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The product logarithm Lambert W function plotted in the complex plane from −2 − 2i to 2 + 2i The graph of y = W(x) for real x < 6 and y > −4. The upper branch (blue) with y ≥ −1 is the graph of the function W 0 (principal branch), the lower branch (magenta) with y ≤ −1 is the graph of the function W −1. The minimum value of x is ...
Suppose z is defined as a function of w by an equation of the form = where f is analytic at a point a and ′ Then it is possible to invert or solve the equation for w, expressing it in the form = given by a power series [1]
Taylor series are used to define functions and "operators" in diverse areas of mathematics. In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the matrix exponential or matrix logarithm.
The test functions used to evaluate the algorithms for MOP were taken from Deb, [4] Binh et al. [5] and Binh. [6] The software developed by Deb can be downloaded, [ 7 ] which implements the NSGA-II procedure with GAs, or the program posted on Internet, [ 8 ] which implements the NSGA-II procedure with ES.
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
An investigation by The Times found that the problem of questionable mushroom products may be far more widespread. Testing done on 33 samples purchased at smoke shops in Los Angeles and San Diego ...
on , and as the Weierstrass M-test shows the series converges uniformly over , the sum and the integral may be interchanged. As the factor ( z − a ) n {\displaystyle (z-a)^{n}} does not depend on the variable of integration w {\displaystyle w} , it may be factored out to yield