Lambert's Transcendental Equation

Author

Eric W. Weisstein
October 17, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/LambertsTranscendentalEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LambertsTranscendentalEquation.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Equation

Solve[(x^α)-(x^β)==(α-β)v (x^(a+β)),x]

Limit[(((x^α)-(x^β))/(α-β)),α->β]

Solve[(x^β) Log[x]==v (x^(2β)),x]//FullSimplify

soln=Solve[Log[x]==v (x^β),x]

Log[(((-((ProductLog[-v β])/(v β))))^(1/β))]//FullSimplify

FullSimplify[Log[x]-v (x^β)/.soln]

FullSimplify[%]

Subtract@@@Table[{Log[x],v (x^β)}/.soln[[1]]/.{v->RandomComplex[],β->RandomComplex[]},{100}]//Chop//Union

Table[Log[(((-((ProductLog[-v β])/(v β))))^(1/β))]-v ((((((-((ProductLog[-v β])/(v β))))^(1/β))))^β) /.{v->RandomComplex[],β->RandomComplex[]},{100}]//Chop//Union

{(((-((ProductLog[-v β])/(v β))))^(1/β)),Exp[-((ProductLog[-β v])/β)]}/.{v->RandomComplex[],β->RandomComplex[]}