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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".
Toggle the table of contents. ... in the Taylor series of a function of w. ... The Lambert W function is the function () ...
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
For any functions and and any real numbers and , the derivative of the function () = + with respect to is ′ = ′ + ′ (). In Leibniz's notation , this formula is written as: d ( a f + b g ) d x = a d f d x + b d g d x . {\displaystyle {\frac {d(af+bg)}{dx}}=a{\frac {df}{dx}}+b{\frac {dg}{dx}}.}
The purple curve and circle is the image of a small circle around the branch point z=0; the red curves are the images of a small circle around the point z=-1/e. The range of W 0 is inside the C-shaped black curve. The range of each of the other branches is a band between two black curves that represent points on the negative real axis (a black ...
Toggle the table of contents. ... In particular, its Taylor formal series diverges: ... is the Lambert W function. This ...
Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As an entire function, it is of order 1 (meaning that log log | 1 / Γ( z ) | grows no faster than log | z | ), but of infinite type (meaning that log | 1 / Γ( z ) | grows faster than any multiple of | z ...
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 ...