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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT
-1 33 "The Lambert W function in Science" }}}{EXCHG {PARA 19 "" 0 ""
{TEXT -1 52 "Robert M. Corless\nORCCA and Applied Mathematics, UWO" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "This worksheet provides a ``hori
zontal slice of science'' by looking at selected applications of the L
ambert W function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 ""
{TEXT -1 49 "We begin with the very simplest encounter with W:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve( y*exp(y) = x, y );" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "alias( W=LambertW );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve( y*exp(y) = q, y );" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "W(1.33);" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 23 "0.676299*exp(0.676299);" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14
"Applications I" }}{PARA 0 "" 0 "" {TEXT -1 57 "Now we spend some time
with a few different applications." }}{SECT 0 {PARA 4 "" 0 "" {TEXT
-1 23 "A diode with resistance" }}{PARA 0 "" 0 "" {TEXT -1 125 "We rep
eat the hand calculations for the diode equations.\nBanwell & Jayakuma
r, Electronics Letters, (36) 4, 2000, pp 291--292." }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 27 "restart; alias(W=LambertW);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "interface( imaginaryunit=_i );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "eqs := \{V_d = V - I*R_s, I \+
= I_sat*(exp( V_d/(eta*V_T)) - 1 ) \};" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 30 "sol := solve( eqs, \{V_d, I\} );" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 61 "R_s := 51.3; eta := 1.00; I_sat := 25.0e-15;
V_T := 1.0/40.0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "exa :=
subs( sol, I );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plots[l
ogplot]( exa, V=0.3..1.0, I=1.0e-9..1.0e-1 );" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 14 "Crysta
l growth" }}{PARA 0 "" 0 "" {TEXT -1 191 "From Braun, Briggs, B\\\"oni
: ``Analytical solution to Matthews' & Blakeslee's critical dislocatio
n formation thickness of epitaxially grown thin films''. J. Crystal Gr
owth (241) 2002 231--234." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27
"restart; alias(W=LambertW);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 27 "MBE := d_c = A*ln( B*d_c );" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 25 "_EnvAllSolutions := true;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "solve( MBE, d_c );" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 26 "d_c := -A*W(-1,-1/(B*A) );" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 89 "a_Cu := 3.615; a_Au := 4.078; a_Ni := 3.5238; # la
ttice copper, gold, & nickel, Angstroms" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 57 "a_Alloy := a_Cu*x + a_Au*(1-x): f := (a_Alloy-a_Ni)/a
_Ni;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "xtest := solve( f, \+
x );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "nu := 153/400.; alp
ha := 3; theta := Pi/2; lambda := 0;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 42 "Burgersvector := a_Ni/2*Vector(3,[1,1,1]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "b := LinearAlgebra:-Norm(Bur
gersvector,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "A := eval
f( b/(4*Pi*f*sin(theta)*cos(lambda))*(1-nu*cos(theta)^2)/(1+nu) );" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "B := alpha/b;" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot( d_c, x=0.7..1.0 );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "
" {TEXT -1 25 "A simple combustion model" }}{PARA 0 "" 0 "" {TEXT -1
433 "The following differential equation has been used (Reif, O'Malley
, Moler) to explore methods for solving combustion problems. It has v
arious physical interpretations, but is chiefly of interest as a model
problem. The idea is that the location of the \"combustion region\" \+
depends on the initial condition; this can be explored by perturbation
methods (Reif, O'Malley) or numerical methods for stiff problems (Mol
er) or analytically." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "rest
art; alias(W=LambertW);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "
combustion := diff( y(t), t ) = y(t)^2*(1-y(t));" }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 43 "dsolve( \{combustion, y(0)=epsilon\}, y(t) );
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "soln := 1/(1+W( exp( 1/
