Tetration

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Glossary

Abel function	An Abel function A(z) is a function associated with a function f (z) that satisfies the Abel functional equation.
Abel functional equation	The Abel functional equation is an equation which the Abel function A(z) must satisfy:         A( f (z)) = A(z) + 1
analytic	A special case of continuous meaning infinitely differentiable, plus some other requirements. The other requirements for a function being analytic differ depending on whether the requirements are for real analytic or complex analytic functions.
Bell matrix	A Bell matrix B[ f ] is a matrix associated with a function f (x) such that:         B[ f (g)] = B[g] B[ f ],         for some other function g(x).
Carleman matrix	A Carleman matrix M[ f ] is a matrix associated with a function f (x) such that:         M[ f (g)] = M[ f ] M[g],         for some other function g(x).
continuous	Continuous things are usually contrasted with discrete things. A good example of something continuous is the set of real numbers.
continuous iteration	Continuous iteration is a generalization of discrete iteration. This effectively turns a function f (x) into a function g(n, x) = f n(x). Two ways of doing this are to satisfy:         g(n, x) = f n(x)         for integer n which is a weaker form of continuous iteration, or:         g(1, x) = f (x)                 and         g(y, x) = f (g(y − 1, x))         for all real y, and         g(y, x) = g(y − 1, f (x))         for all real y which is a stronger form of continuous iteration.
continuously iterated exponentiation	Continuously iterated exponentiation can be interpreted as either:         a continuously iterated exponential function or         a continuously iterated power function, but usually the former is implied, so on this website the former less ambiguous terminology will be used.
discrete	Discrete things are usually contrasted with continuous things. A good example of something discrete is the set of integers.
discrete iteration	See iteration
dynamical system
evolution function
exponential function	An exponential function is any function of the form:         expb(z) = bz
exponentiation	Exponentiation is the process of raising a base to a power. Exponentiation is usually denoted: a^b or ab.
family
flow
hyper-n	The n-th member of the hyper-operation sequence is known as hyper-n.
hyper-operator	A hyper-operator is any member of the hyper-operator sequence.
hyper-operator sequence	The hyper-operator sequence is:         hyper1 = addition,         hyper2 = multiplication,         hyper3 = exponentiation,         hyper4 = tetration,         hyper5 = pentation ... and so on.
hyperpower function	A hyperpower function is any function of the form:         hprn(z) = Tnz
infinitely iterated exponential	The infinitely iterated exponential is T∞x.
inverse function	An inverse function is a function g(x) associated with a function f (x) that satisfies:         f (g(x)) = x         and         g( f (x)) = x over some interval of x. An inverse function is usually denoted: f −1(x).
iterated exponentials	An iterated exponential is a function of the form:         exp b n (z)         for integer n.
iterated powers	An iterated power is a function of the form:         pownb(z)         for integer n.
iterate	The t-fold iterate of a function f (x) is f t(x) where t is constant.
iterate family	An iterate family is a family of iterates f t(x) indexed by t.
iteration	Iteration is the process of applying a function repeatedly, or using the output of a function as its input, a given number of times.
Lambert W-function	The Lambert W-function is the inverse function of:         f (x) = xex
logarithm	A logarithm is an inverse function of an exponential function
matrix	A matrix is a 2-dimensional array of elements. The elements of a matrix are usually integers or real numbers.
Munafo number	A number associated with "how big" a number is. The elements of a matrix are usually integers or real numbers.
orbit	The orbit of a function f (x) from x is f t(x) where x is constant.
orbit family	An orbit family is a family of orbits f t(x) indexed by x.
pentation	Pentation is the orbit of a tetrational function from 1.
power function	A power function is any function of the form:         pown(z) = zn
product-logarithm	A product-logarithm is the inverse function of:         f (x) = xbx
root (function)	A root is an inverse function of a power function.
root (number)	A root is also another name for a zero of a function.
self-map
Schroeder function	An Schroeder function S(z) is a function associated with a function f (z) that satisfies the Schroeder functional equation.
Schroeder functional equation	The Schroeder functional equation is an equation which the Schroeder function S(z) must satisfy:         S( f (z)) = c S(z)
super-exponentiation	Old name for tetration
super-logarithm	A super-logarithm is an inverse function of a tetrational function.
super-root	A super-root is an inverse function of a hyperpower function
tetration	Tetration is the orbit of an exponential function from 1. Tetration is usually denoted: x^^y or yx, but here it is denoted Tyx for clarity, consistency, and usability reasons. For more information on tetration, two really great places to start are: MathWorld and Wikipedia, and the page on this site: Tetration
tetrational function	A tetrational function is any function of the form:         tetb(z) = zb
zero of a function	A zero of a function is a number c such that f (c) = 0