Base Pi Hyper-Operations
#1
Question 
What properties do hyperoperations base pi have?
π+π~6.283.
π*π~9.870.
π^π~36.462.
π^^π~19,924,084,821,713,599,984,983,799,892,180,468,936,939.296.
For \(n\in\Bbb N\), is π[n]π ever an integer/rational/constructible number/algebraic?
Like before:
Quote:The .png file attached to this post is smaller than the .txt file attached to this.
The .txt file attached might not look right on a smart phone.
The text graphs need a monospaced font.
The image attached to this post looks blurry, because of the size of the image.
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Also, the picture is 443 pixels by 314 pixels wide.
Using logarithmic semi-operators base the pith root of pi, π{x}π For some reason, any non real valuedness is not showing up. Isn't the analytic continuation of the Kneser method not real valued at the pith root of pi? Although, I do have some issues with tetration at base the pith root of pi.
Using logarithmic semi-operators base the pith root of pi, here is graph of \(y=\Re(\pi\{x\}\pi)\), from \(-\pi\) to \(\pi\):


Attached Files Thumbnail(s)
   

.txt   Y=Real(π{x}π).txt (Size: 1.71 KB / Downloads: 27)
Please remember to stay hydrated.
ฅ(ミ⚈ ﻌ ⚈ミ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\
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#2
Catullus, be clearer. What is this a graph of?

You write \(\pi[x]\pi\). What does that mean? Please work better. Explain better.
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#3
(06/24/2022, 08:41 AM)JmsNxn Wrote: Catullus, be clearer. What is this a graph of?

You write \(\pi[x]\pi\). What does that mean? Please work better. Explain better.
It is a graph of \(\def\ {\sqrt[\pi]{\pi}}\def\e{\log_\ ^{x-1}(\pi)}y=\Re(\exp_\ ^{x-1}(\e+\e))\), with the analytic continuation of the Kneser method, where x goes from \(-\pi\) to \(\pi\).
Please remember to stay hydrated.
ฅ(ミ⚈ ﻌ ⚈ミ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\
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#4
I managed to make \(\def\ {\sqrt[\pi]{\pi}}\def\e{\log_\ ^{x-1}(\pi)}\def\p{\Im(\exp_\ ^{x-1}(\e+\e))}\p\) show up.
Here is a graph of \(y=\p\), where x goes from \(-\pi\) to \(\pi\):


Attached Files Thumbnail(s)
   

.txt   Y=Imaginary(π{x}π).txt (Size: 1.71 KB / Downloads: 27)
Please remember to stay hydrated.
ฅ(ミ⚈ ﻌ ⚈ミ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\
Reply


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