Estimate logarithm quickly
08-21-2021, 05:41 PM
Post: #3
 Albert Chan Senior Member Posts: 2,551 Joined: Jul 2018
RE: Estimate logarithm quickly
(08-21-2021 03:58 PM)Albert Chan Wrote:  To proof it, it required this inequality (also not proofed, suggestions welcome):

(1-w)^(2-w) * (1+w)^(2+w) < 1, ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ where 0 < w < 1

My attempt to proof this, by showing there is no local extremum for 0 < w < 1
To get the extrema of above is messy. Instead, we solve extrema of ln(expression)

XCAS> f := (2-w)*ln(1-w) + (2+w)*ln(1+w)
XCAS> diff(f,w)

-ln(1-w) - (2-w)/(1-w) + ln(1+w) + (2+w)/(1+w)

XCAS> numer(ans()) // local extrema if df/dw = 0

-ln(w+1) + ln(-w+1) + w^2*ln(w+1) - w^2*ln(-w+1) + 2*w

Divide all by 2, then replace with identity atanh(x) = ln((x+1)/(1-x))/2

(w^2-1) * atanh(w) + w = (w^2-1) * (w + w^3/3 + w^5/5 + ...) + w

All coefficients of w's is positive, 1/(2k-1) - 1/(2k+1) = 2/(4k^2-1) > 0

XCAS> series((w^2-1)*atanh(w) + w, w,0,10)

2*w^3/3 + 2*w^5/15 + 2*w^7/35 + 2*w^9/63 + w^11*order_size(w)

Within 0 < w < 1, there is no local extremum, f is bounded by edges.

XCAS> limit(f, w=0, +1) ﻿ ﻿ ﻿ ﻿ ﻿ → 0
XCAS> limif(f, w=1, -1) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → -infinity

f = ln((1-w)^(2-w) * (1+w)^(2+w)) < 0 ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ → (1-w)^(2-w) * (1+w)^(2+w) < 1
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 Messages In This Thread Estimate logarithm quickly - Albert Chan - 08-21-2021, 03:39 PM RE: Estimate logarithm quickly - Albert Chan - 08-21-2021, 03:58 PM RE: Estimate logarithm quickly - Albert Chan - 08-21-2021 05:41 PM RE: Estimate logarithm quickly - trojdor - 08-21-2021, 06:31 PM RE: Estimate logarithm quickly - Albert Chan - 08-21-2021, 11:42 PM RE: Estimate logarithm quickly - EdS2 - 08-23-2021, 06:44 AM RE: Estimate logarithm quickly - Albert Chan - 10-06-2021, 10:58 PM RE: Estimate logarithm quickly - Albert Chan - 10-18-2021, 01:02 PM RE: Estimate logarithm quickly - Albert Chan - 11-20-2021, 02:02 PM RE: Estimate logarithm quickly - Albert Chan - 11-22-2021, 01:49 AM RE: Estimate logarithm quickly - Albert Chan - 11-23-2021, 09:01 PM RE: Estimate logarithm quickly - Albert Chan - 04-26-2022, 05:01 PM RE: Estimate logarithm quickly - Albert Chan - 11-26-2021, 06:24 PM RE: Estimate logarithm quickly - Csaba Tizedes - 10-07-2021, 06:25 PM RE: Estimate logarithm quickly - Albert Chan - 10-22-2021, 02:15 PM RE: Estimate logarithm quickly - Albert Chan - 04-27-2022, 10:02 PM RE: Estimate logarithm quickly - klesl - 04-26-2022, 05:52 PM RE: Estimate logarithm quickly - vaklaff - 04-28-2022, 04:33 PM RE: Estimate logarithm quickly - klesl - 04-28-2022, 05:02 PM

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