Solving x^y=y^x

[Erwin]

(10-24-2018 10:35 PM)ijabbott Wrote:  

(10-24-2018 09:58 PM)Albert Chan Wrote:  Found a very close solution for infinite tetration c = x^c, or ln(c)/c = ln(x).
From the video (~ 7:00), using basic algebra, he got c = W(-ln(x)) / (-ln(x))

For x^y = y^x, ln(y)/y = ln(x)/x, so just substitute above: c=y, ln(x)=ln(x)/x
--> y = W(-ln(x)/x) / (-ln(x)/x)

I like his series of videos on geometric algebra - something that isn't currently handled very well by our favourite, hand-held, symbolic calculators.

Hallo,

The question in the video for Infinite Tetration was the point for me to put out my old calculator. So I can only show a solution for Infinite Tetration on the HP71b to see if the number converges.
Define the W-Funktion y=x*exp(x) and put the formula on minute 12:10 in the keyboard. Looks like like this:

Definiton of the W-function in BASIC
10 DEF FNW(X) = FNROOT(0,10,FVAR*EXP(FVAR)-X)

and then his example with 1.4 on the keyboard calling there defined W-function:
FNW((-LN(1.4)))/(-LN(1.4))
Result: 1.88666330624

and then his example with SQR(2) on the keyboard calling there defined W-function:
FNW((-LN(SQR(2)))/(-LN(SQR(2)))
Result: 1.99999999998 not exactly 2 but this seems is a limitation on the more then 30 year old calculator.

The possibility for defining a function and using it straight from the keyboard was a revolution in this times.

best regards
Erwin