Newsgroups: sci.math.symbolic,sci.math From: kogeddes@daisy.uwaterloo.ca (Keith O. Geddes) Subject: Re: Problem e^x + a*x + b = 0 Date: Thu, 5 Mar 1998 17:38:56 GMT In article <6dmeru$pk5$1@nnrp1.dejanews.com>, wrote: >In article <34FDCFFB.2968@tel.hr>, > Vlado Pribolsan wrote: >> >> Hi, >> >> I need to find x from >> >> e^x + a*x + b = 0 >> >> (or 1 - e^x = a*x - b) >> >> Is it possible? >> >> Tnx, Vlado. >> > >Yes, but only in terms of the Lambert W function. This is a non-holomorphic >function over C. Whether this constitutes a 'closed form solution' is subject >to interpretation. Well, one can get into a philosophical point. The Lambert W function is, fundamentally, not much more complicated than the logarithm function. It is just not as well known. Whereas the logarithm function is the solution of: solve e^y = x for y similarly, the Lambert W function is the solution of: solve y e^y = x for y . There are issues of branches in the complex plane. For the particular problem you mention above, it can be solved in a Maple session as follows. ============================================================================ > eqn := exp(x) + a*x + b = 0; eqn := exp(x) + a x + b = 0 > soln := solve(eqn,x); exp(- b/a) a LambertW(----------) + b a soln := - -------------------------- a # # You can evaluate the LambertW function, do series expansions, etc. # # As an example, suppose that you were interested in the case b=2 and # you wished to know how the solution varies with a. # Below is a plot of the solution over some range of a. # (I am using dumb ASCII plot mode here, for the purposes of this posting.) # > b := 2; b := 2 > plot(soln, a=1/5..1); -2+ AAAAAAAAAA + AAAAAAAAAAAAAA + AAAAAAAAAA + AAAAAAAA + AAAAAA -4+ AAAAA + AAAA + AAA + AAA + AA -6+ AA + AA + A + AA + AA -8+ AA + AA + AA +AA +A -10-+-+-++-+-+-+-+-+-+-++-+-+-+-+-+-+-+-++-+-+-+-+-+-+-++-+-+-+-+-+-+-++-+-+-+ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ============================================================================ References on the Lambert W function ------------------------------------ I refer you to the home page of Robert Corless at the University of Western Ontario: http://pineapple.apmaths.uwo.ca/~rmc/ He has pointers to information about the Lambert W function. In particular, the URL http://pineapple.apmaths.uwo.ca/~rmc/papers/LambertW/index.html gives references to papers about the Lambert W function. -------------- Professor Keith Geddes Symbolic Computation Group Department of Computer Science University of Waterloo Waterloo ON N2L 3G1 CANADA ============================================================================== From: edgar@math.ohio-state.edu (G. A. Edgar) Newsgroups: sci.math Subject: Re: Lambert-W function Date: Wed, 07 Oct 1998 09:04:44 -0400 In article , Hal Daume III wrote: > What branch of math would the study of the Lambert-W function be > considered? Anyone know of any good books/online sources for information > on it. > Branch of math: probably "special functions". For info (and references) see: R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey and D.E. Knuth, "On The Lambert W Function," in the Maple Share Library. -- Gerald A. Edgar edgar@math.ohio-state.edu ============================================================================== Newsgroups: sci.math From: fbarrett@world.std.com (Fred K Barrett) Subject: Re: Lambert-W function Date: Wed, 7 Oct 1998 15:15:12 GMT The Corless, et al. paper is at ftp://watdragon.uwaterloo.ca/cs-archive/CS-93-03/W.ps.Z It's easy to read, and has many references to follow up. Also note that the book "Concrete Mathematics" by Graham, Knuth, and Patashnik mentions a similar function that is part of the "generalized exponential" family, and they give some nice identities. I haven't seen their notation used elsewhere, though. Good luck -- Fred K. Barrett fbarrett@alum.mit.edu -- ============================================================================== Newsgroups: sci.math From: ddavis@openmarket.com (don davis) Subject: Re: Lambert-W function Date: Wed, 7 Oct 1998 20:36:41 GMT In article , Hal Daume III wrote: > Anyone know of any good books/online sources for information on it? here are some bookmarks i've collected over the past year. - don davis, boston -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- Material on the Lambert W-function (links) http://pineapple.apmaths.uwo.ca/~rmc/papers/LambertW/index.html W-ology, or Some exactly solvable growth models: http://epidem13.plantsci.cam.ac.uk/~kbriggs/W-ology.html Lambert's W-function (short article) http://www.astro.virginia.edu/~eww6n/math/LambertsW-Function.html Iterated Exponential Constants (short article): http://www.astro.virginia.edu/~eww6n/math/IteratedExponentialConstants.html Iterated Exponential Constants (longer article): http://www.mathsoft.com/asolve/constant/itrexp/itrexp.html