{"id":5572,"date":"2023-04-09T10:38:18","date_gmt":"2023-04-09T02:38:18","guid":{"rendered":"https:\/\/fanyuzhao.com\/?p=5572"},"modified":"2023-04-09T10:38:18","modified_gmt":"2023-04-09T02:38:18","slug":"dirac-delta-function","status":"publish","type":"post","link":"https:\/\/fanyuzhao.com\/?p=5572","title":{"rendered":"Dirac Delta Function"},"content":{"rendered":"\n<p>The Dirac Delta Function could be applied to simplify the differential equation. There are three main properties of Dirac Delta Function.<\/p>\n\n\n\n<p>$$\\delta (x-x&#8217;) =\\lim_{\\tau\\to0}\\delta (x-x&#8217;)$$<\/p>\n\n\n\n<p>such that,<\/p>\n\n\n\n<p>$$ \\delta (x-x&#8217;) = \\begin{cases} \\infty &amp; x= x&#8217; \\ 0 &amp; x\\neq x&#8217; \\end{cases} $$<\/p>\n\n\n\n<p>$$\\int_{-\\infty}^{\\infty} \\delta (x-x&#8217;)\\ dx =1$$<\/p>\n\n\n\n<p><strong>Three Properties:<\/strong><\/p>\n\n\n\n<ul><li>Property 1:<\/li><\/ul>\n\n\n\n<p>$$\\delta(x-x&#8217;)=0 \\quad \\quad ,x\\neq x&#8217; $$<\/p>\n\n\n\n<ul><li>Property 2:<\/li><\/ul>\n\n\n\n<p>$$ \\int_{x&#8217;-\\epsilon}^{x&#8217;+\\epsilon} \\delta (x-x&#8217;)dx =1\\quad \\quad ,\\epsilon &gt;0 $$<\/p>\n\n\n\n<ul><li>Property 3:<\/li><\/ul>\n\n\n\n<p>$$\\int_{x&#8217;-\\epsilon}^{x&#8217;+\\epsilon} f(x)\\ \\delta (x-x&#8217;)dx = f(x&#8217;)$$<\/p>\n\n\n\n<p>At <span class=\"katex math inline\">x=x&#8217;<\/span> the Dirac Delta function is sometimes thought of has having an \u201cinfinite\u201d value. So, the Dirac Delta function is a function that is zero everywhere except one point and at that point it can be thought of as either undefined or as having an \u201cinfinite\u201d value.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Dirac Delta Function could be applied to simplify the differential equation. There are three main properties of Dirac Delta Function. $$\\delta (x-x&#8217;) =\\lim_{\\tau\\to0}\\delta (x-x&#8217;)$$ such that, $$ \\delta (x-x&#8217;) = \\begin{cases} \\infty &amp; x= x&#8217; \\ 0 &amp; x\\neq x&#8217; \\end{cases} $$ $$\\int_{-\\infty}^{\\infty} \\delta (x-x&#8217;)\\ dx =1$$ Three Properties: Property 1: $$\\delta(x-x&#8217;)=0 \\quad \\quad &hellip; <a href=\"https:\/\/fanyuzhao.com\/?p=5572\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Dirac Delta Function<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,18,26],"tags":[],"_links":{"self":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/5572"}],"collection":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5572"}],"version-history":[{"count":1,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/5572\/revisions"}],"predecessor-version":[{"id":5573,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/5572\/revisions\/5573"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5572"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5572"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5572"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}