dc.contributor.author Beardon, AF dc.date.accessioned 2022-03-29T15:00:25Z dc.date.available 2022-03-29T15:00:25Z dc.date.issued 2022-03 dc.date.submitted 2019-12-23 dc.identifier.issn 1617-9447 dc.identifier.other s40315-021-00369-6 dc.identifier.other 369 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/335472 dc.description.abstract AbstractThe positive solutions of the equation $$x^y = y^x$$ x y = y x have been discussed for over two centuries. Goldbach found a parametric form for the solutions, and later a connection was made with the classical Lambert function, which was also studied by Euler. Despite the attention given to the real equation $$x^y=y^x$$ x y = y x , the complex equation $$z^w = w^z$$ z w = w z has virtually been ignored in the literature. In this expository paper, we suggest that the problem should not be simply to parametrise the solutions of the equation, but to uniformize it. Explicitly, we construct a pair z(t) and w(t) of functions of a complex variable t that are holomorphic functions of t lying in some region D of the complex plane that satisfy the equation $$z(t)^{w(t)} = w(t)^{z(t)}$$ z ( t ) w ( t ) = w ( t ) z ( t ) for t in D. Moreover, when t is positive these solutions agree with those of $$x^y=y^x$$ x y = y x . dc.language en dc.publisher Springer Science and Business Media LLC dc.subject Article dc.subject Uniformisation dc.subject Complex exponents dc.subject Lambert function dc.subject 30D05 dc.subject 30F10 dc.title The Uniformisation of the Equation $$z^w=w^z$$ dc.type Article dc.date.updated 2022-03-29T15:00:24Z prism.endingPage 134 prism.issueIdentifier 1 prism.publicationName Computational Methods and Function Theory prism.startingPage 123 prism.volume 22 dc.identifier.doi 10.17863/CAM.82903 dcterms.dateAccepted 2020-12-21 rioxxterms.versionofrecord 10.1007/s40315-021-00369-6 rioxxterms.version VoR rioxxterms.licenseref.uri http://creativecommons.org/licenses/by/4.0/ dc.identifier.eissn 2195-3724 cam.issuedOnline 2021-07-26
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