
An Abel function A(z) is a function associated
with a function f (z) that satisfies Abel's functional equation:
A( f (z)) = A(z) + 1 

(NEW!)  A matrix based on the Bell matrix. 

Addition is a binary operation defined on the complex numbers. For more information on this website, click here. 

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.


A Bell matrix B[ f ] is a matrix associated with a function f (x) such that:
for all g(x). 

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A Carleman matrix M[ f ] is a matrix associated with a function f (x) such that:
for all g(x). 

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Continuous things are usually contrasted with discrete things.
A good example of something continuous is the set of real numbers. 

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. 

Discrete things are usually contrasted with continuous things.
A good example of something discrete is the set of integers. 

See iteration 



The prefix endo is usually used with terms with a domain and range like functions, morphisms, and functors. An endofunction is a function whose domain and range are the same set. 



Not to be confused with ordinary factorials. The exponential factorial is defined by EF(x) = x^{EF(x1)}. For more information on this website click here. For more information from other websites, see Wikipedia. 

An exponential function is any function of the form:
exp_{b}(z) = b^{z} 

Exponentiation is the process of raising a base to a power.
Exponentiation is usually denoted: a^b or a^{b}. 

See iteration. 

A functional equation is an equation in which a function is being solved for instead of a variable. Functional equations provide many valuable tools for performing iteration on analytic functions. For more information on this website, click here. For more information from other sources, see MathWorld, PlanetMath, or Wikipedia. 

The nth member of the hyperoperation sequence is known as hypern. 

A hyperoperator is any member of the hyperoperator sequence. 

The hyperoperator sequence is:
hyper1 = addition, hyper2 = multiplication, hyper3 = exponentiation, hyper4 = tetration, hyper5 = pentation ... and so on. 

See tetrate. 

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The infinitely iterated exponential is T^{∞}x. 

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 exponentiation can be interpreted as either:
iterated exponentials or iterated powers, but usually the former is implied, so on this website the former terminology will be used. 

An iterated exponential is a function of the form:
expSBP(b,n)(z) for integer n. 

An iterated power is a function of the form:
pow^{n}_{b}(z) for integer n. 

The tfold iterate of a function f (x) is f ^{t}(x) where t is constant. 

Iteration is the process of applying a function repeatedly, or using the output of a function as its input, a given number of times. 

(NEW!)  The iterational of a function f (x) from x is f ^{t}(x) where x is constant. 

The Lambert Wfunction is the inverse function of:
f (x) = xe^{x} 

A logarithm is an inverse function of an exponential function 

A matrix is a 2dimensional array of elements.
The elements of a matrix are usually integers or real numbers. 

A number associated with "how big" a number is.
The elements of a matrix are usually integers or real numbers. 

See iterational. 

Pentation is the orbit of a tetrational function from 1. 

A power function is any function of the form:
pow_{n}(z) = z^{n} 

A productlogarithm is the inverse function of:
f (x) = xb^{x} 

Involves logarithms. 

A root is an inverse function of a power function. 

A root is also another name for a zero of a function. 

See endofunction. 

An Schroeder function S(z) is a function associated
with a function f (z) that satisfies
the Schroeder functional equation:
S( f (z)) = c S(z). 

See tetrational. 

Old name for tetration coined by Bromer. 

A superlogarithm is an inverse function of a tetrational function. 

See tetrate. 

A superroot is an inverse function of a tetrate function. 

Usual. 

(NEW!)  This term can have two meanings: as a verb, and as a noun.


Tetration is the iterational of an exponential from 1. For more information on tetration, two really great places to start are: MathWorld and Wikipedia, the page on this website about tetration, and of course, this rest of this entire website is about tetration. 

The baseb tetrational is a function from x to btetrax. 

A zero of a function is a number c such that f (c) = 0 