{"paper":{"title":"The fractional derivative of the Dirac delta function and new results on the inverse Laplace transform of irrational functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.MP"],"primary_cat":"math-ph","authors_text":"Nicos Makris","submitted_at":"2020-06-08T22:21:17Z","abstract_excerpt":"Motivated from studies on anomalous diffusion, we show that the memory function $M(t)$ of complex materials, that their creep compliance follows a power law, $J(t)\\sim t^q$ with $q\\in \\mathbb{R}^+$, is the fractional derivative of the Dirac delta function, $\\frac{\\mathrm{d}^q\\delta(t-0)}{\\mathrm{d}t^q}$ with $q\\in \\mathbb{R}^+$. This leads to the finding that the inverse Laplace transform of $s^q$ for any $q\\in \\mathbb{R}^+$ is the fractional derivative of the Dirac delta function, $\\frac{\\mathrm{d}^q\\delta(t-0)}{\\mathrm{d}t^q}$. This result, in association with the convolution theorem, makes "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.04966","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2006.04966/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}