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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Subject: Lambert's W and elementary functions
Date: Sun, 4 May 2025 08:03:22 -0700
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2+2 = 4
2*2 = 4
2^2 = 4


2 = 1+1
2 = root 2 * root 2
2 = x^x

So I'm looking for the form of x^x = y, then the Internet
says it can find the y' th derivative, and then talks about
Lambert W function, yet I'm wondering about x^x=2, because
2 is a lucky number since a bunch of its operations all result
the same product.

1-1=0
1*1=1
1^1=1

5/4 ^ 5/4

1.5596119 ^ 1.5596119 ~ 2.00000413265
1.559610375 ^ 1.559610375 ~ 1.99999972711

1.5596104694 ^ 1.5596104694 ~
1.559610469463 ^ 1.559610469463 ~ 2

Aw, dang, the Inverse Symbolic Calculator is down and has been for some 
time, tragique.

"2 LN LAMBERTW e^x = 1.55961046946"

https://www.hpmuseum.org/cgi-bin/archv018.cgi?read=132639

"x = e^W(log(2)) ~ 1.55961046946..." -- 
http://voodooguru23.blogspot.com/2020/06/the-omega-constant-and-lambert-w.html

https://old.reddit.com/r/calculus/comments/1gr14zd/how_would_an_equation_like_this_be_solved/?rdt=63191


"After reading tons about it for the past few years, I'm pretty much 
convinced
that indeed [Lambert's W]'s a firm candidate for the next "elementary"
transcendental function to be added to the classic trigonometric, 
exponential,
and logarithmic ones (which, essentially, can all be reduced to 
exponentials
and inverse exponentials (i.e. logarithms) of arbitrary complex values)."

Re: Lambert's W on the HP-33s
Message #13 Posted by Valentin Albillo on 12 Feb 2008, 6:46 a.m.,
https://www.hpmuseum.org/cgi-bin/archv018.cgi?read=132639

https://eric.ed.gov/?q=%22A+new+elementary+function+for+our+curricula%22&id=EJ720055

https://en.wikipedia.org/wiki/Lambert_W_function
https://mathworld.wolfram.com/LambertW-Function.html
   "W(1) can be considered ... a sort of golden ratio ... since 
e^(-W(1)) = W(1)"

e to the ..., zero minus ..., these sorts things then can get
windows and boxed about 2, and its terms, the convolutive,
and about zero, and its terms.

"Occurrances of Lambert's Function"

On the Lambert W Function
University of Waterloo
https://cs.uwaterloo.ca › 1993/03 › W.pdf

"The solution of x^x^a = b is
exp(W (a log b)/a)," ....


1.559610469463 + e^(1/e) ~ 3.004


So, finding x^x = 2 sort of gets to oscillating, about 0^0 and 1^1,
then about x^2 +- 1, these kinds of things.

1 = 1
2'nd root 2 = 1.41421356237...
x = 1.55961046946...

Then, I'm looking ar figuring out a different sort of
function than Lambert's W and its branches, "inverse
self exponent", say, figuring that if Lambert's W is a
transcendental function that about deserves to be
an elementary function, then like other sorts elementary
functions there are orthogonal functions, then the entire
body of work of transforms can be applied to that,
figuring out some trajectifolds.

Here though mostly what's of interest is windowing
and boxing about a 2x2 square, in the convolutive setting,
i.e. symmetrically, up and down operations about the
hypergeometric of course, or boxing 0, 1, infinity into
a 2x2 square about 0, 1, 2.