a question on geometry?

It is a transcendent open curve in the 1st quadrant (the condition implies

x > 0 and y > 0), symmetric with respect to the bisector x = y, with 2 asymptotes

x = 1 and y = 1.

Follow the link below to see how it is looking like:

http://farm4.static.flickr.com/3522/3896208375_369...

y^x = x^y implies x lny = y lnx, or lnx / x = lny / y. A little calculus shows the function

z = lnt / t (t > 0)

is increasing for 0 < t ≤ e from -∞ to 1/e, attains its greatest value 1/e when t = e, then for e ≤ t < ∞ decreases from 1/e to +0. The function takes all its non-positive values only once for 0 < t ≤ 1, all positive values, less than 1/e are taken twice (for 2 different values of t, let one of them is x, other y as on the left picture - one of them is less than e, the other - greater), finally the value 1/e is taken once - for t = e.

Take arbitrary z, 0 < z < 1/e. Find both roots t' = x and t" = y of the transcendent equation z = lnt / t and the point (x, y) belongs to the curve /also (y, x) since t' and t" are interchangeable/ - it is a set of all points, obtained that way. Since (t' → 1 + 0) ↔ (t" → ∞) the curve is a graph of a decreasing function with 2 asymptotes, intersecting the ray x = y in (e, e), whose analytical expression is not available using the elementary functions.

Finally a curious fact: (2, 4) and (4, 2) are the only integer points on the curve, since 2 is the only integer between 1 and e.