Abstract
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.
Highlights
The equation x y = yx, where x and y are positive, has been studied by many people over many years but, by contrast, the equation zw = wz, where z and w are complex numbers, has received almost no attention
In 1728 Daniel Bernoulli proved that the only positive integral solutions of x y = yx with x = y are (2, 4) and (4, 2)
For positive x and y, we have x y = yx if and only if loge x/x = loge y/y, where loge is the natural logarithm. It follows from the graph of y =/x that if x y = yx, x > 0, y > 0 and x = y either 1 < x < e < y, or 1 < y < e < x
Summary
Goldbach showed that the set of all positive solutions (x, y) of x y = yx can be parametrised by the functions x(t) and y(t), where t > 0 and x(t ) = t 1/(t−1) = exp loge t t −1. The basic ideas of analytic continuation, and an informal understanding of Riemann surfaces, are enough to show that if Log is analytically continued over the set C∗ of non-zero complex numbers it is defined and single-valued on a Riemann surface which, informally, is an infinite ‘spiral-staircase’ which is constructed by taking an y = −1/e x = −1. A branch W−1 which is a strictly decreasing map of (−1/e, 0) onto (−∞, −1/e) This elementary discussion leads immediately to the result which is often stated in the literature and, for the convenience of the reader, we include a formal proof. Theorem 3 clarifies Lòczi’s remark (Remark 4 in [17, p. 222]) that (in our notation) the function F is not given by W0 β(x) /β(x) throughout the curve C
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