Entire function

Definition

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

Examples

Entires

Examples of entire functions are the polynomials and the exponentials. All sums, products and compositions of these functions also are entire functions.

All the derivatives and some of integrals of entired functions, for example erf, Si, $J_{0}$, also are entired functions.

Non-entires

In general, neither series nor limit of a sequence of entire funcitons needs to be an entire function.

Inverse of an entire function has no need to be entire function. Usually, inverse of a non-trivial function is not entire. (The inverse of the linear function is entire). In particular, inverse 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 always distance until the nearest singularity. Therefore, it is infinite for entire functions.

Any entire function can be expanded in every point to the Tailor 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.

Infinitness

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

Order of an entire function

As all the entire function (except constant) are not bounded, they grow as the argument become large, and can be characterised with the 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.

Cauchi integral

Entire function $~f~$, at any complex $~z~$ and at any contour C evolving point $z$ just once, can be expressed with Cauchi theorem

$f(x)={\frac {1}{2\pi {\rm {i}}}}\oint _{\mathbf {C} }{\frac {f(t)}{t-z}}{\rm {d}}t$

See also

• Cauchi formula
• Tailor series

References

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