epsilon-1-t )*(-epsilon+1)/epsilon ));" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 48 "simplify( eval(soln,t=0) ) assuming epsilon > 0;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "simplify( subs( y(t)=soln, n
ormal( (lhs-rhs)(combustion), expanded ) ) );" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 41 "particular := eval( soln, epsilon=1/30 );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot( particular, t=0..60 );
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "
" 0 "" {TEXT -1 30 " Pigment regeneration kinetics" }}{PARA 0 "" 0 ""
{TEXT -1 193 "From O.A.R. Mahroo & T. D. Lamb, ``Recovery of the human
photopic electroretinogram after bleaching exposures: estimation of p
igment regeneration kinetics'', J. Physiology 554.2 (2004) 417--437" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {URLLINK 17 "Trevor L
amb's home page" 4 "http://www.physiol.cam.ac.uk/staff/lamb/" "" }
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "restart; ali
as(W=LambertW):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "rate_lim
ited := diff( F(t), t ) = -F(t)/tau*(K_m/(F(t)+K_m));" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "dsolve( \{rate_limited, F(0)=B\}, F
(t) ) assuming B>0, K_m > 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 49 "previous := K_m*W( B/K_m*exp(B/K_m)*exp(-t/tau));" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "latex( previous );" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "eval( previous, [B=0.95,K_m=0.2,tau
=0.3] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(1-%,t=0..5
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 32 "eval( previous, exp(-t/tau)=z );" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "series(%,z,2) assuming K_m>0, B > 0
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "dsolve( \{diff( F(t), \+
t) = -F(t)/tau, F(0)=B*exp(B/K_m)\}, F(t) );" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 7 "Histor
y" }}{PARA 0 "" 0 "" {TEXT -1 1028 "The history of the function goes b
ack to J. H. Lambert (1728-1777). This worksheet is not the right med
ium to discuss the life of Lambert, but a short note is appropriate.\n
\nLambert was born the son of a tailor, and was expected by his father
to continue in that profession. His early fight for his education is
a remarkable story---he had to sell drawings and writings to his clas
smates to buy candles for night study, for example---but eventually hi
s talents were recognized and he got a position as tutor in a house wh
ich had a decent library. He then was able to educate himself, and w
ent on to produce fundamental discoveries in cartography (the Lambert \+
projection is still in use), hygrometry, pyrometry, statistics, philos
ophy (where he is actually more famous than as a mathematician), and p
ure mathematics. He is most noted as being the first person to prove \+
that pi; is irrational, which was an important step in proving that th
e classical problem of squaring the circle was impossible by straighte
dge and compass." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {URLLINK 17 "The St. Andrews' page on \+
Lambert" 4 "http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/La
mbert.html" "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {URLLINK 17 "Eric Weisstein's Lambert W page (CRC Handbook 1
998)" 4 "http://mathworld.wolfram.com/LambertW-Function.html" "" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 348 "Fundamental recent work: Richard Bellman, N. Hayes, Sir E. M. \+
Wright (1950's), Fritsch, Schaefer & Crowley, Siewert & Burniston, Com
tet, Gonnet, Corless, Gonnet, Hare, Jeffrey & Knuth (1993, 1996), D. B
arry et al (1995). The function will be taken up in the Digital Libra
ry of Mathematical Functions (Frank Olver is looking after it personal
ly). " }{TEXT 256 26 "My favourite applications:" }{TEXT -1 203 " pig
ment regeneration kinetics (and the quasi-steady state assumption for \+
enzyme kinetics in general), and (a paper I haven't seen yet) \"Rumour
s with general initial conditions\" (Belen & Pearce, 2004). " }{TEXT
257 30 "My least favourite application" }{TEXT -1 409 ": Xyce (a progr
am from Sandia Labs for nuclear weapons design)---Lambert W went in to
version 1.0, and was advertised as a highlight: \"Addition of Lambert
-W function as an option for diode and bipolar transistor devices. Whe
n this option is enabled, Lambert-W functions are used in place of exp
onential functions. Generally, this results in greater numerical robus
tness for notoriously hard-to-solve devices.\"" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0
"" {TEXT -1 26 "Mathematical facts about W" }}{SECT 0 {PARA 4 "" 0 ""
{TEXT -1 14 "A graph or two" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
