jats:titleAbstract</jats:title>jats:pThe positive solutions of the equation jats:inline-formulajats:alternativesjats: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:mix</mml:mi> mml:miy</mml:mi> </mml:msup> mml:mo=</mml:mo> mml:msup mml:miy</mml:mi> mml:mix</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-formulajats:alternativesjats: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:mix</mml:mi> mml:miy</mml:mi> </mml:msup> mml:mo=</mml:mo> mml:msup mml:miy</mml:mi> mml:mix</mml:mi> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, the complex equation jats:inline-formulajats:alternativesjats: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:miz</mml:mi> mml:miw</mml:mi> </mml:msup> mml:mo=</mml:mo> mml:msup mml:miw</mml:mi> mml:miz</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:italicz</jats:italic>(jats:italict</jats:italic>) and jats:italicw</jats:italic>(jats:italict</jats:italic>) of functions of a complex variable jats:italict</jats:italic> that are holomorphic functions of jats:italict</jats:italic> lying in some region jats:italicD</jats:italic> of the complex plane that satisfy the equation jats:inline-formulajats:alternativesjats: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:miz</mml:mi> mml:msup mml:mrow mml:mo(</mml:mo> mml:mit</mml:mi> mml:mo)</mml:mo> </mml:mrow> mml:mrow mml:miw</mml:mi> mml:mo(</mml:mo> mml:mit</mml:mi> mml:mo)</mml:mo> </mml:mrow> </mml:msup> mml:mo=</mml:mo> mml:miw</mml:mi> mml:msup mml:mrow mml:mo(</mml:mo> mml:mit</mml:mi> mml:mo)</mml:mo> </mml:mrow> mml:mrow mml:miz</mml:mi> mml:mo(</mml:mo> mml:mit</mml:mi> mml:mo)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> for jats:italict</jats:italic> in jats:italicD</jats:italic>. Moreover, when jats:italict</jats:italic> is positive these solutions agree with those of jats:inline-formulajats:alternativesjats: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:mix</mml:mi> mml:miy</mml:mi> </mml:msup> mml:mo=</mml:mo> mml:msup mml:miy</mml:mi> mml:mix</mml:mi> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>