lamW (version 2.2.4)

lambertW: Lambert-W Function

Description

The Lambert-W function is defined as the multivalued inverse of the function \(z = f(W) = We^W\). The function has two branches. The principal branch is defined on the interval \(\left[-\frac{1}{e}, +\infty\right)\) and is always greater than or equal to \(-1\). The second branch is defined on the interval \(\left[-\frac{1}{e}, 0\right)\) and is always less than or equal to \(-1\). The real-valued function is not defined for values less than \(-\frac{1}{e}\).

Usage

lambertW0(x)
lambertWm1(x)

Value

Both functions return the appropriate values in the intervals for which they are defined. Outside of those intervals, they will return NaN, except that lambertW0(Inf) will return its limit Inf and lambertWm1(0)

will return its limit -Inf.

Arguments

x

numeric vector of values

Author

Avraham Adler

Details

The Lambert-W function is defined for all real \(x \geq -\frac{1}{e}\). It has two values in the interval \(\left(-\frac{1}{e}, 0\right)\). The values strictly greater than \(-1\) are assigned to the “principal” branch, also referred to as \(W_0\), and the values strictly less than \(-1\) are assigned to the “secondary” branch, referred to as \(W_{-1}\). For non-negative \(x\), only the principal branch exists as the other real-valued branch approaches negative infinity as \(x\) approaches \(0\). The algorithms used to calculate the values predominantly follow those in the reference with some simplifications. There are many applications in which the Lambert-W function is useful, such as combinatorics, physics, and hydrology. The interested reader is directed to the references for more detail.

References

Corless, R. M., Gonnet, G. H., Hare, D. E., Jeffrey, D. J., Knuth, D. E. 1996 "On the Lambert W function", Advances in Computational Mathematics, 5, 329--359, Springer <doi:10.1007/BF02124750>

Fritsch, F. N.; Shafer, R. E. & Crowley, W. P. 1973 "Solution of the transcendental equation (we^w = x)", Communications of the ACM, 16, 123--124, Association for Computing Machinery (ACM) <doi:10.1145/361952.361970>

See Also

This package provides similar functionality to the Lambert functions in the gsl package without having to obtain or install the entire GSL.

Examples

lambertW0(exp(1))      ## Should equal 1, as 1 * exp(1) = e.
lambertW0(0)           ## Should equal 0, as 0 * exp(0) = 0.
lambertW0(-exp(-1))    ## Should equal -1.
lambertWm1(-exp(-1))   ## Should also equal -1.
A <- -2 * exp(-2)
lambertWm1(A)          ## Should equal -2