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{"fact":"If your cat snores, or rolls over on his back to expose his belly, it means he trusts you.","length":90}
{"type":"standard","title":"Liouville's theorem (complex analysis)","displaytitle":"Liouville's theorem (complex analysis)","namespace":{"id":0,"text":""},"wikibase_item":"Q576478","titles":{"canonical":"Liouville's_theorem_(complex_analysis)","normalized":"Liouville's theorem (complex analysis)","display":"Liouville's theorem (complex analysis)"},"pageid":312293,"thumbnail":{"source":"https://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Gamma_abs_3D.png/330px-Gamma_abs_3D.png","width":320,"height":250},"originalimage":{"source":"https://upload.wikimedia.org/wikipedia/commons/3/33/Gamma_abs_3D.png","width":1280,"height":1000},"lang":"en","dir":"ltr","revision":"1283332245","tid":"efc2d25d-0e74-11f0-858e-cc2603ef63b9","timestamp":"2025-03-31T21:13:05Z","description":"Theorem in complex analysis","description_source":"local","content_urls":{"desktop":{"page":"https://en.wikipedia.org/wiki/Liouville's_theorem_(complex_analysis)","revisions":"https://en.wikipedia.org/wiki/Liouville's_theorem_(complex_analysis)?action=history","edit":"https://en.wikipedia.org/wiki/Liouville's_theorem_(complex_analysis)?action=edit","talk":"https://en.wikipedia.org/wiki/Talk:Liouville's_theorem_(complex_analysis)"},"mobile":{"page":"https://en.m.wikipedia.org/wiki/Liouville's_theorem_(complex_analysis)","revisions":"https://en.m.wikipedia.org/wiki/Special:History/Liouville's_theorem_(complex_analysis)","edit":"https://en.m.wikipedia.org/wiki/Liouville's_theorem_(complex_analysis)?action=edit","talk":"https://en.m.wikipedia.org/wiki/Talk:Liouville's_theorem_(complex_analysis)"}},"extract":"In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function for which there exists a positive number such that for all is constant. Equivalently, non-constant holomorphic functions on have unbounded images.","extract_html":"
In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function ![](\"https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61\")
![](\"https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd\")