Entire function

In mathematical analysis and, in particular, in the theory of functions of complex variable, an entire function is a function that is holomorphic in the whole complex plane .

Examples

Entire functions

Examples of entire functions are polynomial and exponential functions. All sums, and products of entire functions are entire, so that the entire functions form a C-algebra. Further, compositions of entire functions are also entire.

All the derivatives and some of the integrals of entire functions, for example the error function erf, Si and the Bessel function J₀ are also entire functions.

Non-entire functions

In general, neither series nor limit of a sequence of entire functions need be an entire function.

The inverse of an entire function has need not be entire. Usually, inverse of a non-trivial function is not entire. (The inverse of a linear function is entire). In particular, inverses of trigonometric functions are not entire.

More non-entire functions: rational function $~f(z)={\frac {a+bx}{c+x}}~$ at any complex $~a~$, $~b~$, $~c~$ , square root, logarithm, function Gamma, tetration.

In particular, non-analytic functions also should be qualified as non-entire: $\Re$, $\Im$, complex conjugation, modulus, argument, Dirichlet function.

Properties

The entire functions have all general properties of other analytic functions, but the infinite range of analyticity enhances the set of the properties, making the entire functions especially beautiful and attractive for applications.

Power series

The radius of convergence of a power series is the distance the nearest singularity. Therefore, it is infinite for entire functions.

Any entire function can be expanded in every point to the Taylor series which converges everywhere.

This does not mean that one can always use the power series for precise evaluation of an entire function, but helps a lot to prove the theorems.

Unboundedness

Liouville's theorem states: an entire function which is bounded must be constant .

Order of an entire function

As all entire functions (except the constants) are unbounded, they grow as the argument become large, and can be characterised by their growth rate, which is called order.

Let $~f~$ be entire function. Positive number $~\alpha ~$ is called order of function $~f~$, if for all positive numbers $~\beta ~$, larger than $~\alpha ~$, there exist positive number $~\rho ~$ such that for all complex $~z~$ such that $~|z|>\rho ~$, the relation $~|f(z)|<\exp {\big (}|z|^{\beta }{\big )}~$ holds .

In particular, all polynomials have order 0; the exponential has order 1; and erf, as the Gaussian exponential, has order 2.

Range of values

Picard's little theorem states: a non-constant entire function takes on every complex number as value, except possibly one .

For example, the exponential never takes on the value 0.

Cauchy integral

Entire function $~f~$, at any complex $~z~$ and at any contour C enclosing the point $z$ just once, can be expressed the Cauchy's theorem $f(x)={\frac {1}{2\pi {\rm {i}}}}\oint _{\mathbf {C} }{\frac {f(t)}{t-z}}{\rm {d}}t$