{"id":75,"date":"2020-08-01T16:27:47","date_gmt":"2020-08-01T16:27:47","guid":{"rendered":"http:\/\/soulofmathematics.com\/?page_id=75"},"modified":"2021-11-21T08:14:20","modified_gmt":"2021-11-21T02:44:20","slug":"integral-transforms","status":"publish","type":"page","link":"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/","title":{"rendered":"INTEGRAL TRANSFORMS"},"content":{"rendered":"\n<p class=\"has-drop-cap\">In&nbsp;mathematics, an&nbsp;<strong>integral transform<\/strong>&nbsp;maps a&nbsp;function&nbsp;from its original&nbsp;function space&nbsp;into another function space via&nbsp;integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the&nbsp;<em>inverse transform<\/em>.<\/p>\n\n\n\n<p>An integral transform is any&nbsp;transform&nbsp;<em>T<\/em>&nbsp;of the following form:<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1d749ab39226b7b3c5e79269db45b8c1e1945cad\" alt=\"{\\displaystyle (Tf)(u)=\\int _{t_{1}}^{t_{2}}f(t)\\,K(t,u)\\,dt}\"\/><\/figure>\n\n\n\n<p>Mathematical notation aside, the motivation behind integral transforms is easy to understand. There are many classes of problems that are difficult to solve\u2014or at least quite unwieldy algebraically\u2014in their original representations. An integral transform &#8220;maps&#8221; an equation from its original &#8220;domain&#8221; into another domain. Manipulating and solving the equation in the target domain can be much easier than manipulation and solution in the original domain. The solution is then mapped back to the original domain with the inverse of the integral transform.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"table-of-transforms\">Table of Transforms<\/h4>\n\n\n\n<figure class=\"wp-block-table aligncenter\"><table><tbody><tr><th scope=\"col\">Transform<\/th><th scope=\"col\">Symbol<\/th><th scope=\"col\"><em>K<\/em><\/th><th scope=\"col\"><em>t<\/em><sub>1<\/sub><\/th><th scope=\"col\"><em>t<\/em><sub>2<\/sub><\/th><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Abel_transform\">Abel transform<\/a><\/td><td><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e230ee38b8f767038b545c54ea4d357da8617ee1\" alt=\"\\frac{2t}{\\sqrt{t^2-u^2}}\"><\/td><td><em>u<\/em><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c26c105004f30c27aa7c2a9c601550a4183b1f21\" alt=\"\\infty \"><\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Fourier_transform\">Fourier transform<\/a><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676\" alt=\"{\\mathcal {F}}\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/edeffbb8cf344049c1d007f9d756fdf02145612e\" alt=\"e^{-2\\pi iut}\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1\" alt=\"-\\infty \"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c26c105004f30c27aa7c2a9c601550a4183b1f21\" alt=\"\\infty \"><\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Fourier_sine_transform\">Fourier sine transform<\/a><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/04ceb66730fd1ff953fbafcc793ce5b70c5b4a82\" alt=\"\\mathcal{F}_s\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0677bccd90f063a507fa8e06c5c4cbe2fa659a85\" alt=\"\\sqrt{\\frac{2}{\\pi}} \\sin(ut)\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2aae8864a3c1fec9585261791a809ddec1489950\" alt=\"{\\displaystyle 0}\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c26c105004f30c27aa7c2a9c601550a4183b1f21\" alt=\"\\infty \"><\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Fourier_cosine_transform\">Fourier cosine transform<\/a><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/94c21edf2d5797258c3db0771809c7e2eecc6177\" alt=\"\\mathcal{F}_c\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9315e8c23a846339c2a510aa103eba5779a6deac\" alt=\"\\sqrt{\\frac{2}{\\pi}} \\cos(ut)\"><\/td><td>0<\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c26c105004f30c27aa7c2a9c601550a4183b1f21\" alt=\"\\infty \"><\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Hankel_transform\">Hankel transform<\/a><\/td><td><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/32ceebc0682b0e2aa4ffc95f98ef8139e68fda5a\" alt=\"t\\,J_\\nu(ut)\"><\/td><td>0<\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c26c105004f30c27aa7c2a9c601550a4183b1f21\" alt=\"\\infty \"><\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Hartley_transform\">Hartley transform<\/a><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/19ef4c7b923a5125ac91aa491838a95ee15b804f\" alt=\"{\\mathcal {H}}\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/107968e3c17cdfa455d23fb78888736a671c681d\" alt=\"\\frac{\\cos(ut)+\\sin(ut)}{\\sqrt{2 \\pi}}\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1\" alt=\"-\\infty \"><\/td><td> <img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c26c105004f30c27aa7c2a9c601550a4183b1f21\" alt=\"\\infty \"><\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Hermite_transform\">Hermite