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The exponential factorial is a positive integer n raised to the power of n − 1, ... For example, 262144 is an exponential factorial since = ...
The exponential factorial is defined recursively as =, =. For example, the exponential factorial of 4 is 4 3 2 1 = 262144. {\displaystyle 4^{3^{2^{1}}}=262144.} These numbers grow much more quickly than regular factorials.
2.2 Exponential function 2.3 Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship 2.4 Modified-factorial denominators
The ratio of the factorial!, that counts all permutations of an ordered set S with cardinality, and the subfactorial (a.k.a. the derangement function) !, which counts the amount of permutations where no element appears in its original position, tends to as grows.
A complex-analysis version of this method [4] is to consider ! as a Taylor coefficient of the exponential function = =!, computed by Cauchy's integral formula as ! = | | = +. This line integral can then be approximated using the saddle-point method with an appropriate choice of contour radius r = r n {\displaystyle r=r_{n}} .
Exponential functions with bases 2 and 1/2. In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. . The exponential of a variable is denoted or , with the two notations used interchangeab
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
In this article, the symbol () is used to represent the falling factorial, and the symbol () is used for the rising factorial. These conventions are used in combinatorics , [ 4 ] although Knuth 's underline and overline notations x n _ {\displaystyle x^{\underline {n}}} and x n ¯ {\displaystyle x^{\overline {n}}} are increasingly popular.