(X^(y )) =  (y^x)
by Debra Woods

A professor at the ISMAA conference came to the booth with a problem that came up in his class. He asked if we can visualize the solution to

(x^y) =  (y^x). 

First he wanted to see a 3D plot of it:

plt = Plot3D[x^y - y^x, {x, 0, 5}, {y, 0, 5}];

Then he asked if we could see where it was positive and negative:

redblack =ContourPlot[x^y - y^x,{x, 0.00001, 10},{y, 0.00001, 10},PlotPoints -> 100,Contours->{0},ColorFunction -> (If[# > 0,  Black,  Red] &),Axes -> True,AxesLabel -> {"x", "y"},AspectRatio -> Automatic];

Where the black areas are positive and the red areas are negative.

At this point, he asked about the values where the equation is = 0, so in other words parameterize the curves. It is clear that equality holds when x=y. It is not clear how to parameterize the curved part however. Given just the little time that we had, I thought that maybe Mathematica would give me a hint on that, so:

Solve[x^2 == 2^x, x]

Solve[x^3 == 3^x, x]

So, I see a pattern and decide to try parameterizing it as follows:

rh = 
        Plot[-((x  ProductLog[-Log[x]/x])/Log)[x], {x, E, 
            10}, PlotStyle -> {Yellow, Thickness[0.02]}];

lh = 
        ParametricPlot[{-((x  ProductLog[-Log[x]/x])/Log)[
                    x], x}, {x, E, 10}, 
          PlotStyle -> {Yellow, Thickness[0.02]}];

Show[redblack, rh, lh];

I personally think that is very pretty and artistic! It also looks better in version 6.0!

Then, he asked about the point where there is only 1 solution. He had tried the following on a graphing calculator:

f[x_, y_] := x^y - y^x

f[2.718, 2.7]

f[2.718, 2.75]

f[2.73, 2.718]

f[2.7, 2.718]

He then said, it is probably E. I said, let's see:

f[E, E]

Aha! In fact it is.

Then I said, let's investigate the asymptote that the curve is approaching:

Limit[-((x  ProductLog[-Log[x]/x])/Log)[x], 
      x -> Infinity]

He was going to take that to class to discuss with them.

Here are some points for further investigation:

a) Find a better parameterization of the curve.
b) Find all integer solutions.
c) Find all rational solutions.