Talk:Entire function

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 $f(z)$ is defined using the limit superior as:

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

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

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

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.