File:InvertW.jpg

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Original file (1,024 × 1,121 pixels, file size: 130 KB, MIME type: image/jpeg)

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Summary

Description
English: Regions of the complex plane for which W(n,z ez) = z. The darker boundaries of a particular region are included in the lighter colored region of the same color. The point at {-1,0} is included in both the n=-1 (blue) region and the n=0 (gray) region. Horizontal grid lines are in multiples of π

The following Mathematica code essentially reproduces the plot:

(* ----- Initialize ----- *)

Clear[x,y,color]

z=x+I y;

xmin=-16;xmax=16;ymin=-16;ymax=16;

W[n_,z_]:=ProductLog[n,z];

color[h_]:={{h,1,h},{h,1,1},{h,h,1},{h,h,h},{1,h,h},{1,0.57+h/2,0},{1,1,h}};

c=color[1/2]; (* light colors *) d=color[0];(* dark colors *)

(* ----- Regions ----- *)

Clear[regn]

regn[n_]:=RegionPlot[Chop[W[n,z Exp[z]]]==z,{x,xmin,xmax},{y,ymin,ymax},PlotStyle->RGBColor[cn+4],PlotPoints->50,FrameLabel->{"x","y"},RotateLabel->False,LabelStyle->(FontSize->10)]

reg=Show[Table[regn[n],{n,-3,3}]];

(* ----- Boundaries ----- *)

Clear[bndn]

bndn[n_]:=ParametricPlot[{-y/Tan[y],y},{y,2ymin,2ymax},RegionFunction->Function[{x,y},(x Cos[y]-y Sin[y])Exp[-x]<=0&& (2n \[Pi]<=y<=(2n+2)\[Pi])],PlotStyle->{RGBColor[dn+4],Thickness[0.015]},PlotRange->{ymin,ymax}]

bnd0=Plot[0,{x,xmin,-1},PlotStyle->{RGBColor[d3],Thickness[0.015]}]; (* xmin < z <= -1 *)

bnd=Show[Table[bndn[n],{n,-3,3}],bnd0];

(* ----- Region Labels ----- *)

Clear[txtn]

txtn[n_,x_,y_]:=Text[Style["n="<>ToString[n],FontSize->15],{x,y}];

xx=-7; tm3=txtn[-3,xx,-15]; tm2=txtn[-2,xx,-10.5]; tm1=txtn[-1,xx,-4]; t0=txtn[0,8,0]; tp1=txtn[1,xx,4]; tp2=txtn[2,xx,10.5]; tp3=txtn[3,xx,15];

txt=Graphics[{tm3,tm2,tm1,t0,tp1,tp2,tp3}];

(* ----- Show all ----- *)

Show[reg,bnd,txt]

Date
Source Own work
Author PAR

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

Regions and boundaries for the simple inverse of the Lambert W function

Items portrayed in this file

depicts

inception

12 July 2020

media type

image/jpeg

File history

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Date/TimeThumbnailDimensionsUserComment
current03:59, 13 July 2020Thumbnail for version as of 03:59, 13 July 20201,024 × 1,121 (130 KB)wikimediacommons>PARManual cleanup

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