{"id":11779,"date":"2023-10-05T17:12:06","date_gmt":"2023-10-05T09:12:06","guid":{"rendered":"https:\/\/blog.iyatt.com\/?p=11779"},"modified":"2025-08-15T22:14:57","modified_gmt":"2025-08-15T14:14:57","slug":"wallis-%e5%85%ac%e5%bc%8f-%e5%ae%9a%e7%a7%af%e5%88%86%e6%b1%82%e5%af%bc","status":"publish","type":"post","link":"https:\/\/blog.iyatt.com\/?p=11779","title":{"rendered":"Wallis \u516c\u5f0f\uff08\u201c\u70b9\u706b\u516c\u5f0f\u201d\uff09 &#8211; \u5b9a\u79ef\u5206\u8ba1\u7b97"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 ez-toc-wrap-center counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">\u76ee\u5f55<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/blog.iyatt.com\/?p=11779\/#Wallis_%E5%85%AC%E5%BC%8F\" >Wallis \u516c\u5f0f<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/blog.iyatt.com\/?p=11779\/#%E5%8F%82%E8%80%83\" >\u53c2\u8003<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/blog.iyatt.com\/?p=11779\/#%E5%85%AC%E5%BC%8F\" >\u516c\u5f0f<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/blog.iyatt.com\/?p=11779\/#%E8%AF%81%E6%98%8E\" >\u8bc1\u660e<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h1><span class=\"ez-toc-section\" id=\"Wallis_%E5%85%AC%E5%BC%8F\"><\/span>Wallis \u516c\u5f0f<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u56e0\u4e3a\u5f20\u5b87\u8001\u5e08\u53d6\u540d\u7684\u539f\u56e0\uff0c\u7f51\u4e0a\u4e5f\u4f20\u53eb\u201c\u70b9\u706b\u516c\u5f0f\u201d\uff0c\u50cf\u5012\u70b9\u706b\u8ba1\u65f6\u4e00\u6837\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E5%8F%82%E8%80%83\"><\/span>\u53c2\u8003<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u300a\u9ad8\u7b49\u6570\u5b66\u300b\u540c\u6d4e\u5927\u5b66 \u7b2c\u4e03\u7248\u4e0a\u518c:<\/p>\n<ul>\n<li>\u7b2c 253 \u9875 \u4f8b 12<\/li>\n<li>\u7b2c 249 \u9875 \u4f8b 6<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"%E5%85%AC%E5%BC%8F\"><\/span>\u516c\u5f0f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\int_0^\\frac{\\pi}{2}\\sin^nxdx=\\int_0^\\frac{\\pi}{2}\\cos^nxdx=\n\\begin{cases}\n\\frac{n-1}{n}\\cdot\\frac{n-3}{n-2}\\cdots\\frac{3}{4}\\cdot\\frac{1}{2}\\cdot\\frac{\\pi}{2}\uff0cn\u4e3a\u6b63\u5076\u6570 \\\\\n\\frac{n-1}{n}\\cdot\\frac{n-3}{n-2}\\cdots\\frac{4}{5}\\cdot\\frac{2}{3}\uff0cn\u4e3a\u6b63\u5947\u6570\n\\end{cases}<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E8%AF%81%E6%98%8E\"><\/span>\u8bc1\u660e<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n\\begin{aligned}\nI_n&amp;=\\int_0^\\frac{\\pi}{2}\\sin^nxdx \\\\\n&amp;=\\int_0^\\frac{\\pi}{2}\\sin^{n-1}xd(-\\cos x) \\\\\n&amp;\\xlongequal{\u5206\u90e8\u79ef\u5206}(-\\sin^{n-1}x\\cos x)|_0^\\frac{\\pi}{2}+(n-1)\\int_0^\\frac{\\pi}{2}\\cos^2 x\\cdot\\sin^{n-2}xdx \\\\\n&amp;=(n-1)\\int_0^\\frac{\\pi}{2}\\sin^{n-2}(1-\\sin^2x)dx \\\\\n&amp;=(n-1)(\\int_0^\\frac{\\pi}{2}\\sin^{n-2}xdx-\\int_0^\\frac{\\pi}{2}\\sin^nxdx) \\\\\n&amp;=(n-1)I_{n-2}-(n-1)I_n \\\\\n\\end{aligned}\n\\\\\n\\Downarrow \\\\\nI_n=\\frac{n-1}{n}I_{n-2} \\\\\n\\\\\nI_{n-2}=\\frac{n-3}{n-2}I_{n-4}\n\\end{array}<\/code><\/pre>\n<p>\u7ee7\u7eed\u9012\u63a8\u53ef\u4ee5\u5f97\u5230\uff1a<\/p>\n<pre><code class=\"language-katex\">\\begin{array}{l}\nI_{2m}=\\frac{2m-1}{2m}\\cdot\\frac{2m-3}{2m-2}\\cdots\\frac{3}{4}\\cdot\\frac{1}{2}\\cdot I_0 \\\\\n\\\\\nI_{2m+1}=\\frac{2m-1}{2m}\\cdot\\frac{2m-3}{2m-2}\\cdots\\frac{4}{5}\\cdot\\frac{2}{3}\\cdot I_1 \\\\\n\\\\\n\u5176\u4e2d \\\\\n\\\\\nI_0=\\int_0^{\\frac{\\pi}{2}}dx=\\frac{\\pi}{2} \\\\\n\\\\\nI_1=\\int_0^{\\frac{\\pi}{2}}\\sin xdx=1\n\\end{array}<\/code><\/pre>\n<p>\u5219\u6709<\/p>\n<pre><code class=\"language-katex\">\\begin{array}{l}\nI_{2m}=\\frac{2m-1}{2m}\\cdot\\frac{2m-3}{2m-2}\\cdots\\frac{3}{4}\\cdot\\frac{1}{2}\\cdot\\frac{\\pi}{2} \\\\\n\\\\\nI_{2m+1}=\\frac{2m-1}{2m}\\cdot\\frac{2m-3}{2m-2}\\cdots\\frac{4}{5}\\cdot\\frac{2}{3}\n\\end{array}<\/code><\/pre>\n<p>\u53e6\u5916<\/p>\n<pre><code class=\"language-katex\">\\int_0^\\frac{\\pi}{2}\\sin^nxdx\\xlongequal{x=\\frac{\\pi}{2}-t}-\\int_\\frac{\\pi}{2}^0\\sin^n(\\frac{\\pi}{2}-t)dt=\\int_0^\\frac{\\pi}{2}\\sin^n(\\frac{\\pi}{2}-t)dt\\xlongequal{\u8bf1\u5bfc\u516c\u5f0f}\\int_0^\\frac{\\pi}{2}\\cos^ntdt<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Wallis \u516c\u5f0f \u56e0\u4e3a\u5f20\u5b87\u8001\u5e08\u53d6\u540d\u7684\u539f\u56e0\uff0c\u7f51\u4e0a\u4e5f\u4f20\u53eb\u201c\u70b9\u706b\u516c\u5f0f\u201d\uff0c\u50cf\u5012\u70b9\u706b\u8ba1\u65f6\u4e00\u6837\u3002 \u53c2\u8003 \u300a\u9ad8\u7b49\u6570\u5b66\u300b\u540c\u6d4e [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"zakra_page_container_layout":"customizer","zakra_page_sidebar_layout":"customizer","zakra_remove_content_margin":false,"zakra_sidebar":"customizer","zakra_transparent_header":"customizer","zakra_logo":0,"zakra_main_header_style":"default","zakra_menu_item_color":"","zakra_menu_item_hover_color":"","zakra_menu_item_active_color":"","zakra_menu_active_style":"","zakra_page_header":true,"_lmt_disableupdate":"no","_lmt_disable":"no","footnotes":""},"categories":[1,612],"tags":[],"class_list":["post-11779","post","type-post","status-publish","format-standard","hentry","category-all","category-612"],"modified_by":"IYATT-yx","_links":{"self":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/11779","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11779"}],"version-history":[{"count":8,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/11779\/revisions"}],"predecessor-version":[{"id":21424,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/11779\/revisions\/21424"}],"wp:attachment":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11779"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11779"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11779"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}