27 "restart; alias(W=LambertW);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 63 "plot( [t*exp(t),t,t=-4..2], view=[-2..4,-4..2], colour=BLACK )
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot( \{W(x),W(-1,x)\}
, x=-2..4, view=[-2..4,-4..2] );" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "
" {TEXT -1 15 "Differentiation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "restart; alias(W=LambertW);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "diff( W(x), x );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "diff( u(x)*exp(u(x))-x , x )
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "isolate(%,diff(u(x),x)
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "normal( diff( W(x), \+
x$5 ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "normal( diff( W
(exp(x)), x$5 ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 6 "Series" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "restart; alias(W=L
ambertW);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "series( W(x), \+
x, 8 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "add( (-n)^(n-1)/
n! * x^n, n=1..7 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seri
es( W(x) + W(x)^2/2, x, 8 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 34 "add( -(-n)^(n-2)/n!*x^n, n=1..7 );" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "series( W(x
), x=-exp(-1), 3 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "alia
s( omega=RootOf( y + ln(y) = z, y ) );" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 22 "asympt( omega, z, 8 ):" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 17 "map( factor, % );" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 "Integration
" }}{PARA 0 "" 0 "" {TEXT -1 523 "``Everybody knows'' that W(z) is not
elementary. If it's true, where's the proof? It turns out to be a c
orollary of a result of Rosenlicht, though Liouville almost certainly \+
knew it. There is a paper by Bronstein, Corless, Davenport and Jeffre
y, ``The Lambert W Function is not Liouvillian'' giving details (to be
published). Dan Kalman has also worked on this.\n\nNonetheless, we m
ay integrate some expressions containing the Lambert W function, by th
e simple change of variable w = W(x) (and hence dx = (w+1)exp(w)dw ).
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "restart; alias(W=Lambert
W);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "int( W(x), x );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "simplify( diff( %, x ) - W(x
) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "int( sin(W(x)), x )
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "int( W(1/x), x );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 14 "Complex values" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "w := u+I*v;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "z := evalc( w*exp(w) );
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalc( Im(z) );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve( %=0, u );" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "divisions := plot( [-v*cot(v), v, \+
v=-5*Pi..5*Pi], view=[-15..15, -5*Pi..5*Pi], discont=true ): plots[dis
play](divisions);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "pts :=
map( t->[Re(t),Im(t)], [seq( LambertW(k,0.3+0.7*I), k=-2..2 )] );" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "ptplot := plot( pts, style=
POINT, symbolsize=40, colour=BLUE):" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 35 "plots[display](\{divisions,ptplot\});" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 3 "" 0 ""
{TEXT -1 15 "Applications II" }}{PARA 0 "" 0 "" {TEXT -1 46 "Now some \+
applications that use complex values." }}{SECT 0 {PARA 4 "" 0 ""
{TEXT -1 48 "A semi-infinite capacitor, or Borda's mouthpiece" }}
{PARA 0 "" 0 "" {TEXT -1 125 "From S.R.Valluri, D.J.Jeffrey and RMC, `
`Some applications of the Lambert W function to Physics'', Canadian J.
Physics, 2000." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N := 5;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "parallel := plot( [[t,Pi,t
=-5..5],[t,-Pi,t=-5..5]], colour=BLUE, thickness=3 ):" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "zpots := plot( [seq( [t,k*Pi/N,t=-5
..5], k=-N+1..N-1 )], linestyle=3, colour=BLACK ):" }}}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 36 "plots[display]( \{parallel, zpots\} );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "equip := theta -> [t+1+exp(
t)*cos(theta),theta+exp(t)*sin(theta),t=-5..2];" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 78 "semicap := plot( [[t,Pi,t=-4..0],[t,-Pi,t=-4..
