Search results
Results from the WOW.Com Content Network
Like tetration, there is currently no accepted method of extension of the exponential factorial function to real and complex values of its argument, unlike the factorial function, for which such an extension is provided by the gamma function. But it is possible to expand it if it is defined in a strip width of 1.
The factorial function of a ... Stirling's formula provides the first term in an asymptotic series that becomes ... The exponential factorial is defined ...
The exponential function (in blue), and the sum of the first n + 1 terms ... is the factorial ... The power series definition of the exponential function makes ...
The logarithm of the gamma function satisfies the following formula due to Lerch: = ′ (,) ′ (), where is the Hurwitz zeta function, is the Riemann zeta function and the prime (′) denotes differentiation in the first variable. The gamma function is related to the stretched exponential function.
In mathematics, the exponential function can be characterized in many ways. This article presents some common characterizations, discusses why each makes sense, and proves that they are all equivalent. The exponential function occurs naturally in many branches of mathematics. Walter Rudin called it "the most important function in mathematics". [1]
The number e is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function.It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted .
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 exponential function e x (in blue), and the sum of the first n + 1 terms of its Taylor series at 0 (in red). The exponential function e x {\displaystyle e^{x}} (with base e ) has Maclaurin series [ 12 ]