Editing Entire function (section)

{{short description|Function that is holomorphic on the whole complex plane}}
In [[complex analysis]], an '''entire function''', also called an '''integral function,''' is a complex-valued [[Function (mathematics)|function]] that is [[holomorphic function|holomorphic]] on the whole [[complex plane]]. Typical examples of entire functions are [[polynomial]]s and the [[exponential function]], and any finite sums, products and compositions of these, such as the [[trigonometric function]]s [[sine]] and [[cosine]] and their [[hyperbolic function|hyperbolic counterparts]] [[hyperbolic sine|sinh]] and [[hyperbolic cosine|cosh]], as well as [[derivative]]s and [[integral]]s of entire functions such as the [[error function]]. If an entire function <math>f(z)</math> has a 
[[root of a function|root]] at <math>w</math>, then <math>f(z)/(z-w)</math>, taking the limit value at <math>w</math>, is an entire function. On the other hand, the [[natural logarithm]], the [[reciprocal function]], and the [[square root]] are all not entire functions, nor can they be [[analytic continuation|continued analytically]] to an entire function.

A '''[[Transcendental function|transcendental]] entire function''' is an entire function that is not a polynomial.

Just as [[Meromorphic function|meromorphic functions]] can be viewed as a generalization of [[Rational function|rational fractions]], entire functions can be viewed as a generalization of polynomials. In particular, if for meromorphic functions one can generalize the factorization into simple fractions (the [[Mittag-Leffler's theorem|Mittag-Leffler theorem]] on the decomposition of a meromorphic function), then for entire functions there is a generalization of the factorization — the [[Weierstrass factorization theorem|Weierstrass theorem]] on entire functions.