Sum[(-n)^(n-1)/n! z^n,{n,\[Inf]}] == ProductLog[z]
John Conway
conway at math.Princeton.EDU
Sun Aug 9 19:46:52 CEST 1998
On Sun, 9 Aug 1998, Wouter Meeussen wrote:
>
> I take it that Lambert_W is the same as what Mathematica calls "ProductLog".
> (quote: ProductLog[z] gives the principal solution for w in z = w e^w )
I thought at first that this was what was traditionally called the
"Zech's logarithm", Z(x), defined by Z(log(x)) = log(x+1), but it doesn't
seem to be. May I ask about series expansions for that?
> I most certainly find its Series development Weird ! it's (-n)^(n-1)
> *************************************************************************
> this means that
>
> Sum[(-n)^(n-1)/n! z^n,{n,\[Infinity]}] == ProductLog[z]
I'm fairly sure that this is one of the standard examples for the
theorems of Burmann and Lagrange that show how to express one function
as a power series in terms of another.
John Conway
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