The Uniformisation of the Equation $$z^w=w^z$$

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> <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> 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> <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>, 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>