One dimension:<\/p>\n
\\[
\nG(x) = \\frac{1}{\\sqrt{2\\pi \\sigma^2}} e^{-\\frac{x^2}{2\\sigma^2}}
\n\\]<\/p>\n
Two dimensions:<\/p>\n
\\[
\nG(x, y) = \\frac{1}{2\\pi \\sigma^2} e^{-\\frac{x^2 + y^2}{2\\sigma^2}}
\n\\]<\/p>\n
This is because:<\/p>\n
\\[
\nI(t)\u00a0 = \\int_{-t}^{t} e^{-x^2}dx \\\\
\nI^2(t)\u00a0 = (\\int_{-t}^{t} e^{-x^2}dx) \\cdot (\\int_{-t}^{t} e^{-y^2}dy) = \\int_{-t}^{t} \\int_{-t}^{t} e^{-(x^2+y^2)}dx dy \\\\
\n\\int_{0}^{t} \\int_{0}^{2\\pi}e^{-r^2}r dr d\\theta < I^2(t) < \\int_{0}^{t\\sqrt 2} \\int_{0}^{2\\pi}e^{-r^2}r dr d\\theta \\\\
\n\\int_{0}^{t} 2\\pi e^{-r^2}\u00a0 r dr d\\theta < I^2(t) < \\int_{0}^{t\\sqrt 2}\u00a02\\pi e^{-r^2}r dr d\\theta \\\\
\n\\pi(1-e^{-t^2}) < I^2(t) < \\pi(1-e^{-2t^2}) \\\\
\n\\lim_{t \\to +\\infty} \\pi(1-e^{-t^2})\u00a0< \\lim_{t \\to +\\infty} I^2(t) < \\lim_{t \\to +\\infty} \\pi(1-e^{-2t^2}) \\\\
\n\\pi < \\lim_{t \\to +\\infty} I^2(t) < \\pi
\n\\]<\/p>\n
so we obtain:<\/p>\n
\\[
\n\\lim_{t \\to +\\infty} I^2(t) = \\int_{-\\infty}^{\\infty}\u00a0\\int_{-\\infty}^{\\infty} e^{-(x^2+y^2)}dx dy = \\pi \\\\
\n\\lim_{t \\to +\\infty} I(t) = \\int_{-\\infty}^{\\infty} e^{-x^2} dx = \\sqrt \\pi \\\\
\n\\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} \\frac{1}{2\\pi \\sigma^2} e^{-\\frac{x^2+y^2}{2\\sigma^2 }} dx dy = 1
\n\\]<\/p>\n","protected":false},"excerpt":{"rendered":"
One dimension: \\[ G(x) = \\frac{1}{\\sqrt{2\\pi \\sigma^2}} e^{-\\frac{x^2}{2\\sigma^2}} \\] Two dimensions: \\[ G(x, y) = \\frac{1}{2\\pi \\sigma^2} e^{-\\frac{x^2 + y^2}{2\\sigma^2}} \\] This is because: \\[ I(t)\u00a0 = \\int_{-t}^{t} e^{-x^2}dx \\\\ I^2(t)\u00a0 = (\\int_{-t}^{t} e^{-x^2}dx) \\cdot (\\int_{-t}^{t} e^{-y^2}dy) = \\int_{-t}^{t} \\int_{-t}^{t} e^{-(x^2+y^2)}dx dy \\\\ \\int_{0}^{t} \\int_{0}^{2\\pi}e^{-r^2}r dr d\\theta < I^2(t) < \\int_{0}^{t\\sqrt 2} \\int_{0}^{2\\pi}e^{-r^2}r dr d\\theta […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-498","post","type-post","status-publish","format-standard","hentry","category-graphics"],"_links":{"self":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/posts\/498","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/comments?post=498"}],"version-history":[{"count":0,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/posts\/498\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/media?parent=498"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/categories?post=498"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/tags?post=498"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}