Editing Liouville's theorem (complex analysis) (section)

== Remarks ==

Let <math>\Complex \cup \{\infty\}</math> be the one-point compactification of the complex plane <math>\Complex</math>. In place of holomorphic functions defined on regions in <math>\Complex</math>, one can consider regions in <math>\Complex \cup \{\infty\}</math>. Viewed this way, the only possible singularity for entire functions, defined on <math>\Complex \subset \Complex \cup \{\infty\}</math>, is the point <math>\infty</math>. If an entire function <math>f</math> is bounded in a neighborhood of <math>\infty</math>, then <math>\infty</math> is a [[removable singularity]] of <math>f</math>, i.e. <math>f</math> cannot blow up or behave erratically at <math>\infty</math>. In light of the [[power series]] expansion, it is not surprising that Liouville's theorem holds.

Similarly, if an entire function has a [[Pole (complex analysis)|pole]] of order <math>n</math> at <math>\infty</math> &mdash;that is, it grows in magnitude comparably to <math>z^n</math> in some neighborhood of <math>\infty</math> 
&mdash;then <math>f</math> is a polynomial. This extended version of Liouville's theorem can be more precisely stated: if <math>|f(z)|\leq M|z|^n</math> for <math>|z|</math> sufficiently large, then <math>f</math> is a polynomial of degree at most <math>n</math>. This can be proved as follows. Again take the Taylor series representation of <math>f</math>,

:<math> f(z) = \sum_{k=0}^\infty a_k z^k.</math>

The argument used during the proof using [[Cauchy estimates]] shows that for all <math>k\geq 0</math>,

:<math>|a_k| \leq Mr^{n-k}.</math>

So, if <math>k > n </math>, then

:<math>|a_k| \leq \lim_{r\to\infty}Mr^{n-k} = 0.</math>

Therefore, <math>a_k = 0</math>.

Liouville's theorem does not extend to the generalizations of complex numbers known as [[split-complex number|double numbers]] and [[dual number]]s.<ref>{{Cite journal|url=https://scholar.rose-hulman.edu/rhumj/vol12/iss2/4/|title=Liouville theorems in the Dual and Double Planes|journal=Rose-Hulman Undergraduate Mathematics Journal|date=15 January 2017|volume=12|issue=2|last1=Denhartigh|first1=Kyle|last2=Flim|first2=Rachel}}</ref>