{"id":108,"date":"2023-05-29T11:31:36","date_gmt":"2023-05-29T10:31:36","guid":{"rendered":"https:\/\/blogs.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.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=\"http:\/\/blogs.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=\"http:\/\/blogs.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=\"http:\/\/blogs.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=\"http:\/\/blogs.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"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Zeros and Poles - Cocteaupedia<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/\" \/>\n<meta property=\"og:locale\" content=\"en_GB\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Zeros and Poles - Cocteaupedia\" \/>\n<meta property=\"og:description\" content=\"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;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/\" \/>\n<meta property=\"og:site_name\" content=\"Cocteaupedia\" \/>\n<meta property=\"article:published_time\" content=\"2023-05-29T10:31:36+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-05-29T22:11:04+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/blogs.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation1-2.png\" \/>\n<meta name=\"author\" content=\"Cocteau Huang\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Cocteau Huang\" \/>\n\t<meta name=\"twitter:label2\" content=\"Estimated reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/2023\\\/05\\\/29\\\/zeros-and-poles\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/2023\\\/05\\\/29\\\/zeros-and-poles\\\/\"},\"author\":{\"name\":\"Cocteau Huang\",\"@id\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/#\\\/schema\\\/person\\\/d57ea00ff9e768fd4a27699d30e027c5\"},\"headline\":\"Zeros and Poles\",\"datePublished\":\"2023-05-29T10:31:36+00:00\",\"dateModified\":\"2023-05-29T22:11:04+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/2023\\\/05\\\/29\\\/zeros-and-poles\\\/\"},\"wordCount\":225,\"commentCount\":0,\"image\":{\"@id\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/2023\\\/05\\\/29\\\/zeros-and-poles\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/files\\\/2023\\\/05\\\/equation1-2.png\",\"articleSection\":[\"Analysis\"],\"inLanguage\":\"en-GB\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/2023\\\/05\\\/29\\\/zeros-and-poles\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/2023\\\/05\\\/29\\\/zeros-and-poles\\\/\",\"url\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/2023\\\/05\\\/29\\\/zeros-and-poles\\\/\",\"name\":\"Zeros and Poles - Cocteaupedia\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/2023\\\/05\\\/29\\\/zeros-and-poles\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/2023\\\/05\\\/29\\\/zeros-and-poles\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/files\\\/2023\\\/05\\\/equation1-2.png\",\"datePublished\":\"2023-05-29T10:31:36+00:00\",\"dateModified\":\"2023-05-29T22:11:04+00:00\",\"author\":{\"@id\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/#\\\/schema\\\/person\\\/d57ea00ff9e768fd4a27699d30e027c5\"},\"breadcrumb\":{\"@id\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/2023\\\/05\\\/29\\\/zeros-and-poles\\\/#breadcrumb\"},\"inLanguage\":\"en-GB\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/2023\\\/05\\\/29\\\/zeros-and-poles\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-GB\",\"@id\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/2023\\\/05\\\/29\\\/zeros-and-poles\\\/#primaryimage\",\"url\":\"http:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/files\\\/2023\\\/05\\\/equation1-2.png\",\"contentUrl\":\"http:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/files\\\/2023\\\/05\\\/equation1-2.png\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/2023\\\/05\\\/29\\\/zeros-and-poles\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Zeros and Poles\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/#website\",\"url\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/\",\"name\":\"Cocteaupedia\",\"description\":\"Notes in Theoretical Physics and Maths\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-GB\"},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/#\\\/schema\\\/person\\\/d57ea00ff9e768fd4a27699d30e027c5\",\"name\":\"Cocteau Huang\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-GB\",\"@id\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/86d23a25bc24e1ee9655f0b3d54afae109abdd6c1a56430c44bf0380db8bc5f8?s=96&d=mm&r=g\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/86d23a25bc24e1ee9655f0b3d54afae109abdd6c1a56430c44bf0380db8bc5f8?s=96&d=mm&r=g\",\"contentUrl\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/86d23a25bc24e1ee9655f0b3d54afae109abdd6c1a56430c44bf0380db8bc5f8?s=96&d=mm&r=g\",\"caption\":\"Cocteau Huang\"},\"url\":\"https:\\\/\\\/blogs.imperial.ac.uk\\\/cocteaupedia\\\/author\\\/yh1721\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Zeros and Poles - Cocteaupedia","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/","og_locale":"en_GB","og_type":"article","og_title":"Zeros and Poles - Cocteaupedia","og_description":"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;]","og_url":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/","og_site_name":"Cocteaupedia","article_published_time":"2023-05-29T10:31:36+00:00","article_modified_time":"2023-05-29T22:11:04+00:00","og_image":[{"url":"http:\/\/blogs.