0]], colour=BLUE, thickness=3 ):" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 69 "zetapots := plot( [seq(equip(Pi*k/N), k=-N+1..N-1 )],
colour=black ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plots[d
isplay]( \{semicap,zetapots\} );" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 92 "phi := zeta -> evalf( Im(zeta-1 - LambertW(ceil((Im(z
eta)-Pi)/(2*Pi)),exp(zeta-1)))/Pi )*Pi;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 17 "phi( 2.0+8.0*I );" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 17 "phi( 2.0-8.0*I );" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 34 "Statistical physics of a Tonks ga
s" }}{PARA 0 "" 0 "" {TEXT -1 192 "From Jean-Michel Caillol, ``Some ap
plications of the Lambert W function to classical statistical mechanic
s'', Journal of Physics A-Mathematical and General, (36) 42 October 20
03, 10431--10442." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 501 "This paper is of great interest---not only does he apply
W to reformulate known results (for a Tonks gas and for a classical f
luid with repulsive pair potentials) from Hauge and Hemmer (1963), but
he derives new mathematical properties of W, and interprets them phys
ically. He derives the dispersion relation in terms of the values of \+
W on its branch cut, and interprets W(z) as the two-dimensional comple
x electric potential created by the distribution of charges g(s) on th
e branch cut (see below)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "restart; alias(W=LambertW);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Im( W(-1.0) ) = (W(0,-1.0)-W
(-1,-1.0))/2.0/I;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "g := s
implify( -1/Pi * diff( (W(0,s)-W(-1,s))/2/I , s )) assuming s < -exp(
-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot( \{(g)\}, s=-5
..-exp(-1), view=[-5..0,0..1] );" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 16 "evalf(-exp(-1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1
124 "We may approximate the dispersion relation (with a little more ca
re we can do it very exactly but that obscures the formula)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Wint := z -> evalf( Int( g*l
n(1-z/s), s=-1000..-0.3678795 ));" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 26 "evalf( eval(g,s=-1000.) );" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 12 "Wint( 3.0 );" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 7 "W(3.0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "%
%-%;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA
4 "" 0 "" {TEXT -1 16 "Delayed dynamics" }}{PARA 0 "" 0 "" {TEXT -1
391 "Delays are ubiquitous; of particular interest for control applica
tions (think of a balky shower faucet, for example) and biological app
lications (e.g. the spread of malaria: after being infected, an organi
sm (human or mosquito) does not become infectious immediately). W com
es up in the simplest of all delay differential equations (and thus ca
n teach us about more complicated situations)." }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 27 "restart; alias(W=LambertW);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 50 "linearDelayDe := diff( y(t), t ) - lambda*y(
t-1) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "subs( y = (t->exp
(s*t)), linearDelayDe );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20
"factor( expand(%) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "so
l := Sum( C[k]*exp(W(k,lambda)*t), k=-N-1..N );" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 40 "diff( sol, t ) - lambda*eval(sol,t=t-1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "combine( % );" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simplify(expand(%));" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "f := evalc(exp(W(-1,-Pi/2)*t) + exp
(W(0,-Pi/2)*t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "diff( \+
f, t ) = -Pi/2* eval(f,t=t-1) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 35 "plot( \{f,2*exp(-Pi/2*t)\}, t=0..3 );" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Some more math" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Related functio
ns" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 54 "TreeT(x) = -W(-x); # More convenient for combinatoric
s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Wright_omega(z) = W( u
nwindK(z), z ); # Quite useful" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 36 "z = log(exp(z)) + 2*Pi*I*unwindK(z);" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 36 "glog(x) = -W(1/x); # From Dan Kalman" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Dilbert_Lambda = solve( y^2*exp(y)=
x, y ); # From Bill Gosper" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA
4 "" 0 "" {TEXT -1 37 "A non self-referential integral for W" }}{PARA
0 "" 0 "" {TEXT -1 298 "I derived the following integral some years ag
o, in order to prove that W(z) was a Stieltjes function. The derivati
