{"id":246992,"date":"2026-04-28T16:44:58","date_gmt":"2026-04-28T21:44:58","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=246992"},"modified":"2026-04-28T16:44:58","modified_gmt":"2026-04-28T21:44:58","slug":"even-series-trick","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/04\/28\/even-series-trick\/","title":{"rendered":"Turning a trick into a technique"},"content":{"rendered":"<p>Someone said a technique is a trick that works twice.<\/p>\n<p>I wanted to see if I could get anything interesting by turning the trick in the <a href=\"https:\/\/www.johndcook.com\/blog\/2026\/04\/28\/circular-arc-approximation\/\">previous post<\/a> into a technique. The trick created a high-order approximation by subtracting a multiple one even function from another. Even functions only have even-order terms, and by using the right multiple you can cancel out the second-order term as well.<\/p>\n<p>For an example, I&#8217;d like to approximate the Bessel function <em>J<\/em><sub>0<\/sub>(<em>x<\/em>) by the better known cosine function. Both are even functions.<\/p>\n<p style=\"padding-left: 40px;\"><em>J<\/em><sub>0<\/sub>(<em>x<\/em>) = 1 \u2212 <em>x<\/em><sup>2<\/sup>\/4 + <em>x<\/em><sup>4<\/sup>\/64 + \u2026<br \/>\ncos(<em>x<\/em>) = 1 \u2212 <em>x<\/em><sup>2<\/sup>\/2 + <em>x<\/em><sup>4<\/sup>\/24 + \u2026<\/p>\n<p>and so<\/p>\n<p style=\"padding-left: 40px;\">2 <em>J<\/em><sub>0<\/sub>(<em>x<\/em>) \u2212 cos(<em>x<\/em>) = 1 \u2212 <em>x<\/em><sup>4<\/sup>\/96 + \u2026<\/p>\n<p>which means<\/p>\n<p style=\"padding-left: 40px;\"><em>J<\/em><sub>0<\/sub>(<em>x<\/em>) \u2248 (1 + cos(<em>x<\/em>))\/2<\/p>\n<p>is an excellent approximation for small <em>x<\/em>.<\/p>\n<p>Let&#8217;s try this for a couple examples.<\/p>\n<p><em>J<\/em><sub>0<\/sub>(0.2) = 0.990025 and (1 + cos(0.2))\/2 = 0.990033.<\/p>\n<p><em>J<\/em><sub>0<\/sub>(0.05) = 0.99937510 and (1 + cos(0.05))\/2 = 0.99937513.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Someone said a technique is a trick that works twice. I wanted to see if I could get anything interesting by turning the trick in the previous post into a technique. The trick created a high-order approximation by subtracting a multiple one even function from another. Even functions only have even-order terms, and by using [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[129],"class_list":["post-246992","post","type-post","status-publish","format-standard","hentry","category-math","tag-special-functions"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/246992","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/comments?post=246992"}],"version-history":[{"count":0,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/246992\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/media?parent=246992"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/categories?post=246992"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/tags?post=246992"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}