Path: news.mathworks.com!newsfeed-00.mathworks.com!panix!bloom-beacon.mit.edu!newsswitch.lcs.mit.edu!nrc-news.nrc.ca!newsflash.concordia.ca!canopus.cc.umanitoba.ca!not-for-mail 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 Organization: Canada Eat The Cookie Foundation Lines: 45 Message-ID: <i4ek4l$pjf$1@canopus.cc.umanitoba.ca> References: <i4a7g4$3kg$1@fred.mathworks.com> <i4abqs$9t7$1@fred.mathworks.com> <i4e9ud$oqj$1@fred.mathworks.com> NNTP-Posting-Host: ibd-nat.ibd.nrc.ca Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit X-Trace: canopus.cc.umanitoba.ca 1282069461 26223 132.246.133.10 (17 Aug 2010 18:24:21 GMT) X-Complaints-To: abuse@cc.umanitoba.ca NNTP-Posting-Date: Tue, 17 Aug 2010 18:24:21 +0000 (UTC) User-Agent: Thunderbird 2.0.0.24 (X11/20100317) In-Reply-To: <i4e9ud$oqj$1@fred.mathworks.com> Xref: news.mathworks.com comp.soft-sys.matlab:662791 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.