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 Conway, John B. (1978). Functions of One Complex Variable I, 2nd edition. Springer. ISBN 0-387-90328-3. </ref>^[1].

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 funcitons, for example erf, Si, $J_{0}$ , also are entired functions.

Every entire function can be represented as a power series or Tailor expansion which converges everywhere.

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.

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

Liouville's theorem establishes an important property of entire functions — an entire function which is bounded must be constant ^[2].

Picard's little theorem states: a non-constant entire function takes on every complex number as value, except possibly one ^[1].

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

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

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

↑ ^1.0 ^1.1 Boas, Ralph P.. Entire Functions. Academic Press. OCLC 847696. Cite error: Invalid <ref> tag; name "ralph" defined multiple times with different content
↑ Conway, John B. (1978). Functions of One Complex Variable I, 2nd edition. Springer. ISBN 0-387-90328-3.