weierstrass factorization theorem]]). The logarithm hits every complex number except possibly one number, which implies that the first function will hit any value other than <math>0</math> an infinite number of times. Similarly, a non-constant, entire function that does not hit a particular value will hit every other value an infinite number of times.

Liouville's theorem is a special case of the following statement:

{{math theorem|math_statement= Assume <math>M,</math> <math>R</math> are positive constants and <math>n</math> is a non-negative integer. An entire function <math>f</math> satisfying the inequality <math>|f(z)| \le M |z|^n</math> for all <math>z</math> with <math>|z| \ge R,</math> is necessarily a polynomial, of [[degree of a polynomial|degree]] at most <math>n ~.</math>{{efn|
The converse is also true as for any polynomial <math display="inline">p(z) = \sum_{k=0}^n a_k z^k</math> of degree <math>n</math> the inequality <math display="inline">|p(z)| \le \left(\ \sum_{k=0}^n|a_k|\ \right) |z|^n</math> holds for any <math>|z|\geq 1 ~.</math>
}}
Similarly, an entire function <math>f</math> satisfying the inequality <math>M |z|^n \le |f(z)|</math> for all <math>z</math> with <math>|z| \ge R,</math> is necessarily a polynomial, of degree at least <math>n</math>.}}