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

===Non-constant elliptic functions cannot be defined on the complex plane===
The theorem can also be used to deduce that the domain of a non-constant [[elliptic function]] <math>f</math> cannot be <math>\Complex</math>. Suppose it was. Then, if <math>a</math> and <math>b</math> are two periods of <math>f</math> such that <math>\tfrac{a}{b}</math> is not real, consider the [[parallelogram]] <math>P</math> whose [[Vertex (geometry)|vertices]] are 0, <math>a</math>, <math>b</math>, and <math>a+b</math>. Then the image of <math>f</math> is equal to <math>f(P)</math>. Since <math>f</math> is [[continuous functions|continuous]] and <math>P</math> is [[Compact space|compact]], <math>f(P)</math> is also compact and, therefore, it is bounded. So, <math>f</math> is constant.

The fact that the domain of a non-constant [[elliptic function]] <math>f</math> cannot be <math>\Complex</math> is what Liouville actually proved, in 1847, using the theory of elliptic functions.<ref>{{Citation|last = Liouville|first = Joseph|author-link = Joseph Liouville|publication-date = 1879|year = 1847|title = Leçons sur les fonctions doublement périodiques|periodical = [[Crelle's Journal|Journal für die Reine und Angewandte Mathematik]]|volume = 88|pages = 277–310|issn = 0075-4102|url = http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=266004|archive-url = https://archive.today/20120711004552/http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=266004|url-status = dead|archive-date = 2012-07-11}}</ref> In fact, it was [[Augustin Louis Cauchy|Cauchy]] who proved Liouville's theorem.<ref>{{Citation|last = Cauchy|first = Augustin-Louis|authorlink = Augustin Louis Cauchy|year = 1844|publication-date = 1882|contribution = Mémoires sur les fonctions complémentaires|contribution-url = http://visualiseur.bnf.fr/StatutConsulter?N=VERESS5-1212867208163&B=1&E=PDF&O=NUMM-90188|title = Œuvres complètes d'Augustin Cauchy|series = 1|volume = 8|place = Paris|publisher = Gauthiers-Villars}}</ref><ref>{{Citation|last = Lützen|first = Jesper|year = 1990|title = Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics|series = Studies in the History of Mathematics and Physical Sciences|volume = 15|publisher = Springer-Verlag|isbn = 3-540-97180-7}}</ref>