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation1-2.png","type":"","width":"","height":""}],"author":"Cocteau Huang","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Cocteau Huang","Estimated reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/#article","isPartOf":{"@id":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/"},"author":{"name":"Cocteau Huang","@id":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/#\/schema\/person\/d57ea00ff9e768fd4a27699d30e027c5"},"headline":"Zeros and Poles","datePublished":"2023-05-29T10:31:36+00:00","dateModified":"2023-05-29T22:11:04+00:00","mainEntityOfPage":{"@id":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/"},"wordCount":225,"commentCount":0,"image":{"@id":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/#primaryimage"},"thumbnailUrl":"http:\/\/blogs.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation1-2.png","articleSection":["Analysis"],"inLanguage":"en-GB","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/","url":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/","name":"Zeros and Poles - Cocteaupedia","isPartOf":{"@id":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/#website"},"primaryImageOfPage":{"@id":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/#primaryimage"},"image":{"@id":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/#primaryimage"},"thumbnailUrl":"http:\/\/blogs.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation1-2.png","datePublished":"2023-05-29T10:31:36+00:00","dateModified":"2023-05-29T22:11:04+00:00","author":{"@id":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/#\/schema\/person\/d57ea00ff9e768fd4a27699d30e027c5"},"breadcrumb":{"@id":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/#breadcrumb"},"inLanguage":"en-GB","potentialAction":[{"@type":"ReadAction","target":["https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/"]}]},{"@type":"ImageObject","inLanguage":"en-GB","@id":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/#primaryimage","url":"http:\/\/blogs.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation1-2.png","contentUrl":"http:\/\/blogs.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation1-2.png"},{"@type":"BreadcrumbList","@id":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/29\/zeros-and-poles\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/"},{"@type":"ListItem","position":2,"name":"Zeros and Poles"}]},{"@type":"WebSite","@id":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/#website","url":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/","name":"Cocteaupedia","description":"Notes in Theoretical Physics and Maths","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-GB"},{"@type":"Person","@id":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/#\/schema\/person\/d57ea00ff9e768fd4a27699d30e027c5","name":"Cocteau Huang","image":{"@type":"ImageObject","inLanguage":"en-GB","@id":"https:\/\/secure.gravatar.com\/avatar\/86d23a25bc24e1ee9655f0b3d54afae109abdd6c1a56430c44bf0380db8bc5f8?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/86d23a25bc24e1ee9655f0b3d54afae109abdd6c1a56430c44bf0380db8bc5f8?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/86d23a25bc24e1ee9655f0b3d54afae109abdd6c1a56430c44bf0380db8bc5f8?s=96&d=mm&r=g","caption":"Cocteau Huang"},"url":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/author\/yh1721\/"}]}},"_links":{"self":[{"href":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts\/108","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/users\/1741"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/comments?post=108"}],"version-history":[{"count":9,"href":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts\/108\/revisions"}],"predecessor-version":[{"id":126,"href":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts\/108\/revisions\/126"}],"wp:attachment":[{"href":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/media?parent=108"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/categories?post=108"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/tags?post=108"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}