{"id":183262,"date":"2025-01-16T06:43:53","date_gmt":"2025-01-16T06:43:53","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=183262"},"modified":"2025-01-16T06:43:55","modified_gmt":"2025-01-16T06:43:55","slug":"give-the-laplace-transform-of-the-unit-step-function","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/16\/give-the-laplace-transform-of-the-unit-step-function\/","title":{"rendered":"Give the Laplace transform of the unit-step function"},"content":{"rendered":"\n<p>Give the Laplace transform of the unit-step function, us(t).<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>The <strong>Laplace transform<\/strong> of the unit-step function us(t)u_s(t) (also denoted u(t)u(t) or H(t)H(t)) is given by: L{u(t)}=1s,fors&gt;0\\mathcal{L}\\{u(t)\\} = \\frac{1}{s}, \\quad \\text{for} \\quad s &gt; 0<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>The unit-step function u(t)u(t), also known as the Heaviside function, is defined as: u(t)={0,t&lt;01,t\u22650u(t) = \\begin{cases} 0, &amp; t &lt; 0 \\\\ 1, &amp; t \\geq 0 \\end{cases}<\/p>\n\n\n\n<p>This function is used to model a sudden change or a &#8220;switch&#8221; that turns on at t=0t = 0 and remains on thereafter.<\/p>\n\n\n\n<p>The <strong>Laplace transform<\/strong> of a time-domain function f(t)f(t) is defined as: L{f(t)}=\u222b0\u221ef(t)e\u2212st\u2009dt\\mathcal{L}\\{f(t)\\} = \\int_{0}^{\\infty} f(t) e^{-st} \\, dt<\/p>\n\n\n\n<p>For the unit-step function, we have f(t)=u(t)f(t) = u(t), so: L{u(t)}=\u222b0\u221eu(t)e\u2212st\u2009dt\\mathcal{L}\\{u(t)\\} = \\int_{0}^{\\infty} u(t) e^{-st} \\, dt<\/p>\n\n\n\n<p>Since u(t)=1u(t) = 1 for t\u22650t \\geq 0, the integral simplifies to: L{u(t)}=\u222b0\u221ee\u2212st\u2009dt\\mathcal{L}\\{u(t)\\} = \\int_{0}^{\\infty} e^{-st} \\, dt<\/p>\n\n\n\n<p>To solve this, we use the standard integral: \u222b0\u221ee\u2212st\u2009dt=1s,fors&gt;0\\int_{0}^{\\infty} e^{-st} \\, dt = \\frac{1}{s}, \\quad \\text{for} \\quad s &gt; 0<\/p>\n\n\n\n<p>Thus, the Laplace transform of the unit-step function is: L{u(t)}=1s,fors&gt;0\\mathcal{L}\\{u(t)\\} = \\frac{1}{s}, \\quad \\text{for} \\quad s &gt; 0<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Key Points:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The unit-step function represents a sudden switch or step at t=0t = 0.<\/li>\n\n\n\n<li>The Laplace transform of u(t)u(t) is 1s\\frac{1}{s}, which is a simple rational function in the frequency domain.<\/li>\n\n\n\n<li>The transform is valid for s>0s > 0 to ensure the integral converges.<\/li>\n<\/ul>\n\n\n\n<p>This result is fundamental in systems and control theory, where it is used to analyze systems that &#8220;start&#8221; at a given time t=0t = 0.<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Give the Laplace transform of the unit-step function, us(t). The Correct Answer and Explanation is : The Laplace transform of the unit-step function us(t)u_s(t) (also denoted u(t)u(t) or H(t)H(t)) is given by: L{u(t)}=1s,fors&gt;0\\mathcal{L}\\{u(t)\\} = \\frac{1}{s}, \\quad \\text{for} \\quad s &gt; 0 Explanation: The unit-step function u(t)u(t), also known as the Heaviside function, is defined as: [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-183262","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183262","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=183262"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183262\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=183262"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=183262"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=183262"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}