{"id":26207,"date":"2025-07-16T22:15:48","date_gmt":"2025-07-16T14:15:48","guid":{"rendered":"http:\/\/www.jdcui.com\/?p=26207"},"modified":"2025-11-25T12:25:18","modified_gmt":"2025-11-25T04:25:18","slug":"%e6%95%b0%e5%ad%a6-%e5%af%bc%e6%95%b0%e5%8f%8a%e4%ba%8c%e9%98%b6%e5%af%bc%e6%95%b0%e7%9a%84laplace%e5%8f%98%e6%8d%a2%e5%85%ac%e5%bc%8f","status":"publish","type":"post","link":"http:\/\/www.jdcui.com\/?p=26207","title":{"rendered":"[\u6570\u5b66] Laplace Transform [\u62c9\u666e\u62c9\u65af\u53d8\u6362\u516c\u5f0f]"},"content":{"rendered":"<p><script><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span>\nMathJax = {\n  tex: {\n    inlineMath: [['$', '$'], ['\\\\(', '\\\\)']]\n  }\n};<span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span>\n<\/script><br \/>\n<script id=\"MathJax-script\" src=\"http:\/\/www.jdcui.com\/wp-content\/MathJax_3_1_2\/MathJax-master\/es5\/tex-mml-chtml.js\" async=\"\"><\/script><\/p>\n<p><span style=\"color: #ff00ff; background-color: #ccffcc;\"><strong>\u5b9e\u5e72\u3001\u5b9e\u8df5\u3001\u79ef\u7d2f\u3001\u601d\u8003\u3001\u521b\u65b0\u3002<\/strong><\/span><\/p>\n<hr \/>\n<p>\u62c9\u666e\u62c9\u65af\u53d8\u6362\uff0c\u5bfc\u6570\u53ca\u4e8c\u9636\u5bfc\u6570\u7684\u62c9\u666e\u62c9\u65af\u53d8\u6362\u516c\u5f0f\u6574\u7406\uff1a<\/p>\n<p>#### 1. \u62c9\u666e\u62c9\u65af\u53d8\u6362\u5b9a\u4e49<br \/>\n\u51fd\u6570 $f(t)$\u7684\u62c9\u666e\u62c9\u65af\u53d8\u6362\u5b9a\u4e49\u4e3a\uff1a<br \/>\n$$<br \/>\n\\mathcal{L}\\{f(t)\\} = F(s) = \\int_0^\\infty e^{-st} f(t) dt<br \/>\n$$<br \/>\n\u5176\u4e2d \\( s = \\sigma + j\\omega \\)\u662f\u590d\u9891\u7387\u53c2\u6570\u3002<\/p>\n<p>#### 2. \u4e00\u9636\u5bfc\u6570\u7684\u62c9\u666e\u62c9\u65af\u53d8\u6362<br \/>\n**\u63a8\u5bfc\u8fc7\u7a0b**\uff1a<br \/>\n$$<br \/>\n\\begin{align*}<br \/>\n\\mathcal{L}\\{f'(t)\\}<br \/>\n&amp;= \\int_0^\\infty e^{-st} f'(t) dt \\\\<br \/>\n&amp;= \\left[ e^{-st} f(t) \\right]_0^\\infty + s \\int_0^\\infty e^{-st} f(t) dt \\quad \\text{(\u5206\u90e8\u79ef\u5206)} \\\\<br \/>\n&amp;= \\lim_{t \\to \\infty} \\left( e^{-st} f(t) \\right) &#8211; f(0) + s F(s) \\\\<br \/>\n&amp;= s F(s) &#8211; f(0) \\quad \\text{(\u6536\u655b\u6761\u4ef6: } \\lim_{t \\to \\infty} e^{-st} f(t) = 0\\text{)}<br \/>\n\\end{align*}<br \/>\n$$<\/p>\n<p>**\u7ed3\u679c**\uff1a<br \/>\n$$<br \/>\n\\boxed{\\mathcal{L}\\{f'(t)\\} = sF(s) &#8211; f(0)}<br \/>\n$$<\/p>\n<p>&nbsp;<\/p>\n<p>#### 3. \u4e8c\u9636\u5bfc\u6570\u7684\u62c9\u666e\u62c9\u65af\u53d8\u6362<br \/>\n**\u63a8\u5bfc\u8fc7\u7a0b**\uff1a<br \/>\n$$<br \/>\n\\begin{align*}<br \/>\n\\mathcal{L}\\{f&#8221;(t)\\}<br \/>\n&amp;= \\mathcal{L}\\left\\{ \\frac{d}{dt}[f'(t)] \\right\\} \\\\<br \/>\n&amp;= s \\cdot \\mathcal{L}\\{f'(t)\\} &#8211; f'(0) \\quad \\text{(\u5e94\u7528\u4e00\u9636\u7ed3\u8bba)} \\\\<br \/>\n&amp;= s \\left[ sF(s) &#8211; f(0) \\right] &#8211; f'(0) \\\\<br \/>\n&amp;= s^2 F(s) &#8211; s f(0) &#8211; f'(0)<br \/>\n\\end{align*}<br \/>\n$$<\/p>\n<p>**\u7ed3\u679c**\uff1a<br \/>\n$$<br \/>\n\\boxed{\\mathcal{L}\\{f&#8221;(t)\\} = s^2F(s) &#8211; sf(0) &#8211; f'(0)}<br \/>\n$$<\/p>\n<p>\/\/<\/p>\n<p><span style=\"color: #ff00ff;\">PS. \u62c9\u666e\u62c9\u65af\u53d8\u6362\uff0c\u8868\u793a\u7684\u662f\u8f93\u5165\u4e0e\u54cd\u5e94\u5728\u590d\u6570\u57df\u5185\u7684\u6620\u5c04\u5173\u7cfb\u3002<\/span><\/p>\n<section>\n<section>\n<section>\n<section>\n<section><\/section>\n<\/section>\n<\/section>\n<\/section>\n<section>\n<section>\n<hr \/>\n<p style=\"text-align: center;\"><span style=\"font-size: 14pt; background-color: #999999; color: #000000;\"><strong>\u5173\u4e8e\u6211\u4eec<\/strong><\/span><\/p>\n<p style=\"text-align: center;\">\u8d85\u9650\u590d\u6742\u9ad8\u5c42\u7ed3\u6784\u8bbe\u8ba1 | \u00a0\u7f8e\u6807\u6b27\u6807\u7ed3\u6784\u8bbe\u8ba1| \u8f6f\u4ef6\u5b9a\u5236\u5f00\u53d1| \u73af\u8bc4\u51cf\u632f\u63a7\u5236 |\u4eba\u884c\u53ca\u98ce\u81f4\u632f\u52a8\u63a7\u5236 | \u51cf\u9694\u9707\u8bbe\u8ba1 | \u65bd\u5de5\u8fc7\u7a0b\u6a21\u62df | \u5c0f\u54c1\u94a2\u7ed3\u6784 | \u6709\u9650\u5143\u4eff\u771f\u5206\u6790 | BIM\u4e0eGH\u53c2\u6570\u5316 | \u5927\u9707\u5f39\u5851\u6027\u5206\u6790<\/p>\n<p style=\"text-align: center;\"><a href=\"http:\/\/www.jdcui.com\/wp-content\/uploads\/2017\/01\/QRCODE.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3636 alignnone\" src=\"http:\/\/www.jdcui.com\/wp-content\/uploads\/2017\/01\/QRCODE.jpg\" alt=\"WeChat_QRCode\" width=\"250\" height=\"255\" \/><\/a><\/p>\n<p style=\"text-align: center;\">https:\/\/www.jdcui.com<\/p>\n<p style=\"text-align: center;\">\u5408\u4f5c\u53ca\u6280\u672f\u54a8\u8be2<\/p>\n<p style=\"text-align: center;\">COOPERATION &amp; CONTACT<\/p>\n<p style=\"text-align: center;\">E-mail\uff1ajidong_cui@163.com<\/p>\n<p style=\"text-align: center;\">WeChat &amp; Tel: 13450468449<\/p>\n<\/section>\n<\/section>\n<\/section>\n","protected":false},"excerpt":{"rendered":"<p>\u5b9e\u5e72\u3001\u5b9e\u8df5\u3001\u79ef\u7d2f\u3001\u601d\u8003\u3001\u521b\u65b0\u3002 \u62c9\u666e\u62c9\u65af\u53d8\u6362\uff0c\u5bfc\u6570\u53ca\u4e8c\u9636\u5bfc\u6570\u7684\u62c9\u666e\u62c9\u65af\u53d8\u6362\u516c\u5f0f\u6574\u7406\uff1a #### 1. \u62c9\u666e\u62c9\u65af\u53d8\u6362\u5b9a\u4e49 \u51fd\u6570 $f(t)$\u7684\u62c9\u666e\u62c9\u65af\u53d8\u6362\u5b9a\u4e49\u4e3a\uff1a $$ \\mathcal{L}\\{f(t)\\} = F(s) = \\int_0^\\infty e^{-st} f(t) dt $$ \u5176\u4e2d \\( s = \\sigma + j\\omega \\)\u662f\u590d\u9891\u7387\u53c2\u6570\u3002 #### 2. \u4e00\u9636\u5bfc\u6570\u7684\u62c9\u666e\u62c9\u65af\u53d8\u6362 **\u63a8\u5bfc\u8fc7\u7a0b**\uff1a $$ &#8230;<\/p>\n","protected":false},"author":1,"featured_media":26217,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[4410,4409,4408,135],"class_list":["post-26207","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-math-geometry","tag-laplace-transform","tag-laplace","tag-4408","tag-135"],"aioseo_notices":[],"views":1230,"_links":{"self":[{"href":"http:\/\/www.jdcui.com\/index.php?rest_route=\/wp\/v2\/posts\/26207","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.jdcui.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.jdcui.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.jdcui.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.jdcui.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=26207"}],"version-history":[{"count":0,"href":"http:\/\/www.jdcui.com\/index.php?rest_route=\/wp\/v2\/posts\/26207\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/www.jdcui.com\/index.php?rest_route=\/wp\/v2\/media\/26217"}],"wp:attachment":[{"href":"http:\/\/www.jdcui.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=26207"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.jdcui.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=26207"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.jdcui.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=26207"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}