Tetration extended to real exponents
- From: "Ingolf Dahl" <ingolf.dahl@xxxxxxxxx>
- Date: 13 Dec 2006 00:36:08 -0800
Tetration (also hyperpower, power tower, super-exponentiation,
and hyper4) can be seen as the fourth operation in the chain addition,
multiplication, power...
I have proposed a way to extend tetration to the range of real numbers,
see
http://forums.wolfram.com/mathgroup/archive/2006/Dec/msg00207.html and
http://forums.wolfram.com/mathgroup/archive/2006/Dec/msg00133.html. I
have also made a Mathematica notebook, with an extended PowerTower
function, available for download on my web page
http://web.telia.com/~u31815170/Mathematica/.
I got into this business by solving the equations f(f(x)) = x^2 +1 and
f(f(x)) = exp (x). (Solving for f(x)). The zero point value for the
first equation is
f(0) = 0.6420945043908285
and for the second equation
f(0) = 0.498743364531670879182837375041686052396842401697695
with derivatives
f'(0) = 0.87682850111901739234161348908444926737095344685410,
f''(0) = 0.489578037438247862041903165991088781934758253721,
f'''(0) = 0.1222616742183662064640316453590024071635142693
I am grateful for any comments on this, and hope that someone can check
if my ideas are sound and relevant.
Thus Pi to the superpower e (or "Pi tetrated to e") is
1885451.906681809677772360465630708698
Math is fun
Ingolf Dahl
ingolf.dahl@xxxxxxxxx
.
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