{"id":31617,"date":"2020-08-08T13:00:00","date_gmt":"2020-08-08T17:00:00","guid":{"rendered":"https:\/\/mathvault.ca\/?p=31617"},"modified":"2020-10-01T15:37:29","modified_gmt":"2020-10-01T19:37:29","slug":"laplace-transform","status":"publish","type":"post","link":"https:\/\/mathvault.ca\/laplace-transform\/","title":{"rendered":"Laplace Transform: A First Introduction"},"content":{"rendered":"\n<p>Let us take a moment to ponder how truly bizarre the <strong><a href=\"https:\/\/mathvault.ca\/hub\/higher-math\/math-symbols\/calculus-analysis-symbols\/#Key_Transforms\" target=\"_blank\" aria-label=\"Laplace transform (opens in a new tab)\" rel=\"noreferrer noopener\" class=\"rank-math-link\">Laplace transform<\/a><\/strong> is.<\/p>\n\n\n\n<p>You put in a sine and get an oddly simple, <strong>arbitrary-looking fraction<\/strong>. Why do we suddenly have squares?<\/p>\n\n\n\n<p>You look at the table of <strong>common Laplace transforms<\/strong> to find a pattern and you see no rhyme, no reason, no obvious link between different functions and their different, very different, results.<\/p>\n\n\n\n<p>What&#8217;s going on here?<\/p>\n\n\n\n<p>Or so we thought when we first encountered the cursive $\\mathcal{L}$ in school.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"888\" height=\"367\" src=\"https:\/\/mathvault.ca\/wp-content\/uploads\/Laplace-Transform.png\" alt=\"Laplace transform of function f\" class=\"wp-image-31622\" title=\"\"><\/figure><\/div>\n\n\n\n<div id=\"toc\"><div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_1 counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">Table of Contents<\/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: #000000;color:#000000\" 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: #000000;color:#000000\" 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-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mathvault.ca\/laplace-transform\/#What_does_the_Laplace_transform_do_really\">What does the Laplace transform do, really?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mathvault.ca\/laplace-transform\/#Some_Preliminary_Examples\">Some Preliminary Examples<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mathvault.ca\/laplace-transform\/#Looking_Inside_the_Laplace_Transform_of_Sine\">Looking Inside the Laplace Transform of Sine<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mathvault.ca\/laplace-transform\/#Diverging_Functions_What_the_Laplace_Transform_is_for\">Diverging Functions: What the Laplace Transform is for<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mathvault.ca\/laplace-transform\/#A_Transform_of_Unfathomable_Power\">A Transform of Unfathomable Power<\/a><\/li><\/ul><\/nav><\/div>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"What_does_the_Laplace_transform_do_really\"><\/span><a href=\"#toc\" class=\"rank-math-link\">What does the Laplace transform do, really?<\/a><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>At a high level, Laplace transform is an <strong><a aria-label=\"integral transform (opens in a new tab)\" href=\"https:\/\/en.wikipedia.org\/wiki\/Integral_transform#:~:text=In%20mathematics%2C%20an%20integral%20transform,in%20the%20original%20function%20space.\" target=\"_blank\" rel=\"noreferrer noopener\" class=\"rank-math-link\">integral transform<\/a><\/strong> mostly encountered in differential equations \u2014 in electrical engineering for instance \u2014 where electric circuits are represented as differential equations.<\/p>\n\n\n\n<p>In fact, it takes a <strong>time-domain function<\/strong>, where $t$ is the variable, and outputs a <strong>frequency-domain function<\/strong>, where $s$ is the variable. Definition-wise, Laplace transform takes a function of real variable $f(t)$ (defined for all $t \\ge 0$) to a function of complex variable $F(s)$ as follows:<\/p>\n\n\n\n\\[\\mathcal{L}\\{f(t)\\} = \\int_0^{\\infty} f(t) e^{-st} \\, dt = F(s) \\]\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Some_Preliminary_Examples\"><\/span><a href=\"#toc\" class=\"rank-math-link\">Some Preliminary Examples<\/a><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>What fate awaits <strong>simple functions<\/strong> as they enter the Laplace transform?