Domain coloring of the holomorphic tetration , with hue representing the function argument and brightness representing magnitude, for n = 2, 3, 4, ..., showing convergence to the infinitely iterated exponential between the two dots
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though Knuth's up arrow notation and the left-exponent xb are common.
Under the definition as repeated exponentiation, means , where n copies of a are iterated via exponentiation, right-to-left, i.e. the application of exponentiation times. n is called the "height" of the function, while a is called the "base," analogous to exponentiation. It would be read as "the nth tetration of a".
It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration.
Tetration is also defined recursively as
allowing for attempts to extend tetration to non-natural numbers such as real and complex numbers.
The two inverses of tetration are called super-root and super-logarithm, analogous to the nth root and the logarithmic functions. None of the three functions are elementary.
Tetration is used for the notation of very large numbers.
mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though Knuth's...
fifth hyperoperation. Pentation is defined to be repeated tetration, similarly to how tetration is repeated exponentiation, exponentiation is repeated multiplication...
Reuben Goodstein after the Greek prefix of n suffixed with -ation (such as tetration (n = 4), pentation (n = 5), hexation (n = 6), etc.) and can be written...
On the positive real numbers, the continuous super-logarithm (inverse tetration) is essentially equivalent: log ∗ n = ⌈ s l o g e ( n ) ⌉ {\displaystyle...
of two are common. The first 20 of them are: Also see Fermat number, tetration and lower hyperoperations. All of these numbers end in 6. Starting with 16...
exponentiation; this operation is sometimes called hyper-4 or tetration. Iterating tetration leads to another operation, and so on, a concept named hyperoperation...
number, larger than what can be represented even using power towers (tetration). However, it can be represented using layers of Knuth's up-arrow notation...
the set of real or rational numbers, is not commutative or associative. Tetration ( ↑↑ {\displaystyle \uparrow \uparrow } ), as a binary operation on the...
mechanical power take-off (PTO) Solar power tower, a type of solar power plant Tetration, a mathematical operation also known as power tower, hyperpower, or superexponentiation...
{e^{e^{\cdot ^{\cdot ^{e}}}}} _{k\ e'{\text{s}}}=e\uparrow \uparrow k} using tetration or Knuth's up-arrow notation. To see the divergence of the series (4)...
by which the magnitude of a googolplex could be represented, such as tetration, hyperoperation, Knuth's up-arrow notation, Steinhaus–Moser notation,...
function is a linear combination of its partial derivatives Euler's infinite tetration theorem – About the limit of iterated exponentiation Euler's rotation...
functions, but much more slowly than double exponential functions. However, tetration and the Ackermann function grow faster. See Big O notation for a comparison...
|x|={\sqrt {x^{2}}}} . Some examples of functions that are not elementary: tetration the gamma function non-elementary Liouvillian functions, including the...
hyperoperations, used to build addition, multiplication, exponentiation, tetration, etc. It was studied in 1986 in an investigation involving generalization...
} This sum is transcendental because it is a Liouville number. Like tetration, there is currently no accepted method of extension of the exponential...
terms, is correct to the first decimal place when n is positive. Also see Tetration: f n(1) = n√2. Using the other fixed point a = f(4) = 4 causes the series...
H-function Hyper operators Iterated logarithm Pentation Super-logarithms Tetration Lambert W function: Inverse of f(w) = w exp(w). Lamé function Mathieu...
Hyperexponential can refer to: The hyperexponential distribution in probability. Tetration, also known as hyperexponentiation. This disambiguation page lists articles...
logit. They are the inverse functions of the double exponential function, tetration, of f(w) = wew, and of the logistic function, respectively. From the perspective...
more classes of growth behavior, like the hyperoperations beginning at tetration, and A ( n , n ) {\displaystyle A(n,n)} , the diagonal of the Ackermann...
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