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 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.

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, invetse 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: [[real part|Failed to parse (syntax error): {\displaystyle \Re</math]], [[imaginary part|<math>\Im</math]], [[complex conjugation]], [[modulus]], [[argument]], [[Dirichlet function]]. ==Properties== ===Infinitness=== [[Liouville's theorem]] establishes an important property of entire functions &mdash; an entire function which is bounded must be constant <ref name="john"> {{cite book|first=John B.|last=Conway|authorlink=John B. Conway|year=1978|title=Functions of One Complex Variable I|edition=2nd edition|publisher=Springer|id=ISBN 0-387-90328-3}}</ref>. ===Range of values=== [[Picard theorem|Picard's little theorem]] states: a non-constant entire function takes on every complex number as value, except possibly one <ref name="ralph">{{cite book |first=Ralph P. last=Boas |uear=1954 |title=Entire Functions |publisher=Academic Press |id=OCLC 847696 }}</ref>. <!-- This property can be used for an elegant proof of the [[fundamental theorem of algebra]]. !--> For example, the [[exponential function|exponential]] never takes on the value 0. ===Cauchi integral=== <!-- I am not sure if this section should be here. Perhaps, it also should be separted article !--> Entire function <math>~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