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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".
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).
The standard Lambert W function expresses exact solutions to what is called ``transcendental algebraic equations of the form:. exp(-c*x) = a_o*(x-r) where a_o, c, and r are real constants.
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).
An explicit expression for the diode current can be obtained in terms of the Lambert W-function (also called the Omega function). [3] A guide to these manipulations follows. A new variable w {\displaystyle w} is introduced as
Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
Short Run Consumption Function, the initial endowed wealth level of a particular individual in the Life-cycle Income Hypothesis; First step of the W0–W6 scale for the classification of meteorites by weathering; W 0 may refer to the principal branch of the Lambert W Function
where W represents Lambert's W function. As the limit y = ∞ x (if existent on the positive real line, i.e. for e −e ≤ x ≤ e 1/e) must satisfy x y = y we see that x ↦ y = ∞ x is (the lower branch of) the inverse function of y ↦ x = y 1/y.