Lambert W function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Lambert W function, also called the omega function or product logarithm, is a <a href="/facts/Multivalued_function/WH5ua6BL">multivalued function</a>, namely the <a href="/facts/Branch_point/lJ7Y6Iht">branches</a> of the <a href="/facts/Converse_relation/EVGlRSz7">converse relation</a> of the function f(w) = wew, where w is any <a href="/facts/Complex_number/2w7mqI7e">complex number</a> and ew is the <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a>. The function is named after <a href="/facts/Johann_Heinrich_Lambert/CG3IHVB9">Johann Lambert</a>, who considered a related problem in 1758. Building on Lambert's work, <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a> described the W function per se in 1783.
For each integer k there is one branch, denoted by Wk(z), which is a complex-valued function of one complex argument. W0 is known as the <a href="/facts/Principal_branch/XrN2gNon">principal branch</a>. These functions have the following property: if z and w are any complex numbers, then

w
        
          e
          
            w
          
        
        =
        z
      
    
    {\displaystyle we^{w}=z}

holds if and only if

w
        =
        
          W
          
            k
          
        
        (
        z
        )
         
         
        
           for some integer 
        
        k
        .
      
    
    {\displaystyle w=W_{k}(z)\ \ {\text{ for some integer }}k.}

When dealing with real numbers only, the two branches W0 and W−1 suffice: for real numbers x and y the equation

y
        
          e
          
            y
          
        
        =
        x
      
    
    {\displaystyle ye^{y}=x}

can be solved for y only if x ≥ −⁠1/e⁠; yields y = W0(x) if x ≥ 0 and the two values y = W0(x) and y = W−1(x) if −⁠1/e⁠ ≤ x < 0.
The Lambert W function's branches cannot be expressed in terms of <a href="/facts/Elementary_function/wVFyDwu8">elementary functions</a>. It is useful in <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>, for instance, in the enumeration of <a href="/facts/Tree_graph/TCWk4Joy">trees</a>. It can be used to solve various equations involving exponentials (e.g. the maxima of the <a href="/facts/Planck%2527s_law/7Umkdb9m">Planck</a>, <a href="/facts/Bose%25E2%2580%2593Einstein_distribution/Utb2jN3n">Bose–Einstein</a>, and <a href="/facts/Fermi%25E2%2580%2593Dirac_distribution/aLo4bton">Fermi–Dirac</a> distributions) and also occurs in the solution of <a href="/facts/Delay_differential_equation/fM5wzXKr">delay differential equations</a>, such as y′(t) = a y(t − 1). In <a href="/facts/Biochemistry/fa2pWEBp">biochemistry</a>, and in particular <a href="/facts/Enzyme_kinetics/LJ0lWoMc">enzyme kinetics</a>, an opened-form solution for the time-course kinetics analysis of <a href="/facts/Michaelis%25E2%2580%2593Menten_kinetics/lYFykW3p">Michaelis–Menten kinetics</a> is described in terms of the Lambert W function.

Lambert W function open-in-new

Lambert W function