TITLE: Explicit components of $y^x=x^y$
QUESTION [5 upvotes]: When graphing the implicit equation $y^x=x^y$ it seems like it consists of two possibly explicit curves. On the one hand there is the linear solution $y=x$. But as it appears there is a second curve monotonously decreasing as x increases, intersecting with the curve $y=x$ at coordinate point $(2,2)$. Is there any way to find out the explicit form of this other curve?

REPLY [5 votes]: You can get a parametric set of equations.  Define $y=ax$  Then
$$x^y=y^x\\x^xa^x=x^{ax}\\a=x^{a-1}\\x=a^{\frac 1{a-1}}\\y=a^{\frac a{a-1}}$$

REPLY [2 votes]: Arthur gave you an explicit form using the Lamber W-function. I present you a more algorithmic approach using power towers. Rearrange $x^y=y^x$ to
$$x=\sqrt[y]{y}^x.$$
So when you define $z:=\sqrt[y]y$, then for a given $y$ you can compute
$$x=z^{z^{z^{\cdots}}}.$$
It is not obivous that this even converges, but check by yourself by using the recursice rule $x_0=1$ and $x_{n+1}=z^{x_n}$.