{"id":13185,"date":"2025-02-09T14:49:59","date_gmt":"2025-02-09T19:49:59","guid":{"rendered":"https:\/\/spinor.info\/weblog\/?p=13185"},"modified":"2025-02-09T14:49:59","modified_gmt":"2025-02-09T19:49:59","slug":"math-is-evil","status":"publish","type":"post","link":"https:\/\/spinor.info\/weblog\/?p=13185","title":{"rendered":"Math is evil"},"content":{"rendered":"

I was reading about Borwein integrals.<\/p>\n

Here’s a nice result:<\/p>\n

$$\\int_0^\\infty dx\\,\\frac{\\sin x}{x}=\\frac{\\pi}{2}.$$<\/p>\n

Neat, is it not. Here’s another:<\/p>\n

$$\\int_0^\\infty dx\\,\\frac{\\sin x}{x}\\frac{\\sin (x\/3)}{x\/3}=\\frac{\\pi}{2}.$$<\/p>\n

Jumping a bit ahead, how about<\/p>\n

$$\\int_0^\\infty dx\\,\\frac{\\sin x}{x}\\frac{\\sin (x\/3)}{x\/3}…\\frac{\\sin (x\/13)}{x\/13}=\\frac{\\pi}{2}.$$<\/p>\n

Shall we conclude, based on these examples, that<\/p>\n

$$\\int_0^\\infty dx\\,\\prod\\limits_{k=0}^\\infty\\frac{\\sin (x\/[2k+1])}{x\/[2k+1]}=\\frac{\\pi}{2}?$$<\/p>\n

Not so fast. First, consider that<\/p>\n

$$\\int_0^\\infty dx\\,\\frac{\\sin x}{x}\\frac{\\sin (x\/3)}{x\/3}…\\frac{\\sin (x\/15)}{x\/15}=\\frac{935615849426881477393075728938}{935615849440640907310521750000}\\frac{\\pi}{2}\\approx\\frac{\\pi}{2}-2.31\\times 10^{-11}.$$<\/p>\n

Or how about<\/p>\n

\\begin{align}
\n\\int_0^\\infty&dx\\,\\cos x\\,\\frac{\\sin x}{x}=\\frac{\\pi}{4},\\\\
\n\\int_0^\\infty&dx\\,\\cos x\\,\\frac{\\sin x}{x}\\frac{\\sin (x\/3)}{x\/3}=\\frac{\\pi}{4},\\\\
\n…\\\\
\n\\int_0^\\infty&dx\\,\\cos x\\,\\frac{\\sin x}{x}…\\frac{\\sin (x\/111)}{x\/111}=\\frac{\\pi}{4},
\n\\end{align}<\/p>\n

but then,<\/p>\n

$$\\int_0^\\infty dx\\,\\cos x\\,\\frac{\\sin x}{x}…\\frac{\\sin (x\/113)}{x\/113}\\approx\\frac{\\pi}{4}-1.1162\\times 10^{-138}.$$<\/p>\n

There is a lot more about Borwein integrals on Wikipedia<\/a>,\u00a0but I think even these few examples are sufficient to convince us that, never mind the actual, physical universe, even the Platonic universe of mathematical truths is fundamentally evil and unreasonable.<\/p>\n

\"\"<\/p>\n<\/fb:like>","protected":false},"excerpt":{"rendered":"

I was reading about Borwein integrals. Here’s a nice result: $$\\int_0^\\infty dx\\,\\frac{\\sin x}{x}=\\frac{\\pi}{2}.$$ Neat, is it not. Here’s another: $$\\int_0^\\infty dx\\,\\frac{\\sin x}{x}\\frac{\\sin (x\/3)}{x\/3}=\\frac{\\pi}{2}.$$ Jumping a bit ahead, how about $$\\int_0^\\infty dx\\,\\frac{\\sin x}{x}\\frac{\\sin (x\/3)}{x\/3}…\\frac{\\sin (x\/13)}{x\/13}=\\frac{\\pi}{2}.$$ Shall we conclude, based on these examples, that $$\\int_0^\\infty dx\\,\\prod\\limits_{k=0}^\\infty\\frac{\\sin (x\/[2k+1])}{x\/[2k+1]}=\\frac{\\pi}{2}?$$ Not so fast. First, consider that $$\\int_0^\\infty dx\\,\\frac{\\sin x}{x}\\frac{\\sin (x\/3)}{x\/3}…\\frac{\\sin […]<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[30],"tags":[],"class_list":["post-13185","post","type-post","status-publish","format-standard","hentry","category-mathematics","category-30-id","post-seq-1","post-parity-odd","meta-position-corners","fix"],"_links":{"self":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts\/13185","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=13185"}],"version-history":[{"count":9,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts\/13185\/revisions"}],"predecessor-version":[{"id":13196,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts\/13185\/revisions\/13196"}],"wp:attachment":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13185"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13185"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13185"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}