{"id":22,"date":"2020-11-10T12:50:01","date_gmt":"2020-11-10T12:50:01","guid":{"rendered":"https:\/\/fooledbyrandomnessdotcom.wordpress.com\/?p=22"},"modified":"2020-11-10T12:50:01","modified_gmt":"2020-11-10T12:50:01","slug":"a-reexpression-of-the-floor-function","status":"publish","type":"post","link":"https:\/\/fooledbyrandomness.com\/blog\/index.php\/2020\/11\/10\/a-reexpression-of-the-floor-function\/","title":{"rendered":"Floor functions using complex logarithms"},"content":{"rendered":"\n

An interesting discovery, thanks to a problem presented by K. Srinivasa Raghava:<\/p>\n\n\n\n

A standard result for \\(x\\) real, where \\(\\lfloor .\\rfloor\\) is the floor function:<\/p>\n\n\n\n

\\(\\lfloor x\\rfloor =\\frac{1}{\\pi }\\sum _{k=1}^{\\infty } \\frac{\\sin (2 \\pi k x)}{k}+x-\\frac{1}{2}\\).<\/p>\n\n\n\n

Now it so happens that:<\/p>\n\n\n\n

\\(\\frac{1}{\\pi }\\sum _{k=1}^{\\infty } \\frac{\\sin (2 \\pi k x)}{k}=\\frac{i \\left(\\log \\left(1-e^{-2 i \\pi x}\\right)-\\log \\left(1-e^{2 i \\pi x}\\right)\\right)}{2 \\pi }\\)<\/p>\n\n\n\n

which is of course intuitive owing to Riemann surfaces produced by the complex logarithm. I could not find the result but I am nearly certain it must be somewhere.<\/p>\n\n\n\n

Now to get an idea, let us examine the compensating function \\(f(x)= \\frac{i \\left(\\log \\left(1-e^{-2 i \\pi x}\\right)-\\log \\left(1-e^{2 i \\pi x}\\right)\\right)}{2 \\pi } \\)<\/p>\n\n\n\n

\"\"<\/a>
<\/a><\/a><\/figcaption><\/figure>\n\n\n\n

 And of course, the complex logarithm (here is the standard function, just to illustrate):<\/p>\n\n\n\n

\"\"<\/a><\/figure>\n","protected":false},"excerpt":{"rendered":"

An interesting discovery, thanks to a problem presented by K. Srinivasa Raghava: A standard result for \\(x\\) real, where \\(\\lfloor .\\rfloor\\) is the floor function: \\(\\lfloor x\\rfloor =\\frac{1}{\\pi }\\sum _{k=1}^{\\infty } \\frac{\\sin (2 \\pi k x)}{k}+x-\\frac{1}{2}\\). Now it so happens that: \\(\\frac{1}{\\pi }\\sum _{k=1}^{\\infty } \\frac{\\sin (2 \\pi k x)}{k}=\\frac{i \\left(\\log \\left(1-e^{-2 i \\pi x}\\right)-\\log … Continue reading “Floor functions using complex logarithms”<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_bbp_topic_count":0,"_bbp_reply_count":0,"_bbp_total_topic_count":0,"_bbp_total_reply_count":0,"_bbp_voice_count":0,"_bbp_anonymous_reply_count":0,"_bbp_topic_count_hidden":0,"_bbp_reply_count_hidden":0,"_bbp_forum_subforum_count":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-22","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/fooledbyrandomness.com\/blog\/index.php\/wp-json\/wp\/v2\/posts\/22","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/fooledbyrandomness.com\/blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/fooledbyrandomness.com\/blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/fooledbyrandomness.com\/blog\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/fooledbyrandomness.com\/blog\/index.php\/wp-json\/wp\/v2\/comments?post=22"}],"version-history":[{"count":0,"href":"https:\/\/fooledbyrandomness.com\/blog\/index.php\/wp-json\/wp\/v2\/posts\/22\/revisions"}],"wp:attachment":[{"href":"https:\/\/fooledbyrandomness.com\/blog\/index.php\/wp-json\/wp\/v2\/media?parent=22"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fooledbyrandomness.com\/blog\/index.php\/wp-json\/wp\/v2\/categories?post=22"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fooledbyrandomness.com\/blog\/index.php\/wp-json\/wp\/v2\/tags?post=22"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}