transform<\/a><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/75a9edddcca2f782014371f75dca39d7e13a9c1b\" alt=\"H\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8fc12dbcf5c1a77d5dc7fdd9af91a87aad78e1ca\" alt=\"{\\displaystyle e^{-x^{2}}H_{n}(x)}\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1\" alt=\"-\\infty \"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c26c105004f30c27aa7c2a9c601550a4183b1f21\" alt=\"\\infty \"><\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Hilbert_transform\">Hilbert transform<\/a><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5b79470948c977172abacfc9311e13998a50c3d7\" alt=\"\\mathcal{H}il\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e4aba46bef4c0fb20d8224ea9dfb35a129738a18\" alt=\"\\frac{1}{\\pi}\\frac{1}{u-t}\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1\" alt=\"-\\infty \"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c26c105004f30c27aa7c2a9c601550a4183b1f21\" alt=\"\\infty \"><\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Jacobi_transform\">Jacobi transform<\/a><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/359e4f407b49910e02c27c2f52e87a36cd74c053\" alt=\"J\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/039b0c1e4af2c10aa7c99fb71d3d438c065d3948\" alt=\"{\\displaystyle (1-x)^{\\alpha }\\ (1+x)^{\\beta }\\ P_{n}^{\\alpha ,\\beta }(x)}\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/704fb0427140d054dd267925495e78164fee9aac\" alt=\"-1\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/92d98b82a3778f043108d4e20960a9193df57cbf\" alt=\"1\"><\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Laguerre_transform\">Laguerre transform<\/a><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8\" alt=\"L\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/85567b33e499e818a726f38c8fd6fae4fe458f8a\" alt=\"{\\displaystyle e^{-x}\\ x^{\\alpha }\\ L_{n}^{\\alpha }(x)}\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2aae8864a3c1fec9585261791a809ddec1489950\" alt=\"{\\displaystyle 0}\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c26c105004f30c27aa7c2a9c601550a4183b1f21\" alt=\"\\infty \"><\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Laplace_transform\">Laplace transform<\/a><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9027196ecb178d598958555ea01c43157d83597c\" alt=\"{\\mathcal {L}}\"><\/td><td><em>e<sup>\u2212ut<\/sup><\/em><\/td><td>0<\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c26c105004f30c27aa7c2a9c601550a4183b1f21\" alt=\"\\infty \"><\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Legendre_transform\">Legendre transform<\/a><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f8fb0b896b1b2a45546779ecafc567f4f1688714\" alt=\"{\\mathcal {J}}\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ced0c8017e2e633e34a03700e554bff311d9a6a0\" alt=\"P_{n}(x)\\,\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/704fb0427140d054dd267925495e78164fee9aac\" alt=\"-1\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/92d98b82a3778f043108d4e20960a9193df57cbf\" alt=\"1\"><\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Mellin_transform\">Mellin transform<\/a><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd\" alt=\"{\\mathcal {M}}\"><\/td><td><em>t<\/em><sup><em>u<\/em>\u22121<\/sup><\/td><td>0<\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c26c105004f30c27aa7c2a9c601550a4183b1f21\" alt=\"\\infty \"><\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Two-sided_Laplace_transform\">Two-sided Laplace<br>transform<\/a><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e5622de88a69f68340f8dcb43d0b8bd443ba9e13\" alt=\"{\\mathcal {B}}\"><\/td><td><em>e<sup>\u2212ut<\/sup><\/em><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1\" alt=\"-\\infty \"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c26c105004f30c27aa7c2a9c601550a4183b1f21\" alt=\"\\infty \"><\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Poisson_kernel\">Poisson kernel<\/a><\/td><td><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/dcdc9f25f057b4d65fd3310f7dcdc798cec94564\" alt=\"\\frac{1-r^2}{1-2r\\cos\\theta +r^2}\"><\/td><td>0<\/td><td>2\u03c0<\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Radon_Transform\">Radon Transform<\/a><\/td><td>R\u0192<\/td><td><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1\" alt=\"-\\infty \"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c26c105004f30c27aa7c2a9c601550a4183b1f21\" alt=\"\\infty \"><\/td><\/tr><tr><td><a href=\"https:\/\/en.wikipedia.org\/wiki\/Weierstrass_transform\">Weierstrass transform<\/a><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6a1cc103563219127f59aec7ed9327a3595566dd\" alt=\"{\\mathcal {W}}\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c8322dcb248d8a49fc40de4abbac286df85061c4\" alt=\"\\frac{e^{-\\frac{(u-t)^2}{4}}}{\\sqrt{4\\pi}}\\,\"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1\" alt=\"-\\infty \"><\/td><td><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c26c105004f30c27aa7c2a9c601550a4183b1f21\" alt=\"\\infty \"><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>We will get into details of few of the  most popular transforms like Laplace Transform, Fourier Transform an many more.