<\/p>\n\n\n\n<p>Take the simplest function: the <strong>constant function<\/strong> $f(t)=1$. In this case, putting $1$ in the transform yields $1\/s$, which means that we went from a constant to a variable-dependent function.<\/p>\n\n\n\n<iframe class=\"iframe\" src=\"https:\/\/www.youtube.com\/embed\/OiNh2DswFt4?start=174\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe>\n\n\n\n<p>(Odd but not too worrying. After all, we&#8217;ve seen $1\/x$ integrating to $\\ln x$ in calculus. Not a constant-to-variable situation of course, but an unexpected transformation nonetheless.)<\/p>\n\n\n\n<p>Let us take it up a notch, with the <strong>linear function<\/strong> $f(t) = t$. After the transformation, it is turned into $1\/s^2$, which means that we went from $1 \\to 1\/s$ to $t \\to 1\/s^2$. A pattern begins to emerge.<\/p>\n\n\n\n<p>Now what about $f(t)=t^n$? With this simple <strong>power function<\/strong>, we end up with: \\[ \\mathcal{L}\\{ t^n \\} = \\frac{n!}{s^{n+1}}\\] So there was a factorial in $\\mathcal{L}\\{t\\}$ all along, hidden by the fact that $1! = 1$. What else is the transform hiding?<\/p>\n\n\n\n<p>Here, a glance at a table of <strong><a aria-label=\"common Laplace transforms (opens in a new tab)\" rel=\"noreferrer noopener\" href=\"https:\/\/tutorial.math.lamar.edu\/pdf\/Laplace_Table.pdf\" target=\"_blank\" class=\"rank-math-link\">common Laplace transforms<\/a><\/strong> would show that the emerging pattern cannot explain other functions easily. Things get weird, and the weirdness escalates quickly \u2014 which brings us back to the sine function.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Looking_Inside_the_Laplace_Transform_of_Sine\"><\/span><a href=\"#toc\" class=\"rank-math-link\">Looking Inside the Laplace Transform of Sine<\/a><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Let us unpack what happens to our sine function as we Laplace-transform it. We begin by noticing that a sine function can be expressed as a <strong>complex exponential<\/strong> \u2014 an indirect result of the celebrated <a href=\"\/euler-formula\/\" target=\"_blank\" aria-label=\" (opens in a new tab)\" rel=\"noreferrer noopener\" class=\"rank-math-link\">Euler&#8217;s formula<\/a>:\\[e^{it} = \\cos t + i \\sin t\\]In fact, a sine is often expressed in terms of exponentials for <strong>ease of calculation<\/strong>, so if we apply that to the function $f(t) = \\sin (at)$, we would get: \\[ \\sin(at) = \\frac{e^{iat}-e^{-iat}}{2i} \\]Thus the <strong>Laplace transform<\/strong> of $\\sin(at)$ then becomes:<br>\\[ \\mathcal{L}\\{\\sin(at)\\} = \\frac{1}{2i} \\int\\limits_0^{\\infty} (e^{iat}-e^{-iat}) e^{-st} \\, dt \\]which means that we have a <strong>product of exponentials<\/strong>. Distributing the terms, we get:<br>\\[ \\mathcal{L}\\{\\sin(at)\\} = \\frac{1}{2i} \\int\\limits_0^{\\infty} e^{iat-st}-e^{-iat-st} \\, dt \\]<\/p>\n\n\n\n<p>Here, <strong>factoring<\/strong> the $t$ in the exponents yields:<br>\\[ \\mathcal{L}\\{\\sin(at)\\} = \\frac{1}{2i} \\int\\limits_0^{\\infty} e^{(ia-s)t}-e^{(-ia-s)t} \\, dt \\]and since $\\mathrm{Re}(s) \\gt 0$ by assumption, we can proceed with the integration from $0$ to $\\infty$ as usual:<br>\\[ \\mathcal{L}\\{\\sin(at)\\} = \\left.\\frac{e^{(ia-s)t}}{2i (ia-s)}\\right|_0^{\\infty}-\\left.\\frac{e^{(-ia-s)t}}{2i (-ia-s)}\\right|_0^{\\infty} \\]<\/p>\n\n\n\n<p>Let us simplify further. <strong>Distributing<\/strong> the $i$ inside the parentheses, we get:<br>\\[ \\mathcal{L}\\{\\sin(at)\\} = \\left.\\frac{e^{(ia-s)t}}{2(-a-is)}\\right|_0^{\\infty}-\\left.