The LambertW function is defined to be the multivalued inverse of the functionw →we

 w
 . It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches ofW, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containingW.

On the Lambert W function

An extension of sinc interpolation on $\mathbb{R}$ to the class of
algebraically decaying functions is developed in the paper. Similarly to the
classical sinc interpolation we establish two types of error estimates. First
covers a wider class of functions with the algebraic order of decay on
$\mathbb{R}$. The second type of error estimates governs the case when the
order of function's decay can be estimated everywhere in the horizontal strip
of complex plane around $\mathbb{R}$. The numerical examples are provided.

Sinc approximation of algebraically decaying functions

Exact analytical solutions of the parameters of real solar cells using Lambert W-function

Analytical approximations for real values of the Lambert W -function

Keen interest in the development and utilization of renewable distributed generation (DG) has been currently observed worldwide. The reliability impact of this highly variable energy source is an important aspect that needs to be assessed as renewable power penetration becomes increasingly significant. Distribution system adequacy assessment including wind-based and solar DG units during different modes of operation is described in this paper. Monte Carlo simulation (MCS) and analytical technique are used in this work with a novel utilization of the clearness index probability density function (pdf) to model the solar irradiance using MCS. The results show that there is no significant difference between the outcomes of the two proposed techniques; however, MCS requires much longer computational time. The effect of islanding appears in the improvement of the loss of load expectation (LOLE) and loss of energy expectation (LOEE).

Adequacy Evaluation of Distribution System Including Wind/Solar DG During Different Modes of Operation

Submittal of an algorithm for consideration for publication in Communications of the ACM implies unrestricted use of the algorithm within a computer is permissible. Description Purpose. WEW solves the transcendental equation we ~ = x for w, given x > 0, by an iteration that converges much more rapidly than either Newton's method or fixed-point iteration. The user provides x = X. The routine returns w = WEW and the last relative correction e, = EN. Two versions are described here. Version A produces CDC 6600 machine accuracy (48 bits), and the relative error should be approximately eJ. Version B produces at least six significant figures, and the relative error should be approximately eJ. Iteration. Assuming x > 0, we may rewrite the equation defining w as w + log(w) = log(x). (1) For a given approximation w. to w, let w~+l = w~ + 8, be a much better approximation. Substitution into (1) yields 8. +log(1-k-8,,/w.) = log x-log w.,-w. Using the approximation [1] log (1 + 8/w) ~ (Sw + 1/6 83)/ (w 2 + 2/3 8w) and clearing fractions yields the following quadratic equation for 8, : Solving for the root that tends to zero as z.-~ 0 gives 2ZnWn 8n (1 + w,-2/3 z,) + ((1 + w, + 2/3 z,) 2-2z,) ½ \" General permission to republish, but not for profit, an algorithm is granted, provided that reference is made to this publication, to its date of issue, and to the fact that reprinting privileges were granted by permission of the Association for Computing Machinery. Work performed under the auspices of the U.S. Atomic Energy Commission. This has a continued fraction expansion [3] 2WnZn 8n = 2Zn 2(1 W w,)-2(1 + w, + 2/3 z,) 2z, 2(1 + w.-1-2/3 z.) for which the third convergent yields sufficient accuracy. If we ignore the quantity 2/3 z, in the third term, we obtain the iteration formula Initial guesses. For small values of x, the given equation has a series solution due to L. Euler [2]. A Pad~ rational fraction approximation to this series is 1 +7/3x+ 5/6x 2\" As computed from (5), wo(x) < w(x), good to within 5 percent if x = 2.5 and much better for smaller values of x. For larger values of x we may use wo = log(x), (6) which has a maximum relative error no greater than 37 percent for x …

Fred N. Fritsch

Papers

Solution of the transcendental equation wew = x

Solution of an Overdetermined System of Equations in the L1 Norm (Remark on Algorithm 478).