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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".
This is a list of special function eponyms in mathematics, to cover the theory of special functions, the differential equations they satisfy, named differential operators of the theory (but not intended to include every mathematical eponym).
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
In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. Lagrange inversion is a special case of the inverse function theorem .
One problem with p/m or +/- is that it is limited to only two branches, but the numerical subscript allows indexing all branches of the Lambert W function (this article fails to note this). The "old" notation is also used in computer algebra systems, which are the most important domain of use for the Lambert W function.
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).
There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease ...
This formula can be derived by expanding the exponential generating function using the Taylor series for the exponential function, and then collecting terms with the same exponent. [10] It allows B n to be interpreted as the n th moment of a Poisson distribution with expected value 1.