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{{subpages}} {{Image|TetrationReal.jpg|right|400px|Fig.1. Tetration <math>\mathrm{tet}_b(x)</math> for <math>b\!=\!\mathrm{e}</math>, <math>b\!=\!2</math>, <math>b\!=\!\exp(1/\mathrm{e})</math>, and <math>b\!=\!\sqrt{2}~</math> versus <math>x</math>.}} '''Tetration''' is a rapidly growing [[mathematical function]], which was introduced in the 20th century and proposed for the representation of huge numbers in the [[mathematics of computation]]. For positive integer values of its argument <math>x</math>, tetration <math>\mathrm{tet}_b(x)</math> on base <math>b</math> can be defined with: <math>{\ \mathrm{tet}_b(x) = \ \atop {\ }} {{\underbrace{b^{b^{\cdot^{\cdot^{b}}}}}} \atop {x}}~</math> For real values of the argument <math>x</math> and various values of the base <math> b </math>, this <math>\mathrm{tet}_b(x)</math> is plotted in Fig.1. Up to year 2008, this function has not been listed among elementary functions, it is not implemented in [[programming languages]] and it is not used for the internal representation of data in computers. In this article, the generalizaiton of tetration for complex (and, in particular, real) values of its argument is described. Tetration is assumed to be a [[holomorphic function]], at least for positive values of the [[real part]] of its argument. This tetration is used to construct the holomorphic extension of the iterated [[Exponential function|exponential]] <math>\exp^c(z)</math> for the case of non-integer values of the number <math>c</math> of iterations.

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