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Just as triangular numbers sum the numbers from to , and factorials take their product, the exponential factorial exponentiates. The exponential factorial is defined recursively as =, =. For example, the exponential factorial of 4 is = These numbers grow much more quickly than regular factorials. [95] Falling factorial
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.
The rising and falling factorials are well defined in any unital ring, and therefore can be taken to be, for example, a complex number, including negative integers, or a polynomial with complex coefficients, or any complex-valued function.
Exponential function: raises a fixed number to a variable power. Hyperbolic functions: formally similar to the trigonometric functions. Inverse hyperbolic functions: inverses of the hyperbolic functions, analogous to the inverse circular functions. Logarithms: the inverses of exponential functions; useful to solve equations involving exponentials.
Comparison of Stirling's approximation with the factorial. In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of .
The limit, should it exist, is a positive real solution of the equation y = x y. Thus, x = y 1/y. The limit defining the infinite exponential of x does not exist when x > e 1/e because the maximum of y 1/y is e 1/e. The limit also fails to exist when 0 < x < e −e. This may be extended to complex numbers z with the definition:
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
The exponential factorials grow much more quickly than regular factorials or even hyperfactorials. The number of digits in the exponential factorial of 6 is approximately 5 × 10 183 230. The sum of the reciprocals of the exponential factorials from 1 onwards is the following transcendental number: