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We provide some examples related to linear perturbations of spinning black holes in general relativity and modified gravity, in which the analytical formulas for the overlap integrals are particularly useful to compute the black-hole spectrum.<\/jats:p>","DOI":"10.1088\/1361-6382\/acb880","type":"journal-article","created":{"date-parts":[[2023,2,2]],"date-time":"2023-02-02T22:45:13Z","timestamp":1675377913000},"page":"057001","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["General spherical harmonic bra-ket overlap integrals of trigonometric functions"],"prefix":"10.1088","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1923-1963","authenticated-orcid":true,"given":"Giuseppe","family":"Lingetti","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4443-1761","authenticated-orcid":true,"given":"Paolo","family":"Pani","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2023,2,10]]},"reference":[{"edition":"2nd edn","year":"1975","author":"Jackson","key":"cqgacb880bib1"},{"edition":"rev. edn","year":"1994","author":"Sakurai","key":"cqgacb880bib2"},{"year":"2014","author":"Poisson","key":"cqgacb880bib3"},{"key":"cqgacb880bib4","doi-asserted-by":"publisher","DOI":"10.1142\/S0217751X13400186","article-title":"Advanced methods in Black-Hole perturbation theory","volume":"28","author":"Pani","year":"2013","journal-title":"Int. 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