Lambert W function

My most recent talk on the Lambert W function

Here is the Maple worksheet that I used instead of slides: stillmore.mws . The talk was given at the Department of Applied Mathematics seminar series on Tuesday, March 21, 2000.

Animations and movies

See Knuth's plenary lecture at the SIAM 45th Annual Meeting for a discussion of material related to the Lambert W function.

An animation of a segment of the Riemann surface for the Lambert W function.

Don Knuth and me with my car, with the W FUN licence plate, when Don visited Waterloo to give the Pascal lectures. Photo kindly taken by Jeff Shallit.

An animation of (some of) the values of W(k,a) as "a" moves from zero down to just below -pi/2. This animation graphically shows the loss of stability in the delay differential equation y'(t) = a y(t-1) as "a" becomes "too negative", in clear contrast with the simple ordinary differential equation y'(t) = a y(t), which just becomes more and more stable as "a" becomes more negative.

The Maple V Release 5 worksheet that generated those pictures.


Pictures

A plot of W, generated parametrically

A fractal related to W . This shows the values of zeta, with z = exp( zeta *exp(-zeta) ), where the iteration a[n] = z^a[n-1] converges to cycles of different lengths, with colours coded as follows: for zeta inside the unit circle, the iteration converges (to zeta = -W(0,-ln(z))/ln(z)). This is coloured a dull turquoise, or greenish-blue. For zeta in the wide light blue ``scalloped'' region that touches the unit circle at -1 (= exp(2*Pi*I/2) ), the iteration goes to a stable 2-cycle. For zeta in the regions coloured a slightly darker blue, two of which touch the unit circle at exp(2*Pi*I/3) and exp(2*Pi*I*2/3), the iteration goes to a stable 3-cycle. An infinite number of other such 3-cycle regions are shown. For zeta in the regions coloured purple, two of which touch the unit circle at exp(2*Pi*I/4) and exp(2*Pi*I*3/4), the iteration goes to a stable 4-cycle. At angles of multiples of 2*Pi/5, regions of 5-cycles (coloured blue-grey) touch the unit circle. At angles of multiples of 2*Pi/6, we see regions which lead to 6-cycles. The 7-cycle regions have been picked out in yellow. The black regions are regions where overflow or underflow occurred in the computation (using Lahey Fortran 95 Express, which allows good access to IEEE arithmetic). We are currently experimenting with other arithmetics to improve this picture. It seems clear that the black regions are just regions of lack of proof---we expect that the 3-cycle regions, for example, continue out to infinity.

A version of this fractal (in the ln(z) plane) appears (in hand-drawn form!) in the paper ``A note on complex iteration'' by I. N. Baker and P. J. Rippon, American Mathematical Monthly, Volume 92, No. 7, August-September 1985, pp. 501--503. That paper proves that the iteration converges exactly for zeta inside the unit circle together with the points on the boundary that have phase angles that are rational multiples of Pi. This fractal picture here gives a graphical confirmation of that proof: at rational points k/m (with gcd(k,m)=1) we see that stable m-cycles are born via Hopf bifurcation; this means that at the boundary point we have an m-cycle coalescing to a 1-cycle.

About the Applied Mathematics Coffee Mug Design

A variant picture of the Riemann Surface, done in Matlab. We have plotted the real part of W(z) as the height, and the imaginary part as the colour, instead of vice versa as above.

A roughly-polished POVray version of the Maple picture of the Riemann Surface for the Lambert W function. This picture was created by Ha Quang Le , a Ph.D. student in the symbolic computation group at the University of Waterloo, and an artist.


Exercises on the Lambert W Function

HTML form generated by LaTeX2html

.dvi form

.ps form


Published Papers on the Lambert W Function

  1. Robert M. Corless, David J. Jeffrey, and Donald E. Knuth, ``A Sequence of Series for the Lambert W Function'' (gzipped PostScript Version) Proceedings ISSAC '97, Maui, Wolfgang W. Kuechlin, editor, pages 197--204. Straight PostScript version
  2. Robert M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, ``On the Lambert W Function", Advances in Computational Mathematics, volume 5, 1996, pp. 329--359. (Compressed Postscript, extension .ps.Z) or Uncompressed Version
  3. D.J. Jeffrey, D.E.G.Hare, Robert M. Corless: Unwinding the branches of the Lambert W function. Math. Scientist. 21, 1--7 (1996) Postscript version
  4. D.J.Jeffrey, R.M. Corless, D.E.G. Hare, D.E. Knuth, Sur l'inversion de y^a e^y au moyen des nombres de Stirling associes. Comptes Rendus Acad. Sci. Paris, t. 320, Serie I, p1449-1452 (1995) French publication (translation courtesy André Boivin); English version
  5. Robert M. Corless, G. H. Gonnet, D. E. G. Hare, and D. J. Jeffrey, ``Lambert's W Function in Maple", Maple Technical Newsletter 9, Spring 1993, pp. 12--22. (NOT QUITE THE SAME AS THE PUBLISHED VERSION)(Compressed Postscript, extension .ps.Z)

Other people's W pages

Keith Briggs' W-ology page