55\alias {lambertW0 }
66\alias {lambertWm1 }
77\title {Lambert - W Function }
8+ \encoding {UTF - 8 }
9+
810\description {
911The Lambert - W function is defined as the multivalued inverse of the function
1012\eqn {z = f(W ) = We ^ W }{z = f(W ) = W * exp(W )}. The function has two branches. The
1113principal branch is defined on the interval
12- \eqn {\left [- \frac {1 }{e }, \infty \right )}{[- 1 / e , Infinity )} and is always greater
14+ \eqn {\left [- \frac {1 }{e }, + \infty \right )}{[- 1 / e , + ∞ )} and is always greater
1315than or equal to \eqn {- 1 }. The second branch is defined on the interval
1416\eqn {\left [- \frac {1 }{e }, 0 \right )}{[- 1 / e , 0 )} and is always less than or equal
1517to \eqn {- 1 }. The real - valued function is not defined for values less than
1618\eqn {- \frac {1 }{e }}{- 1 / e }.
1719}
20+
1821\usage {
1922lambertW0(x )
2023lambertWm1(x )
2124}
25+
2226\arguments {
2327 \item {x }{\strong {numeric } vector of values }}
2428\details {
@@ -35,12 +39,14 @@ simplifications. There are many applications in which the Lambert-W function is
3539useful , such as combinatorics , physics , and hydrology. The interested reader is
3640directed to the references for more detail.
3741}
42+
3843\value {
3944Both functions return the appropriate values in the intervals for which they are
4045defined. Outside of those intervals , they will return \code {NaN }, except that
4146\code {lambertW0(Inf )} will return its limit \code {Inf } and \code {lambertWm1(0 )}
4247will return its limit - \code {Inf }.
4348}
49+
4450\references {
4551Corless , R. M. , Gonnet , G. H. , Hare , D. E. , Jeffrey , D. J. , Knuth , D. E.
46521996 " On the Lambert W function" , \emph {Advances in Computational Mathematics },
@@ -51,14 +57,17 @@ Fritsch, F. N.; Shafer, R. E. & Crowley, W. P.
5157\emph {Communications of the ACM }, \bold {16 }, 123 -- 124 ,
5258Association for Computing Machinery (ACM ) < doi : 10.1145 / 361952.361970 >
5359}
60+
5461\author {
5562Avraham Adler \email {Avraham.Adler @ gmail.com }
5663}
64+
5765\seealso {
5866This package provides similar functionality to the
5967\code {\link [gsl : Lambert ]{Lambert }} functions in the \pkg {gsl } package without
6068having to obtain or install the entire \acronym {GSL }.
6169}
70+
6271\examples {
6372lambertW0(exp(1 )) # # Should equal 1, as 1 * exp(1) = e.
6473lambertW0(0 ) # # Should equal 0, as 0 * exp(0) = 0.
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