{"id":29,"date":"2016-03-15T21:41:45","date_gmt":"2016-03-16T04:41:45","guid":{"rendered":"https:\/\/blogs.ubc.ca\/conwaychen\/?p=29"},"modified":"2016-03-15T21:41:45","modified_gmt":"2016-03-16T04:41:45","slug":"analytic-functions","status":"publish","type":"post","link":"https:\/\/blogs.ubc.ca\/conwaychen\/2016\/03\/15\/analytic-functions\/","title":{"rendered":"Analytic Functions"},"content":{"rendered":"<p>As we are talking about functions, we know that there are functions that are infinitely differentiable. An analytic function is a function that its Taylor series at any point x_0 in its domain converges to the function itself for x of x_0. In other words, if a function is analytic at c, then the function can be expanded to a \u00a0power series around c \u00a0has a positive radius of convergence. For example, if f(x) = e^x is analytic at x=0, then we are able to write out the power series representation\u00a0\u2211((x^n)\/n!) as n\u22650 near 0. So, the major difference between analytic functions and infinitely differentiable functions is that we can always write out a power series representation for analytic functions but not every\u00a0infinitely differentiable function\u00a0has a power series representation on every x in its domain.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>As we are talking about functions, we know that there are functions that are infinitely differentiable. An analytic function is a function that its Taylor series at any point x_0 in its domain converges to the function itself for x of x_0. In other words, if a function is analytic at c, then the function [&hellip;]<\/p>\n","protected":false},"author":35353,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-29","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/blogs.ubc.ca\/conwaychen\/wp-json\/wp\/v2\/posts\/29","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ubc.ca\/conwaychen\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ubc.ca\/conwaychen\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ubc.ca\/conwaychen\/wp-json\/wp\/v2\/users\/35353"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ubc.ca\/conwaychen\/wp-json\/wp\/v2\/comments?post=29"}],"version-history":[{"count":1,"href":"https:\/\/blogs.ubc.ca\/conwaychen\/wp-json\/wp\/v2\/posts\/29\/revisions"}],"predecessor-version":[{"id":30,"href":"https:\/\/blogs.ubc.ca\/conwaychen\/wp-json\/wp\/v2\/posts\/29\/revisions\/30"}],"wp:attachment":[{"href":"https:\/\/blogs.ubc.ca\/conwaychen\/wp-json\/wp\/v2\/media?parent=29"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ubc.ca\/conwaychen\/wp-json\/wp\/v2\/categories?post=29"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ubc.ca\/conwaychen\/wp-json\/wp\/v2\/tags?post=29"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}