Tetration
This site is about tetration. Tetration is an obscure mathematical operation that currently falls under the category of Pure Mathematics, because it has few applications. Even so, many mathematicians have been interested in tetration, because of the historical significance of its relatives in the hyper-operator sequence, and because it is easy to make very large numbers with tetration.
There are many related functions that are not exactly tetration but are inter-connected in some way. When someone thinks about sine or cosine, they generally aren't the only things that come to mind. Usually other related functions such as tangent or the rest of the trigonometric functions also come to mind, or perhaps a vague memory that they have something to do with triangles.
One function that is intimately related to tetration, that
is much more well-known than tetration,
is the Lambert W-function,
also known as the product-logarithm. This function allows
many differential equations and equations involving
exponentials to be solved in closed-form rather than having
to resort to numerical analysis only.
The product-logarithm
is so well known that both Maple (LambertW) and
Mathematica (ProductLog)
provide it as a built-in function.
Tetration and its associated functions have been studied for hundreds of years, so it has quite a history behind it. Most of its history has been the investigation of infinite tetration, also known as the infinitely iterated exponential. This function, along with the square super-root, are isomorphic to the Lambert W-function, so any equation that one can solve can be solved by any of them.
This website is an attempt to bring together those things that are related to tetration, or could possibly be of use in understanding more about tetration, namely various methods that have been used to generalize iteration to continuous iteration, fractional iteration, real-analytic iteration, or complex-analytic iteration (which could also be called holomorphic iteration). This relates to tetration, because the definition (in words) of tetration (used here) is:
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Pages listed alphabetically by title:
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Talks about applications of integer tetration, infinitely iterated exponentials, the Lambert W-function, continuous iteration, analytic iteration, and notations used to express them. |
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Self-explanatory, some Maple and Mathematica code to calculate functions discussed on this website. |
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The definitions of tetration, super-log, super-root, and hyper-ops. |
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About the exponential factorial defined as ef(1) = 1 and ef(x) = xef(x-1). Most discussions of this function involve its definition over integer x, but here we discuss how to extend it to real x. The method used here is described in Functional Equations in more detail. |
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Self-explanatory, some functional equations related to tetration. |
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Self-explanatory, a list of terms used on this website. |
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