tetration

Tetration is the operation of iterated exponentiation and is the fourth operation in the hyperoperation sequence. For a real base a and a positive integer height n, tetration is defined recursively by a^^1 = a and a^^n = a^(a^^(n−1)). Equivalently, a^^n equals a^(a^(...^a)) with n copies of a, evaluated from the top down. In Knuth’s up-arrow notation this is written a↑↑n; in other common notations it is denoted ^n a or a^^n.

Finite tetration yields very large numbers quickly. For example, a^^2 = a^a and a^^3 = a^(a^a); for a = 2, 2^^4 = 2^(2^(2^2)) = 2^16 = 65536.

The infinite power tower, or infinite tetration, refers to the limit of a^^n as n tends to infinity, when it exists. It satisfies the fixed-point equation x = a^x. Real convergence occurs for e^{-e} ≤ a ≤ e^{1/e}. If this range holds, the limit lies between e^{-1} and e, attaining e^{-1} at a = e^{-e} and e at a = e^{1/e}. For bases outside this interval, the tower diverges or fails to converge.

Domain and extensions: Real-valued tetration with non-integer heights typically requires a positive base to avoid complex values; negative bases introduce subtleties and are generally restricted to certain heights or treated within the complex plane. Extensions to complex bases and heights exist through analytic continuation, but remain mathematically subtle and are areas of ongoing study.