\\frac{e^{(-ia-s)t}}{2(a-is)}\\right|_0^{\\infty} \\]By evaluating the $t$ at the <strong>boundaries<\/strong>, we get:<br>\\[ \\mathcal{L}\\{\\sin(at)\\} = \\left( \\frac{e^{(ia-s) \\cdot \\infty}}{2(-a-is)}-\\frac{e^{(ia-s) \\cdot 0}}{2 (-a-is)}\\right)-\\left(\\frac{e^{(-ia-s)\\cdot\\infty}}{2(a-is)}-\\frac{e^{(-ia-s)\\cdot 0}}{2(a-is)}\\right) \\]And because $\\mathrm{Re}(s) &gt; 0$ by assumption, both $e^{(ia-s) \\cdot \\infty}$ and $e^{(-ia-s)\\cdot\\infty}$ oscillate to $0$ (i.e., <strong><a aria-label=\"vanish at infinity (opens in a new tab)\" rel=\"noreferrer noopener\" href=\"https:\/\/mathvault.ca\/math-glossary\/#vanish\" target=\"_blank\" class=\"rank-math-link\">vanish at infinity<\/a><\/strong>), after which we are then left with:\\[ \\mathcal{L}\\{\\sin(at)\\} = \\frac{1}{-2(-a-is)} + \\frac{1}{2(a-is)} \\]Once there, merging the <strong>fractions<\/strong> together would yield:\\begin{align*} \\mathcal{L}\\{\\sin(at)\\} &amp; = \\frac{2(a-is)-2(-a-is)}{-4 (a-is)(-a-is)} \\\\ &amp; = \\frac{2a-2is + 2a+2is}{4 (a^2 + isa-isa + s^2)} \\\\ &amp; = \\frac{4a}{4(a^2 + s^2)} \\\\ &amp; = \\frac{a}{a^2 + s^2} \\end{align*}which shows that after Laplace transform, a sine is turned into a more tractable <strong>geometric function<\/strong>. By following similar reasoning, the Laplace transform of cosine can be shown to be equal to the following expression as well: \\[ \\mathcal{L}\\{\\cos (at)\\} = \\frac{s}{a^2 + s^2} \\qquad (\\mathrm{Re}(s) &gt; 0) \\] But then, one might argue &#8220;Why do we need to transform trigonometric functions like this when we can just <strong>integrate<\/strong> them?&#8221;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Diverging_Functions_What_the_Laplace_Transform_is_for\"><\/span><a href=\"#toc\" class=\"rank-math-link\">Diverging Functions: What the Laplace Transform is for<\/a><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>What if we throw a wrench in there by introducing a <strong>diverging function<\/strong>, say, $f(t)=e^{at}$? As it turns out, the Laplace transform of the exponential $e^{at}$ is actually deceptively simple: \\begin{align*} \\mathcal{L}\\{e^{at}\\} &amp; = \\int_0^{\\infty} e^{at}e^{-st} \\, dt \\\\ &amp; = \\int_0^{\\infty} e^{(a-s)t} \\, dt \\end{align*}Here, we see that so long as $\\mathrm{Re}(s) \\gt a$, we would get that: \\begin{align*} \\int_0^{\\infty} e^{(a-s)t} \\, dt &amp; = \\left. \\frac{e^{(a-s)t}}{a-s} \\right|_0^{\\infty} \\\\ &amp; = 0-\\frac{1}{a-s}  \\\\ &amp; = \\frac{1}{s-a} \\end{align*} That is, as long as $\\mathrm{Re}(s) &gt; a$, the Laplace transform of $e^{at}$ is a simple $1\/(s-a)$. Here&#8217;s a <strong>video version<\/strong> of the derivation for the record.<\/p>\n\n\n\n<iframe class=\"iframe\" src=\"https:\/\/www.youtube.com\/embed\/33TYoybjqPg?start=50\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe>\n\n\n\n<p>On the other hand, if we mix the exponential $e^{at}$ with the <strong>power function<\/strong> $t^n$, we would then have: \\[ \\mathcal{L}\\{t^n e^{at}\\} = \\int\\limits_0^{\\infty} t^n e^{at} e^{-st} \\, dt \\] which, after a bit of <a aria-label=\"recursion (opens in a new tab)\" href=\"\/math-glossary\/#recursion\" target=\"_blank\" rel=\"noreferrer noopener\" class=\"rank-math-link\">recursion<\/a> and <a aria-label=\"integration by parts (opens in a new tab)\" href=\"https:\/\/mathvault.ca\/integration-overshooting-method\/#Integration_By_Parts\" target=\"_blank\" rel=\"noreferrer noopener\" class=\"rank-math-link\">integration by parts<\/a>, would become:\\[ \\frac{n!}{(s-a)^{n+1}} \\]Here, notice how the transforms of exponential and power function are both <strong>represented<\/strong> in the expression, with the factorial $n!$, the $1\/(s-a)$ fraction, and the $n + 1$ exponent.<\/p>\n\n\n\n<p>In fact, it turns out that we can integrate <em>any<\/em> function with the Laplace transform, as long as it does not <strong>diverge<\/strong> faster than the $e^{at}$ exponential. In the tables of Laplace transforms, you might have noticed the $\\mathrm{Re}(s) \\gt a$ condition. That is what the condition is alluding to.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"A_Transform_of_Unfathomable_Power\"><\/span><a href=\"#toc\" class=\"rank-math-link\">A Transform of Unfathomable Power<\/a><span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>However, what we have seen is only the tip of the iceberg, since we can also use Laplace transform to transform the <strong>derivatives<\/strong> as well. In goes $f^{(n)}(t)$. Something happens. Then out goes:\\[ s^n \\mathcal{L}\\{f(t)\\}-\\sum_{r=0}^{n-1} s^{n-1-r} f^{(r)}(0) \\]For example, when $n=2$, we have that:\\[ \\mathcal{L}\\{f^{\\prime\\prime}(t)\\} = s^2 \\mathcal{L}\\{f(t)\\}-sf(0)-f'(0) \\]In addition to the derivatives, the $\\mathcal{L}$ can also process some <strong>integrals<\/strong>: the integral sine, cosine and exponential, as well as the <a href=\"\/hub\/higher-math\/math-symbols\/probability-statistics-symbols\/#Continuous_Probability_Distributions_and_Associated_Functions\" target=\"_blank\" aria-label=\"error function (opens in a new tab)\" rel=\"noreferrer noopener\" class=\"rank-math-link\">error function<\/a> \u2014 to name a few.