{"id":19860,"date":"2023-07-05T10:20:21","date_gmt":"2023-07-05T17:20:21","guid":{"rendered":"https:\/\/www.kith.org\/jed\/?page_id=19860"},"modified":"2024-09-21T23:32:35","modified_gmt":"2024-09-22T06:32:35","slug":"i-to-the-i-power","status":"publish","type":"page","link":"https:\/\/www.kith.org\/jed\/hodgepodge\/nonfiction\/i-to-the-i-power\/","title":{"rendered":"i to the i power"},"content":{"rendered":"\r\n\r\n<style>\r\np, p+p {\r\n  text-indent: 0;\r\n}\r\n\r\np.equation {\r\n  margin: 1em;\r\n}\r\n<\/style>\r\n\r\n<div style=\"float:right; background-color:#eee; width: 30%; margin: em; padding: 1em;\">\r\n<p><b>Sidebar<\/b>: What is ln(-1)?<\/p>\r\n<p>Euler's Equation states that<\/p>\r\n<p class=\"equation\">e<sup>i&pi;<\/sup> + 1 = 0 <\/p>\r\n<p>Thus,<\/p>\r\n<p class=\"equation\">e<sup>i&pi;<\/sup> = -1<\/p>\r\n<p>So<\/p>\r\n<p class=\"equation\">i&pi; = ln(-1)<\/p>\r\n<\/div>\r\n\r\n<h3>Question<\/h3>\r\n<p>What is i<sup>i<\/sup>, expressed as a power of e?<\/p>\r\n<h3>Answer<\/h3>\r\n<p>Since we want to express it as a power of e, we can start by saying that<\/p>\r\n<p class=\"equation\">e<sup><var>x<\/var><\/sup> = i<sup>i<\/sup><\/p>\r\n<p>for some <var>x<\/var>. So<\/p>\r\n<p class=\"equation\"><var>x<\/var> = ln(i<sup>i<\/sup>)<\/p>\r\n<p class=\"equation\"><var>x<\/var> = i ln(i)<\/p>\r\n<p>Now, i is the square root of -1, so<\/p>\r\n<p class=\"equation\"><var>x<\/var> = i ln(-1<sup>1\/2<\/sup>)<\/p>\r\n<p class=\"equation\"><var>x<\/var> = (i\/2) ln(-1)<\/p>\r\n<p>But as demonstrated in the sidebar,<\/p>\r\n<p class=\"equation\">ln(-1) = i&pi;<\/p>\r\n<p>So<\/p>\r\n<p class=\"equation\"><var>x<\/var> = (i\/2) &middot; i&pi;<\/p>\r\n<p class=\"equation\"><var>x<\/var> = (i &middot; i &middot; &pi;) \/2<\/p>\r\n<p>but i &middot; i is -1, so<\/p>\r\n<p class=\"equation\"><var>x<\/var> = -&pi;\/2<\/p>\r\n<p>so<\/p>\r\n<p class=\"equation\">i<sup>i<\/sup> = e<sup>-&pi;\/2<\/sup><\/p>\r\n<p>(Note that this is a real number! It's about 0.208.)<\/p>\r\n\r\n<hr width=\"25%\" \/>\r\n\r\n<p>Thanks to Peter Hartman for showing me this elegant derivation.<\/p>\r\n\r\n\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":5,"featured_media":0,"parent":5482,"menu_order":180,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-19860","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/www.kith.org\/jed\/wp-json\/wp\/v2\/pages\/19860","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.kith.org\/jed\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.kith.org\/jed\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.kith.org\/jed\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/www.kith.org\/jed\/wp-json\/wp\/v2\/comments?post=19860"}],"version-history":[{"count":12,"href":"https:\/\/www.kith.org\/jed\/wp-json\/wp\/v2\/pages\/19860\/revisions"}],"predecessor-version":[{"id":19882,"href":"https:\/\/www.kith.org\/jed\/wp-json\/wp\/v2\/pages\/19860\/revisions\/19882"}],"up":[{"embeddable":true,"href":"https:\/\/www.kith.org\/jed\/wp-json\/wp\/v2\/pages\/5482"}],"wp:attachment":[{"href":"https:\/\/www.kith.org\/jed\/wp-json\/wp\/v2\/media?parent=19860"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}