$x^y = y^x$ Closed form solution??

Question

I looked and could not find... How can I show the solutions for $x^y = y^x$?

There (as far as I can tell) are three curves satisfying this. One is $x=y$, one curves through $(2,4) (4,2) (e,e)$ and goes asymptotic in the $++$ and $+-$ quadrants and its mirror image reflected across $x=y$.

But it is not easy to find a closed form expression for the solutions...at least not for me. Thanks

As I explain in this answer, the solution space can be parametrized as $(a^{1/(a-1)},a^{a/(a-1)})$ where $a \neq 1$ (with $a > 0$, if we're looking for real numbers). — Ben Grossmann, Dec 19 '16 at 01:23

score 1 · Answer 1 · answered Dec 19 '16 at 01:29

1

This requires the Lambert W function.

$$y^x=x^y$$

$$y^{1/y}=x^{1/x}$$

Take the reciprocal of both sides:

$$(1/y)^{1/y}=(1/x)^{1/x}$$

$$\frac1y=e^{W\left(-\frac{\ln(x)}x\right)}$$

$$\large y=e^{-W\left(-\frac{\ln(x)}x\right)}$$

If we take the $W_0$ branch, we get $y=x$. If we take the $W_{-1}$ branch, we get the other weird looking line.

answered Dec 19 '16 at 01:29

Simply Beautiful Art

74,685

And so I go down the Lambert function rabbit hole.... gracias – The fracman Dec 19 '16 at 03:35
XD your welcome @Thefracman – Simply Beautiful Art Dec 19 '16 at 11:56

$x^y = y^x$ Closed form solution??

1 Answers1