File:Plot of x^y = y^x.png

var('x y')
f = x^y-y^x
range_x = (x,0.01,8)
range_y = (y,0.01,8)

height = range_y[2]-range_y[1]
width = range_x[2]-range_x[1]
p = plot([], dpi=150, gridlines=False)
#p += implicit_plot(f, range_x, range_y)
p += region_plot(f>0, range_x, range_y, incol='lightblue', bordercol='gray', axes_labels=['$x$','$y$'])
e = exp(1)
p += text("$x^y > y^x$", [e, 4], rgbcolor='black', fontsize='large')
p += text("$x^y > y^x$", [e, 1.5], rgbcolor='black', fontsize='large')
p += text("$x^y < y^x$", [1, e], rgbcolor='black', fontsize='large')
p += text("$x^y < y^x$", [4, e], rgbcolor='black', fontsize='large')

point1 = (e, pi)
point2 = (2, pi)
point3 = (e, e)

p += scatter_plot([point1], facecolor='green')
p += text("$"+latex(point1)+"$", (point1[0],point1[1]+height/30), fontsize='medium', rgbcolor='black')

p += scatter_plot([point2], facecolor='red')
p += text("$"+latex(point2)+"$", (point2[0],point2[1]+height/30), fontsize='medium', rgbcolor='black')

p += scatter_plot([point3])
p += text("$"+latex(point3)+"$", (point3[0],point3[1]-height/30), fontsize='medium', rgbcolor='black')

g_latex = r"$y = \frac{-xW_0\left(-\ln(x)/x\right)}{\ln(x)}$"  # -x*lambert_w(0, -ln(x)/x)/ln(x)
g = lambda x: -x*lambert_w(0, -ln(x)/x)/ln(x)
h_latex = r"$y = \frac{-xW_{-1}\left(-\ln(x)/x\right)}{\ln(x)}$"  # -x*lambert_w(-1, -ln(x)/x)/ln(x)
h = lambda x: -x*lambert_w(-1, -ln(x)/x)/ln(x)
p += plot(g, range_x, thickness=2)
p += plot(h, (N(g(8)), exp(1)), thickness=2, color='green')
p += text(g_latex, (range_x[1]+width*0.8,N(g(x=range_x[1]+width*0.8))+height/15), fontsize='xx-large')
p += text(h_latex, (3.5, 7), fontsize='xx-large', color='green')
p += text("$y = x$", (7, 6), fontsize='xx-large', color='gray')
show(p)