Editing Entire function (section)

==Growth==

Entire functions may grow as fast as any increasing function: for any increasing function 
<math>g:[0,\infty)\to[0,\infty)</math> there exists an entire function <math>f</math> such that 
<math>f(x)>g(|x|)</math> for all real <math>x</math>. Such a function <math>f</math> may be easily found of the form:

<math display="block">f(z)=c+\sum_{k=1}^{\infty}\left(\frac{z}{k}\right)^{n_k}</math>

for a constant <math>c</math> and a strictly increasing sequence of positive integers <math>n_k</math>. Any such sequence defines an entire function <math>f(z)</math>, and if the powers are chosen appropriately we may satisfy the inequality <math>f(x)>g(|x|)</math> for all real <math>x</math>. (For instance, it certainly holds if one chooses <math>c:=g(2)</math> and, for any integer <math>k \ge 1</math> one chooses an even exponent <math> n_k </math> such that  <math>\left(\frac{k+1}{k}\right)^{n_k} \ge g(k+2)</math>).