[{"department":[{"_id":"GradSch"},{"_id":"ChWo"}],"OA_type":"green","corr_author":"1","type":"preprint","abstract":[{"text":"In calculus, l'Hopital's rule provides a simple way to evaluate the limits of quotient functions when both the numerator and denominator vanish. But what happens when we move beyond real functions on a real interval? In this article, we study when the quotient of two complex-valued functions in higher dimension can be defined continuously at the points where both functions vanish. Surprisingly, the answer is far subtler than in the real-valued setting. We provide a complete characterization for the continuity of the quotient function. We also point out why extending this result to smoother quotients remains an intriguing challenge.","lang":"eng"}],"citation":{"mla":"Chern, Albert, and Sadashige Ishida. “L’Hopital Rules for Complex-Valued Functions in Higher Dimensions.” <i>ArXiv</i>, 2602.09958, doi:<a href=\"https://doi.org/10.48550/ARXIV.2602.09958\">10.48550/ARXIV.2602.09958</a>.","chicago":"Chern, Albert, and Sadashige Ishida. “L’Hopital Rules for Complex-Valued Functions in Higher Dimensions.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/ARXIV.2602.09958\">https://doi.org/10.48550/ARXIV.2602.09958</a>.","ista":"Chern A, Ishida S. L’Hopital rules for complex-valued functions in higher dimensions. arXiv, 2602.09958.","ama":"Chern A, Ishida S. L’Hopital rules for complex-valued functions in higher dimensions. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/ARXIV.2602.09958\">10.48550/ARXIV.2602.09958</a>","apa":"Chern, A., &#38; Ishida, S. (n.d.). L’Hopital rules for complex-valued functions in higher dimensions. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/ARXIV.2602.09958\">https://doi.org/10.48550/ARXIV.2602.09958</a>","ieee":"A. Chern and S. Ishida, “L’Hopital rules for complex-valued functions in higher dimensions,” <i>arXiv</i>. .","short":"A. Chern, S. Ishida, ArXiv (n.d.)."},"_id":"21737","month":"02","publication_status":"submitted","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"date_created":"2026-04-15T16:28:24Z","external_id":{"arxiv":["2602.09958"]},"arxiv":1,"publication":"arXiv","project":[{"grant_number":"101045083","_id":"34bc2376-11ca-11ed-8bc3-9a3b3961a088","name":"Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","OA_place":"repository","ddc":["510"],"day":"10","doi":"10.48550/ARXIV.2602.09958","file":[{"file_id":"21771","date_updated":"2026-04-28T10:53:27Z","relation":"main_file","checksum":"6a76591c723d3e949ad5afa9f7dbb2ee","content_type":"application/pdf","file_size":867109,"creator":"dernst","date_created":"2026-04-28T10:53:27Z","file_name":"2026_arXiv_2602.09958.pdf","success":1,"access_level":"open_access"}],"oa":1,"date_published":"2026-02-10T00:00:00Z","article_number":"2602.09958","year":"2026","acknowledgement":"This project was funded in part by the European Research Council (ERC Consolidator Grant 101045083 CoDiNA) and the National Science Foundation CAREER Award 2239062.\r\n","title":"L'Hopital rules for complex-valued functions in higher dimensions","author":[{"full_name":"Chern, Albert","last_name":"Chern","first_name":"Albert"},{"id":"6F7C4B96-A8E9-11E9-A7CA-09ECE5697425","full_name":"Ishida, Sadashige","last_name":"Ishida","orcid":"0000-0002-3121-3100","first_name":"Sadashige"}],"date_updated":"2026-04-28T10:56:30Z","status":"public","language":[{"iso":"eng"}],"file_date_updated":"2026-04-28T10:53:27Z","article_processing_charge":"No","has_accepted_license":"1","oa_version":"Preprint","keyword":["l’Hopital theorem","complex functions"]}]