{"id":933,"date":"2021-01-10T13:48:38","date_gmt":"2021-01-10T13:48:38","guid":{"rendered":"http:\/\/adrianbell.me\/?p=933"},"modified":"2021-01-10T13:48:40","modified_gmt":"2021-01-10T13:48:40","slug":"the-power-of-complex-numbers","status":"publish","type":"post","link":"https:\/\/adrianbell.me\/?p=933","title":{"rendered":"The Power of Complex Numbers"},"content":{"rendered":"\n

I've reached a part of my learning which starts to touch upon exactly why I signed up for this module. Had<\/em> to share...<\/p>\n\n\n\n

How would you normally ever go about finding the sum of this infinite series of natural numbers?<\/p>\n\n\n\n

\"\sum^{\infty}_{n=0}\frac{1}{2^{n}}\sin nx\"<\/p>\n\n\n\n

???<\/p>\n\n\n\n

Well! Complex numbers to the rescue!<\/p>\n\n\n\n

\"\sum^{\infty}_{n=0}\frac{1}{2^{n}}\sin nx\"<\/p>\n\n\n\n

\"=\text{Im}\left(\sum^{\infty}_{n=0}\frac{1}{2^{n}}e^{inx}\right)\"<\/p>\n\n\n\n

\"=\text{Im}\left(\frac{1}{1-\frac{1}{2}e^{ix}}\right)\"<\/p>\n\n\n\n

\"=\ldots\" (simplification steps)<\/p>\n\n\n\n

\"=\frac{2\sin x}{5-4\cos x}\"<\/p>\n\n\n\n

How cool is that!<\/p>\n","protected":false},"excerpt":{"rendered":"

I've reached a part of my learning which starts to touch upon exactly why I signed up for this module. Had to share... How would you normally ever go about finding the sum of this infinite series of natural numbers? ??? Well! Complex numbers to the rescue! (simplification steps) How cool is that!<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[46],"tags":[47],"class_list":["post-933","post","type-post","status-publish","format-standard","hentry","category-complex-analysis","tag-complex-analysis"],"_links":{"self":[{"href":"https:\/\/adrianbell.me\/index.php?rest_route=\/wp\/v2\/posts\/933","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/adrianbell.me\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/adrianbell.me\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/adrianbell.me\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/adrianbell.me\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=933"}],"version-history":[{"count":10,"href":"https:\/\/adrianbell.me\/index.php?rest_route=\/wp\/v2\/posts\/933\/revisions"}],"predecessor-version":[{"id":943,"href":"https:\/\/adrianbell.me\/index.php?rest_route=\/wp\/v2\/posts\/933\/revisions\/943"}],"wp:attachment":[{"href":"https:\/\/adrianbell.me\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=933"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/adrianbell.me\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=933"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/adrianbell.me\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=933"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}