Lambert W function

In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function f ( w ) = w e w {\displaystyle f(w)=we^{w}} , where w {\displaystyle w} is any complex number and e w {\displaystyle e^{w}} is the exponential function. The function is named after Johann Lambert, who considered a related problem in 1758. Building on Lambert's work, Leonhard Euler described the W function per se in 1783. For each integer k {\displaystyle k} there is one branch, denoted by W k ( z ) {\displaystyle W_{k}\left(z\right)} , which is a complex-valued function of one complex argument. W 0 {\displaystyle W_{0}} is known as the principal branch. These functions have the following property: if z {\displaystyle z} and w {\displaystyle 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 W 0 {\displaystyle W_{0}} and W − 1 {\displaystyle W_{-1}} suffice: for real numbers x {\displaystyle x} and y {\displaystyle y} the equation y e y = x {\displaystyle ye^{y}=x} can be solved for y {\displaystyle y} only if x ≥ − 1 e {\textstyle x\geq {\frac {-1}{e}}} ; yields y = W 0 ( x ) {\displaystyle y=W_{0}\left(x\right)} if x ≥ 0 {\displaystyle x\geq 0} and the two values y = W 0 ( x ) {\displaystyle y=W_{0}\left(x\right)} and y = W − 1 ( x ) {\displaystyle y=W_{-1}\left(x\right)} if − 1 e ≤ x < 0 {\textstyle {\frac {-1}{e}}\leq x<0} . The Lambert W function's branches cannot be expressed in terms of elementary functions. It...