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From: Walter Roberson <roberson@hushmail.com>
Newsgroups: comp.soft-sys.matlab
Subject: Re: Finding all roots of a logarithm function (with parameters)
Date: Tue, 17 Aug 2010 13:23:40 -0500
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summersyu Yu wrote:

> I found that the two roots have the following forms:
> 
> root1=-a/c*lambertw(-c/a*exp(b-c)/a)-1;
> root2=-a/c*lambertw(-1,-c/a*exp(b-c)/a)-1;

Were you able to prove that, or did you extract that from my earlier response 
where I indicated without proof that those were the only forms I found?


> What is the branch argument -1? In the help, it says "the K-th branch of 
> this multi-valued function". But I do not understand why it is -1 not 
> other values.

I don't know why -1 and not something else; the -1 is what I found experimentally.

As to what it means, see the following from the Maple help for LambertW:


o The principal branch and the pair of branches LambertW(-1, x) and 
LambertW(1, x) share an order 2 branch point at -exp(-1). The branch cut 
dividing these branches is the subset of the real line from -infinity to 
-exp(-1), and the values of the branches of LambertW on this branch cut are 
assigned using the rule of counter-clockwise continuity around the branch 
point. This means that LambertW(x) is real-valued for x in the range -exp(-1) 
.. infinity, while the image of -infinity .. -exp(-1) under LambertW(x) is the 
curve -y*cot(y)+I*y, for y in 0 .. Pi.

   Similarly, the branch corresponding to -1, LambertW(-1, x), is real-valued 
on the interval -exp(-1) .. 0, while the image of -infinity .. -exp(-1) under 
this branch is the curve -y*cot(y)+I*y, for y in -Pi .. 0.

o For all the branches other than the principal branch, the branch cut 
dividing them is the negative real axis. The branches are numbered up and down 
from the real axis (this is very similar to the way the branches of the 
logarithm are indexed by the multiple of (2*I)*Pi which must be subtracted 
from the imaginary part to recover the principal branch). Again, the values of 
the branches of LambertW along the branch cut are determined by the rule of 
counter-clockwise continuity around the branch point at 0. Thus, the image of 
the negative real axis under the branch LambertW(k, x) is the curve 
-y*cot(y)+I*y, for y in 2*k*Pi .. (2*k+1)*Pi if 0 < k and y in (2*k+1)*Pi .. 
(2*k+2)*Pi if k < -1. These curves, therefore, bound the ranges of the 
branches of LambertW, and in each case, the upper boundary of the region is 
included in the range of the corresponding branch.