<\/p>\n\n\n\n<p>But that&#8217;s not all. There is also the <strong><a aria-label=\"inverse Laplace transform (opens in a new tab)\" rel=\"noreferrer noopener\" href=\"https:\/\/mathvault.ca\/hub\/higher-math\/math-symbols\/calculus-analysis-symbols\/#Key_Transforms\" target=\"_blank\" class=\"rank-math-link\">inverse Laplace transform<\/a><\/strong>, which takes a frequency-domain function and renders a time-domain function.<\/p>\n\n\n\n<p>In fact, performing the transform from time to frequency and back once introduces a factor of $1\/2\\pi$. Sometimes, you&#8217;ll see the whole fraction in front of the inverse function, while other times, the transform and its inverse share a factor of $1\/\\sqrt{2\\pi}$.<\/p>\n\n\n\n<p>This is as if the Kraken could restitute the boat intact \u2014 but only for a factor of $1\/2\\pi$.<\/p>\n\n\n\n<p>The Laplace transform, even after all those years, never ceases to bring us awe with its power. Here&#8217;s a <strong>table<\/strong> summarizing the transforms we&#8217;ve discussed thus far: <\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Function<\/th><th class=\"has-text-align-center\" data-align=\"center\">Laplace Transform<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">$1$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{1}{s}$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$t$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{1}{s^2}$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$t^n$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{n!}{s^{n+1}}$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$e^{at}$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{1}{s-a}$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$\\sin(at)$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{a}{a^2+s^2}$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$\\cos(at)$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{s}{a^2+s^2}$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$t^n e^{at}$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\dfrac{n!}{(s-a)^{n+1}}$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$f^{(2)}(t)$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\displaystyle s^2 \\mathcal{L}\\{f(t)\\}-sf(0)-f'(0)$<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">$f^{(n)}(t)$<\/td><td class=\"has-text-align-center\" data-align=\"center\">$\\displaystyle s^n \\mathcal{L}\\{f(t)\\}-\\sum_{r=0}^{n-1} s^{n-1-r} f^{(r)}(0)$<\/td><\/tr><\/tbody><\/table><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>A gentle, concise introduction to the concept of Laplace transform, along with 9 basic examples to illustrate its derivations and usage.<\/p>\n","protected":false},"author":117,"featured_media":31622,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[367,17,257],"tags":[],"class_list":["post-31617","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-college-math","category-calculus","category-complex-number","post-wrapper","thrv_wrapper"],"_links":{"self":[{"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/posts\/31617","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/users\/117"}],"replies":[{"embeddable":true,"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/comments?post=31617"}],"version-history":[{"count":0,"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/posts\/31617\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/media\/31622"}],"wp:attachment":[{"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/media?parent=31617"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/categories?post=31617"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathvault.ca\/wp-json\/wp\/v2\/tags?post=31617"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}