The Gamma function is an amazing integral: <\/p>\n\n\n\n
Gamma(x) = INTEGRALt=0 to infinity<\/sub> tx-1<\/sup>e<\/em>-t<\/sup> dt .<\/p>\n\n\n\n Using integration by parts, you can show that this function satisfies the property<\/p>\n\n\n\n Gamma(x) = (x-1) Gamma(x-1).<\/p>\n\n\n\n Using Gamma(1)=1, you can calculate Gamma(2), Gamma(3),… Surprise: the Gamma function satisfies Gamma(n) = Factorial(n-1). So you can think of the Gamma function as being a continuous form of the\u00a0factorial\u00a0function. It satisfies lots of cool properties; here is just one:\u00a0<\/p>\n\n\n\n Gamma(1\/2) = Sqrt[Pi].<\/p>\n\n\n\n Presentation Suggestions:<\/strong> The Math Behind the Fact:<\/strong> How to Cite this Page:<\/strong> Fun Fact suggested by: <\/strong> The Gamma function is an amazing integral: Gamma(x) = INTEGRALt=0 to infinity tx-1e-t dt . Using integration by parts, you can show...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[20,7,115,4],"class_list":["post-377","page","type-page","status-publish","hentry","tag-analysis","tag-calculus","tag-continuous-factorial","tag-medium"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/377","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=377"}],"version-history":[{"count":4,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/377\/revisions"}],"predecessor-version":[{"id":1458,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/377\/revisions\/1458"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=377"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=377"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Does this remind you of anything?<\/p>\n\n\n\n
(I would have used the notation “!” but you might think I was just excited!)<\/p>\n\n\n\n
Calculus students might be challenged to compute Gamma(2), Gamma(3), etc. and discover the connection with the factorial function. You may wish to assign the integration by parts as a homework exercise prior to presenting this Fun Fact.<\/p>\n\n\n\n
The Gamma function is an important function in analysis, complex analysis, combinatorics, and probability.<\/p>\n\n\n\n
Su, Francis E., et al. “Gamma Function.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n
Michael Moody <\/p>\n","protected":false},"excerpt":{"rendered":"