Solving x^y=y^x

[Albert Chan]

(10-26-2018 04:54 PM)Erwin Wrote:  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.

Unless W function is built-in, it might be easier solve directly for infinite tetration.
So, for any x, solve for y, such that x^y = y

Below solved with my Casio FX-115MS solver, using x as initial guess for y:

For x=1.4, y = 1.886663306
For x= √2, y = 2
For x=1.5, y does not converge

Last x > e^(1/e), thus diverged. see Exponential Reiterated