The Uniformisation of the Equation $$z^w=w^z$$
| Autor: | Alan F. Beardon |
|---|---|
| Přispěvatelé: | Apollo - University of Cambridge Repository |
| Rok vydání: | 2021 |
| Předmět: |
4902 Mathematical Physics
Applied Mathematics 4901 Applied Mathematics Holomorphic function 4904 Pure Mathematics symbols.namesake Computational Theory and Mathematics Lambert W function Goldbach's conjecture 49 Mathematical Sciences symbols Euler's formula Connection (algebraic framework) Parametric equation Complex plane Analysis Variable (mathematics) Mathematics Mathematical physics |
| Zdroj: | Computational Methods and Function Theory. 22:123-134 |
| ISSN: | 2195-3724 1617-9447 |
| DOI: | 10.1007/s40315-021-00369-6 |
| Popis: | The 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 . |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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