New Sum of Powers Log Function

[Albert Chan]

(03-31-2021 01:27 PM)Namir Wrote:  I do beleive we must use the solution path of -1 with the Lambert function. The W0(x) does not give the same answers.

Although not a proof, I had shown the reason for this here

Here is another way. If x = 0, we have s = Σ(1, k=1..n) = n
This is when both branches of LambertW gives the same value.

p = -W(-ln(N)/s) / ln(N), where N = n+1/2

With s=n ⇒ p=x+1=1, we do not need the +1/2 correction:

s = ∫(1, t=1/2 .. n+1/2) = ∫(1, t=0 .. n)

p = -W(-ln(n)/n) / ln(n) = ln(n) / ln(n) = 1       // W both branches, see identities

For n≠s, we have p≠1, but 2 solutions for p.
Dropped term 0.5**p/p is a decreasing function, so we want the maximum p.

In other words, we pick most negative value of W(-ln(N)/s), i.e. -1 branch.