The Uniformisation of the Equation $$z^w=w^z$$
Authors
Beardon, AF
Publication Date
2022-03Journal Title
Computational Methods and Function Theory
ISSN
1617-9447
Publisher
Springer Science and Business Media LLC
Volume
22
Issue
1
Pages
123-134
Language
en
Type
Article
This Version
VoR
Metadata
Show full item recordCitation
Beardon, A. (2022). The Uniformisation of the Equation $$z^w=w^z$$. Computational Methods and Function Theory, 22 (1), 123-134. https://doi.org/10.1007/s40315-021-00369-6
Abstract
<jats:title>Abstract</jats:title><jats:p>The positive solutions of the equation <jats:inline-formula><jats:alternatives><jats:tex-math>$$x^y = y^x$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
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</mml:math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$x^y=y^x$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
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<mml:mi>y</mml:mi>
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</mml:math></jats:alternatives></jats:inline-formula>, the complex equation <jats:inline-formula><jats:alternatives><jats:tex-math>$$z^w = w^z$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msup>
<mml:mi>z</mml:mi>
<mml:mi>w</mml:mi>
</mml:msup>
<mml:mo>=</mml:mo>
<mml:msup>
<mml:mi>w</mml:mi>
<mml:mi>z</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> 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 <jats:italic>z</jats:italic>(<jats:italic>t</jats:italic>) and <jats:italic>w</jats:italic>(<jats:italic>t</jats:italic>) of functions of a complex variable <jats:italic>t</jats:italic> that are holomorphic functions of <jats:italic>t</jats:italic> lying in some region <jats:italic>D</jats:italic> of the complex plane that satisfy the equation <jats:inline-formula><jats:alternatives><jats:tex-math>$$z(t)^{w(t)} = w(t)^{z(t)}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>z</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>w</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:msup>
<mml:mo>=</mml:mo>
<mml:mi>w</mml:mi>
<mml:msup>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>z</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> for <jats:italic>t</jats:italic> in <jats:italic>D</jats:italic>. Moreover, when <jats:italic>t</jats:italic> is positive these solutions agree with those of <jats:inline-formula><jats:alternatives><jats:tex-math>$$x^y=y^x$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msup>
<mml:mi>x</mml:mi>
<mml:mi>y</mml:mi>
</mml:msup>
<mml:mo>=</mml:mo>
<mml:msup>
<mml:mi>y</mml:mi>
<mml:mi>x</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula>.</jats:p>
Keywords
Article, Uniformisation, Complex exponents, Lambert function, 30D05, 30F10
Identifiers
s40315-021-00369-6, 369
External DOI: https://doi.org/10.1007/s40315-021-00369-6
This record's URL: https://www.repository.cam.ac.uk/handle/1810/335472
Rights
Licence:
http://creativecommons.org/licenses/by/4.0/
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