{"id":108,"date":"2023-05-29T11:31:36","date_gmt":"2023-05-29T10:31:36","guid":{"rendered":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/?p=108"},"modified":"2023-05-29T23:11:04","modified_gmt":"2023-05-29T22:11:04","slug":"zeros-and-poles","status":"publish","type":"post","link":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/","title":{"rendered":"Zeros and Poles"},"content":{"rendered":"<p>We will briefly discuss zeros and poles of meromorphic functions here. We assume the <strong>Laurent series<\/strong> exist in the vicinity of a point <em>z<\/em><sub>0<\/sub>:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1375\" height=\"56\" class=\"alignnone size-full wp-image-110\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation1-2.png\" alt=\"\" \/><\/p>\n<p>Clearly if we want <em>f(z<\/em><sub>0<\/sub><em>)<\/em> is zero we require <em>a<sub>k <\/sub><\/em>is zero for <em>k<\/em> smaller <strong><em>and equals to<\/em><\/strong> zero. Consequently we define that <em>z<\/em><sub>0 <\/sub>is a zero of order <em>n<\/em> if:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1375\" height=\"53\" class=\"alignnone size-full wp-image-114\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation2-2.png\" alt=\"\" \/><\/p>\n<p>We can observe some useful properties from (2). Firstly, an<em> n<\/em> order zero implies that up to the (<em>n<\/em>-1)<sup>th <\/sup>derivative of <em>f(z) are<\/em>\u00a0also zero at <em>z<\/em><sub>0<\/sub> and vice versa. We can use this property to determine the order of zeros of a function, in the case that they are not so obvious. Secondly, The zeros of <em>f(z)<\/em> are the poles of 1\/<em>f(z)<\/em> for obvious reason, provided that <em>f(z)<\/em> is not identically zero. We would like to find the properties of the Laurent series of \u00a01\/<em>f(z)<\/em>:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1368\" height=\"62\" class=\"alignnone size-full wp-image-123\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation3-3.png\" alt=\"\" \/><\/p>\n<p>The value of <em>m<\/em> can be determined by multiply the denominator of right hand side : We obtain a series that constantly equals to one. This requires <em>m<\/em>=<em>n<\/em>, and we see the coefficients <em>b<sub>k<\/sub><\/em> are fixed by the values of <em>a<sub>k<\/sub><\/em> .<\/p>\n<p>In summary we found that<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1368\" height=\"68\" class=\"alignnone size-full wp-image-116\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation4-2.png\" alt=\"\" \/><\/p>\n<p><strong><em>In particular, if f(z) is analytic and non-zero at z<sub>0\u00a0<\/sub>, we now from (2) know that n=0 and thus from (4) 1\/f(z) is also analytic and non-zero at z<sub>0.<\/sub><\/em><\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We will briefly discuss zeros and poles of meromorphic functions here. We assume the Laurent series exist in the vicinity of a point z0: Clearly if we want f(z0) is zero we require ak is zero for k smaller and equals to zero. Consequently we define that z0 is a zero of order n if: [&hellip;]<\/p>\n","protected":false},"author":1741,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-108","post","type-post","status-publish","format-standard","hentry","category-analysis"],"_links":{"self":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts\/108","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/users\/1741"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/comments?post=108"}],"version-history":[{"count":9,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts\/108\/revisions"}],"predecessor-version":[{"id":126,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts\/108\/revisions\/126"}],"wp:attachment":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/media?parent=108"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/categories?post=108"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/tags?post=108"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}