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.

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.

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.

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- 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
- 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
- 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
- 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
- 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)