New Sum of Powers Log Function
03-31-2021, 04:06 PM
Post: #15
 Albert Chan Senior Member Posts: 1,800 Joined: Jul 2018
RE: New Sum of Powers Log Function
(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.

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 Messages In This Thread New Sum of Powers Log Function - Namir - 03-29-2021, 04:53 PM RE: New Sum of Powers Log Function - C.Ret - 03-29-2021, 08:39 PM RE: New Sum of Powers Log Function - Albert Chan - 03-29-2021, 10:47 PM RE: New Sum of Powers Log Function - Albert Chan - 03-30-2021, 01:44 AM RE: New Sum of Powers Log Function - Albert Chan - 03-30-2021, 08:35 PM RE: New Sum of Powers Log Function - Albert Chan - 03-30-2021, 10:26 PM RE: New Sum of Powers Log Function - Namir - 03-30-2021, 11:05 AM RE: New Sum of Powers Log Function - Albert Chan - 03-30-2021, 04:35 PM RE: New Sum of Powers Log Function - Paul Dale - 03-30-2021, 11:41 AM RE: New Sum of Powers Log Function - Gene - 03-30-2021, 01:43 PM RE: New Sum of Powers Log Function - C.Ret - 03-30-2021, 04:01 PM RE: New Sum of Powers Log Function - Namir - 03-30-2021, 05:56 PM RE: New Sum of Powers Log Function - Namir - 03-31-2021, 01:27 PM RE: New Sum of Powers Log Function - Albert Chan - 03-31-2021 04:06 PM RE: New Sum of Powers Log Function - Albert Chan - 03-31-2021, 09:25 PM RE: New Sum of Powers Log Function - Namir - 03-31-2021, 02:19 PM RE: New Sum of Powers Log Function - Albert Chan - 04-01-2021, 02:56 PM RE: New Sum of Powers Log Function - Namir - 04-01-2021, 06:05 PM RE: New Sum of Powers Log Function - Albert Chan - 04-01-2021, 11:05 PM RE: New Sum of Powers Log Function - Namir - 04-01-2021, 11:55 PM RE: New Sum of Powers Log Function - Albert Chan - 04-02-2021, 01:29 AM RE: New Sum of Powers Log Function - Albert Chan - 04-02-2021, 01:18 PM RE: New Sum of Powers Log Function - Namir - 04-04-2021, 03:41 PM

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