on starts the same way as Jean-Michel Caillol's dispersion relation (u
sing the Cauchy formula) but we can make a further change of variables
to get an entirely real integral formula." }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "restart; alias(W=L
ambertW);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "W_over_z := z
-> evalf( Int( ((1-v*cot(v))^2 + v^2)/(z + v*csc(v)*exp(-v*cot(v))), \+
v=-Pi..Pi )/(2*Pi ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "W
_over_z( 3.832 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "W(3.83
2)/3.832;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 199 "This is not the fir
st integral for W. Steven Finch pointed out an entry in Bierens de Ha
an's tables, from an 1823 paper of Poisson. Recently, a number of oth
er integrals for W have been discovered." }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "
" {TEXT -1 27 "Non-harmonic Fourier series" }}{PARA 0 "" 0 "" {TEXT
-1 173 "We have already seen these in the solution of the delay differ
ential equation. They have many interesting properties (too many to g
o into here). We give a diagram, though." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 27 "restart; alias(W=LambertW);" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 100 "Harmonic := plot( [seq( [0,k], k=-4..4)], style
=POINT, symbol=DIAMOND, symbolsize=30, colour=BLUE ):" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "nonHarmonic := plot( [seq( [Re(W(k
,-0.2)),Im(W(k,-0.2)/2/Pi)], k=-4..4)], style=POINT, symbolsize=30, co
lour=RED, symbol=CIRCLE ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
41 "plots[display]( \{Harmonic,nonHarmonic\} );" }}}{EXCHG {PARA 0 ""
0 "" {TEXT -1 267 "What kinds of functions can you express by non-Harm
onic Fourier series? Most of them! But on finite intervals only; tho
ugh this is enough to solve delay differential equations with initial \+
conditions given on (say) -1 <= t <= 0. See ``Essential Maple'', chap
ter 3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23
"Iterated exponentiation" }}{PARA 0 "" 0 "" {TEXT -1 137 "In some sens
e this is what started it all. Euler was interested in this, and many
people continue to be interested; the topic is on the " }{URLLINK 17
"Math FAQ" 4 "http://www.faqs.org/faqs/sci-math-faq/specialnumbers/fxt
ofxeqx/" "" }{TEXT -1 270 ". R. Arthur Knoebel won the Chauvenet priz
e for his paper ``Exponentials Reiterated''; Baker and Rippon settled \+
the question of convergence in the complex plane (extending a result o
f Carlsson). Recently, Devaney and co-workers have investigated this \+
(see their paper " }{URLLINK 17 "Tying Hairs" 4 "http://citeseer.ist.p
su.edu/424410.html" "" }{TEXT -1 34 ".) More pictures can be found at
" }{URLLINK 17 "my outdated website" 4 "http://www.apmaths.uwo.ca/~rc
orless/frames/PAPERS/LambertW/index.html" "" }{TEXT -1 1 "." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "restart; alias(W=LambertW);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "x^(x^(x^(x^(`...`)))) = 2;"
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "What is x?" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 8 "x^2 = 2;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 22 "x := evalf( sqrt(2) );" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 9 "b := 1.0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
32 "for i to 30 do b := x^b; end do;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 9 "x := 'x';" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
26 "x^(x^(x^(x^(`...`)))) = y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 8 "x^y = y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(%,
y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "evalf( eval(%,x=sqrt
(2)) );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 185 "Try it with y=4, and \+
you get x^4 = 4, or x = sqrt(2)...oops, convergence actually matters :
-) Euler showed that the iteration converges for real x between exp(-
exp(1)) and exp(exp(-1))" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31
"exp(-exp(1.0)) ; exp(exp(-1.));" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 147 "plot( \{[(t*log(t)/W(-1,t*log(t)))^(1/t),t*log(t)/W(
-1,t*log(t)),t=-10..10],[(t*log(t)/W(0,t*log(t)))^(1/t),t*log(t)/W(0,t
*log(t)),t=-10..exp(1)]\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA
3 "" 0 "" {TEXT -1 18 "Concluding remarks" }}{PARA 0 "" 0 "" {TEXT -1
62 "The point of this talk is that simple functions occur often. " }
{TEXT 258 55 "Having a name and a standard helps to avoid reinvention
" }{TEXT -1 99 ". The pleasure of the talk (for me) was learning some
interesting science \"through the back door\"." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "I thank David Jeffrey, D
on Knuth, Dave Hare, Gaston Gonnet, Bill Gosper, Hualiang Zhong, Hui D
ing, and John Wright. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 ""
}}}}{MARK "0 1 0" 33 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS
0 1 2 33 1 1 }