{"id":804,"date":"2013-08-29T06:56:33","date_gmt":"2013-08-28T22:56:33","guid":{"rendered":"http:\/\/www.shuizilong.com\/house\/?p=804"},"modified":"2014-06-02T23:54:38","modified_gmt":"2014-06-02T15:54:38","slug":"complexifying-the-integral","status":"publish","type":"post","link":"https:\/\/www.shuizilong.com\/house\/archives\/complexifying-the-integral\/","title":{"rendered":"\u79ef\u5206\u590d\u5316"},"content":{"rendered":"

<\/p>\n

\"\"\/<\/a><\/p>\n

$$!\\int e^{-x}\\cos{x} \\mathrm{d}x$$<\/p>\n

\u3002\u3002\u3002\u9996\u5148\u6559\u6388\u628a\u5206\u90e8\u79ef\u5206\u9119\u89c6\u4e86\u4e00\u756a\u3002\u3002\u663e\u7136\u8fd9\u4e2a\u4e1c\u897f\u5206\u90e8\u79ef\u5206\u4e24\u6b21\u3001\u51fa\u73b0\u5faa\u73af\u5f97\u5230\u65b9\u7a0b\u7136\u540e\u5c31\u53ef\u4ee5\u89e3\u51fa\u6765\u3002\u3002<\/p>\n

\u5206\u90e8\u79ef\u5206<\/h4>\n

$$!\\int u\\mathrm{d}v = uv – \\int v\\mathrm{d}u$$<\/p>\n

\uff08\u3002\u3002\u8fd9\u4e2a\u4e1c\u897f\u5c45\u7136\u6b63\u597d\u5c31\u662f wiki<\/a> \u9875\u7684\u4f8b\u9898\u3002\u3002\u3002\u3002\u5176\u5b9e\u4e5f\u6ca1\u6559\u6388\u8bf4\u7684\u90a3\u4e48 tricky \u5566\u3002\u3002
\n\uff08\u3002\u3002\u6211\u4eec\u770b\u5230\u5206\u90e8\u79ef\u5206\u7684\u540e\u534a\u90e8\u5206\u3002\u3002\u3002\u5176\u5b9e\u5f62\u5f0f\u4e0a\u548c\u5f00\u59cb\u8981\u79ef\u7684\u4e1c\u897f\u4e00\u6837\u3002\u3002\u3002\u56e0\u6b64\u5206\u90e8\u79ef\u5206\u516c\u5f0f\u53ef\u4ee5\u53cd\u590d\u8fed\u4ee3\u3002\u3002\u66f4\u6613\u4e8e\u64cd\u4f5c\u7684\u65b9\u6cd5\u662f\u3002\u3002\u5bf9\u5176\u4e2d\u4e00\u4e2a\u51fd\u6570\u4e0d\u505c\u7684\u5fae\u5206\u3002\u3002\u53e6\u4e00\u4e2a\u51fd\u6570\u4e0d\u505c\u7684\u79ef\u5206\u3002\u3002\u628a\u5bf9\u5e94\u9879\u4e58\u8d77\u6765\u3002\u3002\u7136\u540e + – + – \u3002\u3002\u3002\u3002\u76f4\u5230\u51fa\u73b0 0 \u6216\u8005\u5faa\u73af\u3002\u3002\u3002\u8fd9\u4e2a\u65b9\u6cd5\u53ef\u4ee5\u53c2\u89c1\u3002\u3002 \u8208\u5927\u5fae\u79ef\u5206\u8bb2\u4e49<\/a>\u3002\u3002\u6211\u4eec\u770b\u5230\u5176\u5b9e\u5206\u90e8\u79ef\u5206\u5728\u79bb\u6563\u7684\u60c5\u51b5\u4e0b\u5bf9\u5e94\u7684\u5c31\u662f Abel \u6c42\u548c\u3002\u3002\u3002\u7136\u540e\u4e24\u8fb9\u5206\u522b\u5fae\u5206\u79ef\u5206\u6362\u6210\u5206\u522b\u5dee\u5206\u7d2f\u548c\u5c31\u884c\u4e86ww\u3002\u3002\u3002<\/p>\n

\u5fae\u5206\u9879 \uff08\u672c\u4f8b\u4e2d\u9009\u54ea\u4e2a\u5fae\u5206\u54ea\u4e2a\u79ef\u5206\u5f71\u54cd\u4e0d\u5927\u3002\u3002\u3002<\/p>\n

$$! \\cos{x} \\rightarrow -\\sin{x} \\rightarrow -\\cos{x} \\rightarrow … $$<\/p>\n

\u79ef\u5206\u9879\u3002\u3002<\/p>\n

$$! e^{-x} \\rightarrow -e^{-x} \\rightarrow e^{-x} \\rightarrow … $$<\/p>\n

\u3002\u3002\u4e24\u9879\u4e4b\u540e\u5c31\u5faa\u73af\u4e86\u3002\u3002\u3002\u4e8e\u662f\u8bbe\u8fd9\u4e2a\u79ef\u5206\u7b49\u4e8e $$S$$ \u3002\u3002\u3002\u6709\u3002\u3002<\/p>\n

$$!S = cos{x}-e^{-x} – -\\sin{x}e^{-x} – S $$<\/p>\n

$$! 2S = e^{-x}(\\sin{x} – \\cos{x})$$<\/p>\n

\u79ef\u5206\u590d\u5316<\/h4>\n

\u3002\u3002\u56e0\u4e3a $$\\cos{x}$$ \u53ef\u4ee5\u770b\u6210\u662f $$e^{ix}$$ \u4e5f\u5c31\u662f $$\\cos{x} + i\\sin{x}$$ \u7684\u5b9e\u90e8\u3002\u3002\u3002\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u5148\u628a\u8fd9\u4e2a\u8fd8\u539f\u6210\u539f\u6765\u7684\u5b9e\u53d8\u91cf\u590d\u503c\u51fd\u6570\u3002\u3002\u3002<\/p>\n

$$!\\int e^{-x}\\cos{x} \\mathrm{d}x = \\mathrm{Re}(\\int e^{(-1+i)x} \\mathrm{d}x) $$<\/p>\n

\u3002\u7136\u540e\u5c31\u53ea\u8981\u5bf9\u90a3\u4e2a\u6307\u6570\u51fd\u6570\u79ef\u5206\u3002\u3002\u518d\u53d6\u5b9e\u90e8\u5c31\u884c\u4e86\u3002\u3002\u3002\u3002<\/p>\n

$$! \\int e^{(-1+i)x} \\mathrm{d}x = \\frac{1}{-1+i} e^{(-1+i)x} = \\frac{-1-i}{2} e^{-x} (\\cos{x} + i\\sin{x})$$<\/p>\n

$$!Re(\\frac{-1-i}{2} e^{-x} (\\cos{x} + i\\sin{x})) = \\frac{e^{-x}}{2} (\\sin{x}-\\cos{x})$$<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false}}},"categories":[122],"tags":[],"class_list":["post-804","post","type-post","status-publish","format-standard","hentry","category-122"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p2tdP7-cY","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/804","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/comments?post=804"}],"version-history":[{"count":1,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/804\/revisions"}],"predecessor-version":[{"id":805,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/804\/revisions\/805"}],"wp:attachment":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/media?parent=804"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/categories?post=804"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/tags?post=804"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}