Re: Bringing back an old tetration curiosity?

On Sep 17, 6:12 pm, theronr...@xxxxxxxxx wrote:
On Sep 17, 10:12 am, mike3 <mike4...@xxxxxxxxx> wrote:

I dug up an interesting thread here where a method was given that
might be able to extend
tetration to the reals, at least for a base of sqrt(2). You can see
the thread at this link:

http://groups.google.com/group/sci.math/browse_frm/thread/39a7019f905...

The proposed method in this thread is flawed. The generated "curve"
has branches, so it is not a function. This is easily observable in hi-
res plot of the "curve".

This proves experimentally that there is no function f: (-2; +oo) -> R
such that:
f(x) = y <=> f(-y) = -x (1)
f(x + 1) = sqrt(2)^f(x) (2)
f(0) = 1 (3)

After rethinking I think that the above statement is a bit
speculative. However I'm sure that there is no continuous f: (-2; +oo)
-> R that satisfies (1), (2) and (3). Here is why:

http://img403.imageshack.us/img403/1466/sqrttetrxjx1.png

This is a 5000x5000 plot of f(x) for -1 <= x <= 0 <= y <= 1.

Theron

.

Relevant Pages

  • Re: Bringing back an old tetration curiosity?
    ... might be able to extend ... tetration to the reals, at least for a base of sqrt. ... The proposed method in this thread is flawed. ...
    (sci.math)
  • Re: Bringing back an old tetration curiosity?
    ... might be able to extend ... tetration to the reals, at least for a base of sqrt. ... the method says that to generalize tetration to real ... to a non-integer tower, ...
    (sci.math)
  • Re: Bringing back an old tetration curiosity?
    ... might be able to extend ... tetration to the reals, at least for a base of sqrt. ... to a non-integer tower, ...
    (sci.math)
  • Re: are imaginary exponents "truly" defined that way?
    ... definition in order to extend it to cases where b is negative, ... Other extensions would preserve different rules. ... sqrtis a FUNCTION, defined on the non-negative reals, such that ... Your grasp of the "laws of exponents" is inadequate. ...
    (sci.math)
  • Re: very complex numbers
    ... be impossible to extend complex numbers into three ... finite dimensional over R and which strictly extend C." Quaternions ... qualification that might lead the OP into thinking that the rationals, ... reals, and complex numbers are the only fields possible. ...
    (sci.math)