Posts: 94

Threads: 15

Joined: Apr 2009

12/15/2009, 08:52 PM
(This post was last modified: 12/15/2009, 11:15 PM by Base-Acid Tetration.)
Go in the negative direction from z=-1 in the complex plane. Set the cut lines from even negative numbers run to negative IMAGINARY infinity; and odd negative numbers to positive IMAGINARY (screwed that up, sorry) infinity. what happens then? your path can just zigzag between the upper and lower half planes without circling any of the singularities once. I may be wrong about how the cut lines of the double logarithmic singularities can be set.

Posts: 1,616

Threads: 102

Joined: Aug 2007

(12/15/2009, 08:52 PM)Base-Acid Tetration Wrote: Go in the negative direction from z=-1 in the complex plane. Set the cut lines from even negative numbers run to negativeinfinity; and odd negative numbers to positive infinity. what happens then? your path can just zigzag between the upper and lower half planes without circling any of the singularities once.

If the domain is not simply connected (i.e. each two paths in the domain are homotopic) you probably can not define tetration there.

Posts: 94

Threads: 15

Joined: Apr 2009

12/15/2009, 11:08 PM
(This post was last modified: 12/15/2009, 11:16 PM by Base-Acid Tetration.)
(12/15/2009, 10:56 PM)bo198214 Wrote: If the domain is not simply connected (i.e. each two paths in the domain are homotopic) you probably can not define tetration there.

You can see that the domain that I defined is indeed simply connected.

I meant negative and positive IMAGINARY infinities. screwed that up...

slalom.bmp (Size: 243.63 KB / Downloads: 483)

Posts: 1,616

Threads: 102

Joined: Aug 2007

(12/15/2009, 11:08 PM)Base-Acid Tetration Wrote: You can see that the domain that I defined is indeed simply connected. is there something I missed?

Ah, ok, then your description was wrong (you said negative and positive infinity but you meant negative and positive imaginary infinity).

Hm, ok, but what do you want to show with this construction?

Posts: 94

Threads: 15

Joined: Apr 2009

12/15/2009, 11:23 PM
(This post was last modified: 12/15/2009, 11:42 PM by Base-Acid Tetration.)
(12/15/2009, 11:15 PM)bo198214 Wrote: Hm, ok, but what do you want to show with this construction?

i meant to show that you didn't have to circle the singularities an integer number of times.

but by the arbitrariness of definition of the principal branch of the logarithm, the fact was utterly trivial!

the branch I am giving is pieced from these branches, written in mike3's notation:

for the nth branch point.

Also tetration is now defined for the whole real line \ integers < -2, but those are useless as the while riemann surface has everything that maps to each of the values in [-2,-1].