<\/p>\n\n\n\n<div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link\" href=\"https:\/\/soulofmathematics.com\/index.php\/laplace-transform\/\">Laplace Transform<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>In&nbsp;mathematics, an&nbsp;integral transform&nbsp;maps a&nbsp;function&nbsp;from its original&nbsp;function space&nbsp;into another function space via&nbsp;integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the&nbsp;inverse transform. An integral transform is any&nbsp;transform&nbsp;T&nbsp;of the following form: Mathematical notation aside, the motivation behind integral transforms is easy to understand. There are many classes of problems that are difficult to solve\u2014or at least quite unwieldy algebraically\u2014in their original representations. An integral transform &#8220;maps&#8221; an equation from its original &#8220;domain&#8221; into another domain. Manipulating and solving the equation in the target domain can be much easier than manipulation and solution in the original domain. The solution is then mapped back to the original domain with the inverse of the integral transform. Table of Transforms Transform Symbol K t1 t2 Abel transform u Fourier transform Fourier sine transform Fourier cosine transform 0 Hankel transform 0 Hartley transform Hermite transform Hilbert transform Jacobi transform Laguerre transform Laplace transform e\u2212ut 0 Legendre transform Mellin transform tu\u22121 0 Two-sided Laplacetransform e\u2212ut Poisson kernel 0 2\u03c0 Radon Transform R\u0192 Weierstrass transform We will get into details of few of the most popular transforms like Laplace Transform, Fourier Transform an many more.<\/p>\n","protected":false},"author":1,"featured_media":447,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"jetpack_post_was_ever_published":false,"footnotes":""},"class_list":["post-75","page","type-page","status-publish","has-post-thumbnail","hentry"],"ams_acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v23.6 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>INTEGRAL TRANSFORMS - SOUL OF MATHEMATICS<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"INTEGRAL TRANSFORMS - SOUL OF MATHEMATICS\" \/>\n<meta property=\"og:description\" content=\"In&nbsp;mathematics, an&nbsp;integral transform&nbsp;maps a&nbsp;function&nbsp;from its original&nbsp;function space&nbsp;into another function space via&nbsp;integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the&nbsp;inverse transform. An integral transform is any&nbsp;transform&nbsp;T&nbsp;of the following form: Mathematical notation aside, the motivation behind integral transforms is easy to understand. There are many classes of problems that are difficult to solve\u2014or at least quite unwieldy algebraically\u2014in their original representations. An integral transform &#8220;maps&#8221; an equation from its original &#8220;domain&#8221; into another domain. Manipulating and solving the equation in the target domain can be much easier than manipulation and solution in the original domain. The solution is then mapped back to the original domain with the inverse of the integral transform. Table of Transforms Transform Symbol K t1 t2 Abel transform u Fourier transform Fourier sine transform Fourier cosine transform 0 Hankel transform 0 Hartley transform Hermite transform Hilbert transform Jacobi transform Laguerre transform Laplace transform e\u2212ut 0 Legendre transform Mellin transform tu\u22121 0 Two-sided Laplacetransform e\u2212ut Poisson kernel 0 2\u03c0 Radon Transform R\u0192 Weierstrass transform We will get into details of few of the most popular transforms like Laplace Transform, Fourier Transform an many more.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/\" \/>\n<meta property=\"og:site_name\" content=\"SOUL OF MATHEMATICS\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-21T02:44:20+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/AffectionateDirectAmazontreeboa-size_restricted.gif?fit=640%2C320&ssl=1\" \/>\n\t<meta property=\"og:image:width\" content=\"640\" \/>\n\t<meta property=\"og:image:height\" content=\"320\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/gif\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/\",\"url\":\"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/\",\"name\":\"INTEGRAL TRANSFORMS - SOUL OF MATHEMATICS\",\"isPartOf\":{\"@id\":\"https:\/\/soulofmathematics.com\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/AffectionateDirectAmazontreeboa-size_restricted.gif?fit=640%2C320&ssl=1\",\"datePublished\":\"2020-08-01T16:27:47+00:00\",\"dateModified\":\"2021-11-21T02:44:20+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/#primaryimage\",\"url\":\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/AffectionateDirectAmazontreeboa-size_restricted.gif?fit=640%2C320&ssl=1\",\"contentUrl\":\"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/AffectionateDirectAmazontreeboa-size_restricted.gif?fit=640%2C320&ssl=1\",\"width\":640,\"height\":320},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/soulofmathematics.