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LambertW - The Lambert W function

Calling Sequence

LambertW(x)

LambertW(k, x)

Parameters

x

-

algebraic expression

k

-

algebraic expression, understood to be an integer

Description

• 

The LambertW function satisfies

LambertWxⅇLambertWx=x

• 

As the equation yⅇy=x has an infinite number of solutions y for each (non-zero) value of x, LambertW has an infinite number of branches. Exactly one of these branches is analytic at 0. In Maple this branch is referred to as the principal branch of LambertW, and is denoted by LambertW(x).  The other branches all have a branch point at 0, and these branches are denoted in Maple by LambertW(k, x), where k is any non-zero integer.  (The principal branch can also be referred to as LambertW(0, x).)

• 

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 ⅇ1, 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 ⅇ1..infinity, while the image of infinity..ⅇ1 under LambertW(x) is the curve ycoty+Iy, for y in 0..Pi.

  

Similarly, the branch corresponding to -1, LambertW(-1, x), is real-valued on the interval ⅇ1..0, while the image of infinity..ⅇ1 under this branch is the curve ycoty+Iy, for y in -Pi .. 0.

• 

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 2IPi 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 ycoty&plus;Iy, for y in 2kPi..2k&plus;1Pi if 0<k and y in 2k&plus;1Pi..2k&plus;2Pi 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.

• 

The asymptotic behavior of LambertW at complex infinity and at 0 (for the non-principal branches) is given by

LambertWk,xlogk,xloglogk,x+m=0n=0cm,nloglogk,xm+1logk,xm+n+1

  

where logx denotes the principal branch of the logarithm, logk&comma;x&equals;logx&plus;2IkPi and the cm&comma;n are constants independent of k. The expansion for LambertW(-1, x) is not valid for x tending to 0 along the negative real axis (the effect of the branch point at -exp(-1) must be considered), but holds otherwise.

• 

The LambertW function is closely related to the tree generating function Tx popularized in the analysis of algorithms discipline.  When Tn counts the number of distinct oriented trees with n labeled vertices and Tx&equals;n&equals;1infinityTnxnn&excl;, then Tx&equals;LambertWx.

Examples

LambertW0

0

(1)

LambertW&ExponentialE;1

1

(2)

LambertW1.5&plus;2.5I

0.9698925542&plus;0.5301457549I

(3)

LambertW1&comma;0.1

3.577152064

(4)

LambertW1&comma;5&pi;2

52I&pi;

(5)

LambertW100&comma;&pi;&plus;&ExponentialE;I

LambertW100&comma;&pi;&plus;I&ExponentialE;

(6)

evalf

5.017543738&plus;627.4530224I

(7)

The alias command can be used to shorten the name, if desired

aliasW&equals;LambertW

W

(8)

solve3x<&ExponentialE;x&comma;x

RealRange&infin;&comma;OpenW13&comma;RealRangeOpenW1&comma;13&comma;&infin;

(9)

&DifferentialD;&DifferentialD;xWx

Wx1&plus;Wxx

(10)

&int;cosWk&comma;x&DifferentialD;x

12Wk&comma;x&plus;12xcosWk&comma;xWk&comma;x&plus;12xsinWk&comma;x

(11)

See Also

alias, initialfunctions, Wrightomega

References

  

Corless, R.M.; Gonnet, G.H.; Hare, D.E.G.; Jeffrey, D.J.; and Knuth, D.E. "On the Lambert W Function." Advances in Computational Mathematics, Vol. 5, (1996): 329-359.


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