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Lambert W-function

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1: 4.13 Lambert W -Function
§4.13 Lambert W -Function
The Lambert W -function W ( x ) is the solution of the equation … We call the solution for which W ( x ) W ( - 1 / e ) the principal branch and denote it by Wp ( x ) . … Properties include: … For integral representations of all branches of the Lambert W -function see Kheyfits (2004). …
2: 4.44 Other Applications
For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). For an application of the Lambert W -function to generalized Gaussian noise see Chapeau-Blondeau and Monir (2002). For other applications of the Lambert W -function see Corless et al. (1996).
3: 4.45 Methods of Computation
§4.45(iii) Lambert W -Function
For x [ - 1 / e , ) the principal branch Wp ( x ) can be computed by solving the defining equation W e W = x numerically, for example, by Newton’s rule (§3.8(ii)). … Similarly for Wm ( x ) in the interval [ - 1 / e , 0 ) . …
4: 27.7 Lambert Series as Generating Functions
§27.7 Lambert Series as Generating Functions
Lambert series have the form …
5: 4.48 Software
Links to research literature for the Lambert W -function and for test software are included also. …
§4.48(iv) Lambert W -Function
6: 33.22 Particle Scattering and Atomic and Molecular Spectra
For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function W - η , + 1 2 ( 2 ρ ) . …
7: 26.7 Set Partitions: Bell Numbers
or, specifically, N = e Wp ( n ) , with properties of the Lambert function Wp ( n ) given in §4.13. …
8: 12.19 Tables
  • Fox (1960) includes modulus and phase functions for W ( a , x ) and W ( a , - x ) , and several auxiliary functions for x - 1 = 0 ( .005 ) 0.1 , a = - 10 ( 1 ) 10 , 8S.

  • 9: 4.1 Special Notation
    k , m , n integers.
    10: 28.8 Asymptotic Expansions for Large q
    28.8.9 W m ± ( x ) = e ± 2 h sin x ( cos x ) m + 1 { ( cos ( 1 2 x + 1 4 π ) ) 2 m + 1 , ( sin ( 1 2 x + 1 4 π ) ) 2 m + 1 ,