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(arithmetic) The arithmetic operator consisting of repeated exponentiation, by analogy with exponentiation being repeated multiplication and multiplication being repeated addition, ${\displaystyle ^{b}a}$ denoting ${\displaystyle a}$ to the power of ${\displaystyle a}$ to the power of ${\displaystyle \ldots }$ to the power of ${\displaystyle a}$, in which ${\displaystyle a}$ appears ${\displaystyle b}$ times.

Domain coloring of the holomorphic tetration ${\displaystyle {}^{z}e}$, with hue representing the function argument and brightness representing magnitude

(arithmetic) The arithmetic operator consisting of repeated exponentiation, by analogy with exponentiation being repeated multiplication and multiplication being repeated addition, ${\displaystyle ^{b}a}$ denoting ${\displaystyle a}$ to the power of ${\displaystyle a}$ to the power of ${\displaystyle \ldots }$ to the power of ${\displaystyle a}$, in which ${\displaystyle a}$ appears ${\displaystyle b}$ times.

Tetration by period

(arithmetic) The arithmetic operator consisting of repeated exponentiation, by analogy with exponentiation being repeated multiplication and multiplication being repeated addition, ${\displaystyle ^{b}a}$ denoting ${\displaystyle a}$ to the power of ${\displaystyle a}$ to the power of ${\displaystyle \ldots }$ to the power of ${\displaystyle a}$, in which ${\displaystyle a}$ appears ${\displaystyle b}$ times.

Tetration by escape