{"id":249665,"date":"2025-07-09T18:36:22","date_gmt":"2025-07-09T18:36:22","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=249665"},"modified":"2025-07-09T18:36:24","modified_gmt":"2025-07-09T18:36:24","slug":"laplace-transform-of-periodic-functions-if-satisfies-for-all-where-is-some-fixed-positive-number-then-is-called-a-periodic-function-with-period","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/09\/laplace-transform-of-periodic-functions-if-satisfies-for-all-where-is-some-fixed-positive-number-then-is-called-a-periodic-function-with-period\/","title":{"rendered":"Laplace Transform of Periodic Functions If\u00a0\u00a0satisfies\u00a0\u00a0for all\u00a0, where\u00a0\u00a0is some fixed positive number, then\u00a0\u00a0is called a periodic function with period\u00a0."},"content":{"rendered":"\n<pre id=\"preorder-ask-header-text\" class=\"wp-block-preformatted\">. Laplace Transform of Periodic Functions If&nbsp;&nbsp;satisfies&nbsp;&nbsp;for all&nbsp;, where&nbsp;&nbsp;is some fixed positive number, then&nbsp;&nbsp;is called a periodic function with period&nbsp;.&nbsp;<\/pre>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/image-348.png\" alt=\"\" class=\"wp-image-249666\"\/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The problem requires us to prove the formula for the Laplace Transform of a periodic function. A function f(t) is periodic with a period T if f(t + T) = f(t) for all t \u2265 0. The formula to be proven is:<\/p>\n\n\n\n<p>L{f(t)} = [\u222b\u2080\u1d40 e\u207b\u02e2\u1d57 f(t) dt] \/ [1 &#8211; e\u207b\u02e2\u1d40]<\/p>\n\n\n\n<p>Here is the step by step proof and explanation.<\/p>\n\n\n\n<p>The proof begins with the fundamental definition of the Laplace transform, which is an improper integral from zero to infinity:<\/p>\n\n\n\n<p>L{f(t)} = \u222b\u2080^\u221e e\u207b\u02e2\u1d57 f(t) dt<\/p>\n\n\n\n<p>The key insight for a periodic function is to decompose this single infinite integral into a sum of integrals over each period. We can break the interval [0, \u221e) into an infinite sequence of intervals: [0, T], [T, 2T], [2T, 3T], and so on. This allows us to rewrite the transform as an infinite series:<\/p>\n\n\n\n<p>L{f(t)} = \u03a3\u2099\u208c\u2080^\u221e \u222b\u2099\u209c^\u207d\u207f\u207a\u00b9\u207e\u1d40 e\u207b\u02e2\u1d57 f(t) dt<\/p>\n\n\n\n<p>To evaluate this series, we can simplify each integral term. We use a change of variables to shift the integration interval for each term back to [0, T]. Let us define a new variable u such that t = u + nT. From this, we get dt = du. We also update the limits of integration: when t = nT, u = 0, and when t = (n+1)T, u = T.<\/p>\n\n\n\n<p>Substituting these into the integral gives:<br>\u222b\u2080\u1d40 e\u207b\u02e2(\u1d58\u207a\u207f\u1d40) f(u + nT) du<\/p>\n\n\n\n<p>Now we use the given property that the function is periodic. Since f(u + nT) = f(u), we can simplify the expression. We also use the property of exponents to separate e\u207b\u02e2(\u1d58\u207a\u207f\u1d40) into e\u207b\u02e2\u1d58 * e\u207b\u02e2\u207f\u1d40. The integral becomes:<br>\u222b\u2080\u1d40 e\u207b\u02e2\u1d58 * e\u207b\u02e2\u207f\u1d40 f(u) du<\/p>\n\n\n\n<p>The term e\u207b\u02e2\u207f\u1d40 is a constant with respect to the integration variable u, so we can factor it outside the integral:<br>e\u207b\u02e2\u207f\u1d40 \u222b\u2080\u1d40 e\u207b\u02e2\u1d58 f(u) du<\/p>\n\n\n\n<p>Substituting this simplified form back into our infinite series, we get:<br>L{f(t)} = \u03a3\u2099\u208c\u2080^\u221e [e\u207b\u02e2\u207f\u1d40 \u222b\u2080\u1d40 e\u207b\u02e2\u1d58 f(u) du]<\/p>\n\n\n\n<p>The integral \u222b\u2080\u1d40 e\u207b\u02e2\u1d58 f(u) du is a constant value with respect to the summation index n, so it can be factored out of the series:<br>L{f(t)} = (\u222b\u2080\u1d40 e\u207b\u02e2\u1d58 f(u) du) * (\u03a3\u2099\u208c\u2080^\u221e e\u207b\u02e2\u207f\u1d40)<\/p>\n\n\n\n<p>The remaining summation, \u03a3\u2099\u208c\u2080^\u221e (e\u207b\u02e2\u1d40)\u207f, is a standard geometric series with a common ratio r = e\u207b\u02e2\u1d40. For the Laplace transform to converge, s must be sufficiently large and positive, which ensures that |r| &lt; 1. The sum of an infinite geometric series is 1 \/ (1 &#8211; r). Therefore:<br>\u03a3\u2099\u208c\u2080^\u221e (e\u207b\u02e2\u1d40)\u207f = 1 \/ (1 &#8211; e\u207b\u02e2\u1d40)<\/p>\n\n\n\n<p>Finally, we substitute this result back into our equation and replace the dummy integration variable u with t to yield the final formula:<\/p>\n\n\n\n<p>L{f(t)} = [\u222b\u2080\u1d40 e\u207b\u02e2\u1d57 f(t) dt] \/ [1 &#8211; e\u207b\u02e2\u1d40]<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-25.jpeg\" alt=\"\" class=\"wp-image-249667\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>. Laplace Transform of Periodic Functions If&nbsp;&nbsp;satisfies&nbsp;&nbsp;for all&nbsp;, where&nbsp;&nbsp;is some fixed positive number, then&nbsp;&nbsp;is called a periodic function with period&nbsp;.&nbsp; The Correct Answer and Explanation is: The problem requires us to prove the formula for the Laplace Transform of a periodic function. A function f(t) is periodic with a period T if f(t + T) [&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-249665","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/249665","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=249665"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/249665\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=249665"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=249665"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=249665"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}