Entire function

In <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, an entire function, also called an integral function, is a complex-valued <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> that is <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a> on the whole <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>. Typical examples of entire functions are <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> and the <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a>, and any finite sums, products and compositions of these, such as the <a href="/facts/Trigonometric_function/czej7ur0">trigonometric functions</a> <a href="/facts/Sine/gQ6LXK8z">sine</a> and <a href="/facts/Cosine/czej7ur0">cosine</a> and their <a href="/facts/Hyperbolic_function/Y4Cb5S35">hyperbolic counterparts</a> <a href="/facts/Hyperbolic_sine/Y4Cb5S35">sinh</a> and <a href="/facts/Hyperbolic_cosine/Y4Cb5S35">cosh</a>, as well as <a href="/facts/Derivative/7Ito70Dh">derivatives</a> and <a href="/facts/Integral/V7lyV997">integrals</a> of entire functions such as the <a href="/facts/Error_function/Ca2H7pGr">error function</a>. If an entire function 
 
 
 
 f
 (
 z
 )
 
 
 {\displaystyle f(z)}
 
 has a 
<a href="/facts/Root_of_a_function/mlK3dhoT">root</a> at 
 
 
 
 w
 
 
 {\displaystyle w}
 
, then 
 
 
 
 f
 (
 z
 )
 
 /
 
 (
 z
 −
 w
 )
 
 
 {\displaystyle f(z)/(z-w)}
 
, taking the limit value at 
 
 
 
 w
 
 
 {\displaystyle w}
 
, is an entire function. On the other hand, the <a href="/facts/Natural_logarithm/PnreKEzI">natural logarithm</a>, the <a href="/facts/Reciprocal_function/yczdYyCw">reciprocal function</a>, and the <a href="/facts/Square_root/AfrzfBdQ">square root</a> are all not entire functions, nor can they be <a href="/facts/Analytic_continuation/aBCuEgvt">continued analytically</a> to an entire function.
A <a href="/facts/Transcendental_function/yMkeq4Eo">transcendental</a> entire function is an entire function that is not a polynomial.
Just as <a href="/facts/Meromorphic_function/LeNBd6Sh">meromorphic functions</a> can be viewed as a generalization of <a href="/facts/Rational_function/2nJGu0Ko">rational fractions</a>, 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 <a href="/facts/Mittag-Leffler%2527s_theorem/CnLkL3a8">Mittag-Leffler theorem</a> on the decomposition of a meromorphic function), then for entire functions there is a generalization of the factorization — the <a href="/facts/Weierstrass_factorization_theorem/atdInY5m">Weierstrass theorem</a> on entire functions.

Entire function open-in-new

Entire function