The positive solutions of the equation \(x^y = y^x\) 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 \(x^y=y^x\), the complex equation \(z^w = w^z\) 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 \(z(t)^{w(t)} = w(t)^{z(t)}\) for t in D. Moreover, when t is positive these solutions agree with those of \(x^y=y^x\).

方程\(x^y = y^x\)的正解已经讨论了两个多世纪。哥德巴赫找到了解的参数形式，后来又与经典的朗伯函数建立了联系，欧拉也研究了这一点。尽管对实际方程\(x^y=y^x\)给予了关注，但在文献中实际上忽略了复杂方程\(z^w = w^z\)。在这篇说明性论文中，我们建议问题不应该只是简单地对方程的解进行参数化，而应该对其进行统一。明确地，我们构造了复变量t的函数对z ( t ) 和w ( t )是全纯函数吨躺在一些区域d复平面满足方程的\（Z（t）的^ {值w（t）} =值w（t）^ {Z（t）的} \）为吨在d . 此外，当t为正时，这些解与\(x^y=y^x\) 的解一致。