# How to extend this extension of tetration? [closed]

if $0\le b<1$, then $a↑↑b = a^b$

if $b\ge1$, then $a↑↑b = a^{a↑↑(b-1)}$

if $b<0$, then $a↑↑b = \log_a(a↑↑(b+1))$

so for example,

$2↑↑\pi = 2^{2^{2^{2^{0.1415926...}}}} = 21.5963561$

How can I extend this to complex numbers?

https://www.desmos.com/calculator/p8qcvngczb3

## closed as unclear what you're asking by user8795, Shailesh, Arnaldo, JonMark Perry, mlcMay 19 '17 at 3:29

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• I suppose you could do it that way, but does it have nice continuity/analyticity characteristics? – Oscar Lanzi May 19 '17 at 0:31
• I just want to ask how to extend this, if you could tell me what you are unsure what is being asked that would be appreciated. – Theoretical May 19 '17 at 15:24
• Are these edits acceptable? – Theoretical May 19 '17 at 16:29
• Presuming your question is focused on $a↑↑z$ where $a$ is real and $z$ is complex. I think this is interesting and will try applying this to an old question of mine concerning the nature of $i↑↑n$. – Simply Beautiful Art May 19 '17 at 23:19
• Here's the graph I made of $i↑↑x$ for $-2<x<9$. – Simply Beautiful Art May 19 '17 at 23:52