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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".
Lambert-W step-potential [3] Multistate Landau–Zener Models [4] See also. List of quantum-mechanical systems with analytical solutions;
Heaviside step function: ... Lambert W function: Inverse of f(w) = w exp(w). ... is an example of continuous function that is nowhere differentiable;
Nontrivial solutions can be written explicitly using the Lambert W function. The idea is to write the equation as a e b = c {\displaystyle ae^{b}=c} and try to match a {\displaystyle a} and b {\displaystyle b} by multiplying and raising both sides by the same value.
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).
Many mathematicians exclude non-analytic functions such as the absolute value function or discontinuous functions such as the step function, [9] [6] but others allow them. Some have proposed extending the set to include, for example, the Lambert W function. [10] Some examples of functions that are not elementary: tetration; the gamma function
The Lambert W function is the function () ... For example, substituting −1 for ... the function inversion is taken as a crucial step.
For example, for , the solutions of = are approximately those of () / =, namely = and =. For a graphical solution, one method is to set each side of a single-variable transcendental equation equal to a dependent variable and plot the two graphs , using their intersecting points to find solutions (see picture).