{"id":923,"date":"2015-12-07T04:09:13","date_gmt":"2015-12-07T04:09:13","guid":{"rendered":"http:\/\/quant-math.org\/wp\/?p=923"},"modified":"2019-01-08T11:25:12","modified_gmt":"2019-01-08T11:25:12","slug":"the-cauchy-principal-alue-and-the-finite-part-integral-as-values-of-absolutely-convergent-integrals-e-a-galapon-arxiv1512-01323","status":"publish","type":"post","link":"https:\/\/quant-math.org\/wp\/2015\/12\/07\/the-cauchy-principal-alue-and-the-finite-part-integral-as-values-of-absolutely-convergent-integrals-e-a-galapon-arxiv1512-01323\/","title":{"rendered":"The Cauchy principal value and the finite part integral as values of absolutely convergent integrals (E.A. Galapon , arXiv:1512.01323)"},"content":{"rendered":"<h3>Abstract<\/h3>\n<blockquote><p>The divergent integral $\\int_a^b f(x)(x-x_0)^{-n-1}\\mathrm{d}x$, for $-\\infty&lt;a&lt;x_0&lt;b&lt;\\infty$ and $n=0,1,2,\\dots$, is assigned, under certain conditions, the value equal to the simple average of the contour integrals $\\int_{C^{\\pm}} f(z)(z-x_0)^{-n-1}\\mathrm{d}z$, where $C^+$ ($C^-$) is a path that starts from $a$ and ends at $b$, and which passes above (below) the pole at $x_0$. It is shown that this value, which we refer to as the Analytic Principal Value, is equal to the Cauchy principal value for $n=0$ and to the finite-part of the divergent integral for positive integer $n$. This implies that, where the conditions apply, the Cauchy principal value and the finite-part integral are in fact values of absolutely convergent integrals. Moreover, it leads to the replacement of the boundary values in the Sokhotski-Plemelj-Fox Theorem with integrals along some arbitrary paths. The utility of the Analytic Principal Value in the numerical, analytical and asymptotic evaluation of the Cauchy principal value and the finite-part integral is discussed and demonstrated.<\/p><\/blockquote>\n<p>Download a preprint of the paper <a href=\"http:\/\/arxiv.org\/abs\/1512.01323\" target=\"_blank\">here<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Abstract The divergent integral $\\int_a^b f(x)(x-x_0)^{-n-1}\\mathrm{d}x$, for $-\\infty&lt;a&lt;x_0&lt;b&lt;\\infty$ and $n=0,1,2,\\dots$, is assigned, under certain conditions, the value equal to the simple average of the contour integrals $\\int_{C^{\\pm}} f(z)(z-x_0)^{-n-1}\\mathrm{d}z$, where $C^+$ ($C^-$) is a path that starts from $a$ and ends at $b$, and which passes above (below) the pole at $x_0$. It is shown that\u2026 <span class=\"read-more\"><a href=\"https:\/\/quant-math.org\/wp\/2015\/12\/07\/the-cauchy-principal-alue-and-the-finite-part-integral-as-values-of-absolutely-convergent-integrals-e-a-galapon-arxiv1512-01323\/\">Read More &raquo;<\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[34,8,16],"tags":[],"class_list":["post-923","post","type-post","status-publish","format-standard","hentry","category-divergent-integrals","category-mathematical-physics","category-preprints"],"_links":{"self":[{"href":"https:\/\/quant-math.org\/wp\/wp-json\/wp\/v2\/posts\/923","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/quant-math.org\/wp\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/quant-math.org\/wp\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/quant-math.org\/wp\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/quant-math.org\/wp\/wp-json\/wp\/v2\/comments?post=923"}],"version-history":[{"count":6,"href":"https:\/\/quant-math.org\/wp\/wp-json\/wp\/v2\/posts\/923\/revisions"}],"predecessor-version":[{"id":930,"href":"https:\/\/quant-math.org\/wp\/wp-json\/wp\/v2\/posts\/923\/revisions\/930"}],"wp:attachment":[{"href":"https:\/\/quant-math.org\/wp\/wp-json\/wp\/v2\/media?parent=923"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/quant-math.org\/wp\/wp-json\/wp\/v2\/categories?post=923"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/quant-math.org\/wp\/wp-json\/wp\/v2\/tags?post=923"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}