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The notation convention chosen here (with W 0 and W −1) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth. [3]The name "product logarithm" can be understood as follows: since the inverse function of f(w) = e w is termed the logarithm, it makes sense to call the inverse "function" of the product we w the "product logarithm".
The range of the Lambert W function, showing all branches. The orange curves are images of either the positive or the negative imaginary axis. The black curves are images of the positive or negative real axis (except for the one that intersects −1, which is the image of part of the negative real axis).
If the assertions about analyticity are omitted, the formula is also valid for formal power series and can be generalized in various ways: It can be formulated for functions of several variables; it can be extended to provide a ready formula for F(g(z)) for any analytic function F; and it can be generalized to the case ′ =, where the inverse ...
In fact, the set of functions with a convergent Taylor series is a meager set in the Fréchet space of smooth functions. Even if the Taylor series of a function f does converge, its limit need not be equal to the value of the function f (x). For example, the function
The Wright omega function along part of the real axis In mathematics , the Wright omega function or Wright function , [ note 1 ] denoted ω , is defined in terms of the Lambert W function as: ω ( z ) = W ⌈ I m ( z ) − π 2 π ⌉ ( e z ) . {\displaystyle \omega (z)=W_{{\big \lceil }{\frac {\mathrm {Im} (z)-\pi }{2\pi }}{\big \rceil }}(e^{z}).}
It is the value of W(1), where W is Lambert's W function. The name is derived from the alternate name for Lambert's W function, the omega function. The numerical value of Ω is given by Ω = 0.56714 32904 09783 87299 99686 62210... (sequence A030178 in the OEIS). 1/Ω = 1.76322 28343 51896 71022 52017 76951... (sequence A030797 in the OEIS).
Function () = =, represented as a Matplotlib plot, using a version of the domain coloring method [1]. In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form
The inverse gamma function also has the following asymptotic formula [7] + ( ()), where () is the Lambert W function. The formula is found by inverting the Stirling approximation , and so can also be expanded into an asymptotic series.