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The Lambert function

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lambertW(k, x)


For integer k, the values represent the solutions of the equation yey = x.

lambertW is the inverse function of .

In the complex plane, the equation yey = x has a countably infinite number of solutions. They are represented by lambertW(k, x) with k ranging over the integers.

For all real x ≥ 0, the equation has exactly one real solution. It is represented by y=lambertW(x) or, equivalently, y=lambertW(0, x).

For all real x in the range -exp(-1) < x < 0, there are exactly two real solutions. The larger one is represented by y=lambertW(x), the smaller one by y=lambertW(-1, x).

Exactly one real solution lambertW(0, -exp(-1))= lambertW(-1, -exp(-1))= -1 exists for .

For , lambertW(k, x) takes no real value.

The values lambertW(-1, 0)=- infinity and lambertW(0, 0)=0 are implemented. Further, the result y is returned for some exact arguments of the form . For floating-point arguments a floating-point value is returned. For all other arguments, unevaluated function calls are returned.

The float attributes are kernel functions, i.e., floating-point evaluation is fast.

Environment Interactions

When called with a floating-point argument, the function is sensitive to the environment variable DIGITS which determines the numerical working precision.


Example 1

We demonstrate some calls with exact and symbolic input data:

lambertW(-3), lambertW(-1, -5/2), lambertW(1/2),
lambertW(5, I), lambertW(3, 1 + I), lambertW(-1, x + 1)

Some exact values are found:

lambertW(-1, -exp(-1)), lambertW(-1, -2*exp(-2)), 
lambertW(-1, -3/2*exp(-3/2)), lambertW(exp(1)), 
lambertW(2*exp(2)), lambertW(5/2*exp(5/2)),
lambertW(1, (3+4*I)*exp(3+4*I))

Floating point values are computed for floating-point arguments:

lambertW(-1, -0.3), lambertW(2000.0)

lambertW(-3, -0.277), lambertW(1, 2345.6)

Example 2

The functions diff, float, and series handle expressions involving the Lambert function:

diff(lambertW(k, x), x)

float(ln(3 + lambertW(sqrt(PI))))

series(lambertW(x), x = 0);
series(lambertW(x), x = -1/exp(1), 3);
series(lambertW(-1, x), x = -1/exp(1), 3);



An arithmetical expression, the "argument"


An arithmetical expression representing an integer, the "branch"

Return Values

Arithmetical expression.


R.M. Corless, D.J. Jeffrey and D.E. Knuth: "A sequence of Series for the Lambert W Function", in: Proceedings of ISSAC'97, Maui, Hawaii. W.W. Kuechlin (ed.). New York: ACM, pp. 197-204, 1997.

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