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"INTEGRAL TRANSFORMS\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/soulofmathematics.com\/#website\",\"url\":\"https:\/\/soulofmathematics.com\/\",\"name\":\"SOUL OF MATHEMATICS\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/soulofmathematics.com\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"INTEGRAL TRANSFORMS - SOUL OF MATHEMATICS","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/","og_locale":"en_US","og_type":"article","og_title":"INTEGRAL TRANSFORMS - SOUL OF MATHEMATICS","og_description":"In&nbsp;mathematics, an&nbsp;integral transform&nbsp;maps a&nbsp;function&nbsp;from its original&nbsp;function space&nbsp;into another function space via&nbsp;integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the&nbsp;inverse transform. An integral transform is any&nbsp;transform&nbsp;T&nbsp;of the following form: Mathematical notation aside, the motivation behind integral transforms is easy to understand. There are many classes of problems that are difficult to solve\u2014or at least quite unwieldy algebraically\u2014in their original representations. An integral transform &#8220;maps&#8221; an equation from its original &#8220;domain&#8221; into another domain. Manipulating and solving the equation in the target domain can be much easier than manipulation and solution in the original domain. The solution is then mapped back to the original domain with the inverse of the integral transform. Table of Transforms Transform Symbol K t1 t2 Abel transform u Fourier transform Fourier sine transform Fourier cosine transform 0 Hankel transform 0 Hartley transform Hermite transform Hilbert transform Jacobi transform Laguerre transform Laplace transform e\u2212ut 0 Legendre transform Mellin transform tu\u22121 0 Two-sided Laplacetransform e\u2212ut Poisson kernel 0 2\u03c0 Radon Transform R\u0192 Weierstrass transform We will get into details of few of the most popular transforms like Laplace Transform, Fourier Transform an many more.","og_url":"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/","og_site_name":"SOUL OF MATHEMATICS","article_modified_time":"2021-11-21T02:44:20+00:00","og_image":[{"url":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/AffectionateDirectAmazontreeboa-size_restricted.gif?fit=640%2C320&ssl=1","width":640,"height":320,"type":"image\/gif"}],"twitter_card":"summary_large_image","twitter_misc":{"Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/","url":"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/","name":"INTEGRAL TRANSFORMS - SOUL OF MATHEMATICS","isPartOf":{"@id":"https:\/\/soulofmathematics.com\/#website"},"primaryImageOfPage":{"@id":"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/#primaryimage"},"image":{"@id":"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/#primaryimage"},"thumbnailUrl":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/AffectionateDirectAmazontreeboa-size_restricted.gif?fit=640%2C320&ssl=1","datePublished":"2020-08-01T16:27:47+00:00","dateModified":"2021-11-21T02:44:20+00:00","breadcrumb":{"@id":"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/#primaryimage","url":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/AffectionateDirectAmazontreeboa-size_restricted.gif?fit=640%2C320&ssl=1","contentUrl":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/AffectionateDirectAmazontreeboa-size_restricted.gif?fit=640%2C320&ssl=1","width":640,"height":320},{"@type":"BreadcrumbList","@id":"https:\/\/soulofmathematics.com\/index.php\/integral-transforms\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/soulofmathematics.com\/"},{"@type":"ListItem","position":2,"name":"INTEGRAL TRANSFORMS"}]},{"@type":"WebSite","@id":"https:\/\/soulofmathematics.com\/#website","url":"https:\/\/soulofmathematics.com\/","name":"SOUL OF MATHEMATICS","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/soulofmathematics.com\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"}]}},"jetpack_sharing_enabled":true,"jetpack-related-posts":[],"jetpack_shortlink":"https:\/\/wp.me\/Pcfs4y-1d","_links":{"self":[{"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/pages\/75","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/comments?post=75"}],"version-history":[{"count":4,"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/pages\/75\/revisions"}],"predecessor-version":[{"id":3006,"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/pages\/75\/revisions\/3006"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/media\/447"}],"wp:attachment":[{"href":"https:\/\/soulofmathematics.com\/index.php\/wp-json\/wp\/v2\/media?parent=75"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}