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

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Summary

Description
English: The boundary of the blue and white regions are the points where . The blue region corresponds to and the white region corresponds to . The bottom blue curve can be written explicitly by the expression , where is the Lambert W function. The top green curve is , and the dashed black curve is . From the plot, we can see that and .
Date
Source Own work
Author Hugo Spinelli
Other versions Plot of x^y = y^x.png
SVG development
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This vector image was created with Inkscape.
Source code
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Try it here (permalink). Cleaned up with Inkscape afterwards.
SageMath code
var('x y')
f = x^y-y^x
max_x, max_y = 8, 8
range_x = (x, 0.01, max_x)
range_y = (y, 0.01, max_y)

p = plot([], dpi=150, gridlines=False)
p += region_plot(f>0, range_x, range_y, incol='lightblue', bordercol='gray', axes_labels=['$x$','$y$'])
e = exp(1)
p += text('$x^y > y^x$', [3.5, 5], rgbcolor='dimgray', fontsize='large')
p += text('$x^y > y^x$', [3.5, 1], rgbcolor='dimgray', fontsize='large')
p += text('$x^y < y^x$', [1.0, 4], rgbcolor='dimgray', fontsize='large')
p += text('$x^y < y^x$', [6.0, 4], rgbcolor='dimgray', fontsize='large')

point1 = (exp(1), pi)
point2 = (2, pi)
point3 = (exp(1), exp(1))

p += scatter_plot([point1], markersize=25, edgecolor='gray', facecolor='lightgray')
p += text(r'$(e,\,\pi)$', (point1[0], point1[1] + .25), fontsize='medium', rgbcolor='gray')

p += scatter_plot([point2], markersize=25, edgecolor='gray', facecolor='lightgray')
p += text(r'$(2,\,\pi)$', (point2[0] - .1, point2[1] + .25), fontsize='medium', rgbcolor='gray')

p += scatter_plot([point3], facecolor='red')
p += text(r'$(e,\,e)$', (point3[0], point3[1] - .5), fontsize='large', rgbcolor='black')

g_latex = r'$y = \exp\left(-W_0\left(\frac{-\ln x}{x}\right)\right)$'
g = lambda x: exp(-lambert_w(0, -ln(x)/x))
h_latex = r'$y = \exp\left(-W_{-1}\left(\frac{-\ln x}{x}\right)\right)$'
h = lambda x: exp(-lambert_w(-1, -ln(x)/x))
p += plot(g, range_x, thickness=2)
p += plot(h, (N(g(8)), 8), thickness=2, color='green')
p += text(g_latex, (6.2, 2.2), fontsize='x-large')
p += text(h_latex, (3.6, 7.0), fontsize='x-large', color='green')
p += text('$y = x$', (7.2, 6.4), fontsize='xx-large', color='black')
p += plot(x, range_x, linestyle=':', thickness=1, color='black')

show(p)
p.save('Plot_of_x^y_=_y^x.svg')

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

Captions

Plot of equation x^y = y^x with inequility regions highlighted.

30 September 2023

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Date/TimeThumbnailDimensionsUserComment
current12:07, 30 September 2023Thumbnail for version as of 12:07, 30 September 2023419 × 424 (45 KB)Hugo SpinelliUploaded own work with UploadWizard

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