# Talk:Entire function

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 Definition:  is a function that is holomorphic in the whole complex plane. [d] [e]
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## Missed section

Most of wikilinks are red... a lot of background articles should be written to make it usable. I did not copy the section about the order, because I think that it should be independent article; I just repeat it below: Dmitrii Kouznetsov 00:27, 17 May 2008 (CDT)

The order of an entire function ${\displaystyle f(z)}$ is defined using the limit superior as:

${\displaystyle \rho =\limsup _{r\rightarrow \infty }{\frac {\ln(\ln(M(r)))}{\ln(r)}},}$

where ${\displaystyle r}$ is the distance from ${\displaystyle 0}$ and ${\displaystyle M(r)}$ is the maximum absolute value of ${\displaystyle f(z)}$ when ${\displaystyle \left|z\right|=r.}$ If ${\displaystyle 0<\rho <\infty ,}$ one can also define the type:

${\displaystyle \sigma =\limsup _{r\rightarrow \infty }{\frac {\ln(M(r))}{r^{\rho }}}.}$

Note that an entire function may have a singularity or even an essential singularity at the complex point at infinity. In the latter case, it is called a transcendental entire function. As a consequence of Liouville's theorem, a function which is entire on the whole Riemann sphere (complex plane and the point at infinity) is constant.

J. E. Littlewood chose the Weierstrass sigma function as a 'typical' entire function in one of his books.