In mathematics, the exponential function can be characterized in many ways. This article presents some common characterizations, discusses why each makes...

the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. The exponential of a...

The stretched exponential function f β ( t ) = e − t β {\displaystyle f_{\beta }(t)=e^{-t^{\beta }}} is obtained by inserting a fractional power law into...

Artin–Hasse exponential Bacterial growth Baker–Campbell–Hausdorff formula Cell growth Barometric formula Beer–Lambert law Characterizations of the exponential function...

for real values of x may be defined in a few different equivalent ways (see Characterizations of the exponential function). Several of these methods may...

the characterization, as a smooth manifold, is up to diffeomorphism. Characterizations of the category of topological spaces Characterizations of the...

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...

not a root of any polynomial with rational coefficients, as is eα for any non-zero algebraic α. Characterizations of the exponential function Transcendental...

K_{\text{x}}}} where the limit definition of the exponential has been used (see also Characterizations of the exponential function). More generally B (...

sets. Gaussian functions arise by composing the exponential function with a concave quadratic function: f ( x ) = exp ⁡ ( α x 2 + β x + γ ) , {\displaystyle...

multi-valued function. For example, the complex logarithm is the multi-valued inverse of the complex exponential function. Similarly, the discrete logarithm...

Maclaurin series. The exponential function is analytic. Any Taylor series for this function converges not only for x close enough to x0 (as in the definition)...

mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear...

exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function...

an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: Γ ( z ) = M { e − x } ( z )...

and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang...

interpretation of the exponential form and the various limitations upon the function f necessary for its application extended over several centuries. The problems...

Characterizations of the exponential function). If the observed value of X is 100, then the estimate is 1, although the true value of the quantity being...

sets is the exponential of the exponential generating function for connected structures. The exponential formula is a power series version of a special...

number), a quadratic function c x 2 {\displaystyle cx^{2}} ( c {\displaystyle c} as a nonnegative real number) and an exponential function c e x {\displaystyle...

exactly an exponential distribution scaled by 1/2. The probability density function of the Laplace distribution is also reminiscent of the normal distribution;...

For example, addition and division, the factorial and exponential function, and the function which returns the nth prime are all primitive recursive...

order of time: exponential convergence to zero convergence to a non-zero fixed value (see Exponential function or Characterizations of the exponential function...

shifted exponential with the weight being a function of the normal distribution. The probability density function (pdf) of the exponentially modified...

{\displaystyle k} ⁠, the coefficient of ⁠ t k / k ! {\displaystyle t^{k}/k!} ⁠ in the moment generating function (expressed as an exponential power series in...

\zeta >0} Characterizations of the exponential function Exponential growth Exponentiation Generalised logistic function Logistic distribution...

one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient...

. This is the survival function of the exponential distribution. If a discrete probability distribution is memoryless, then it must be the geometric distribution...

occurrence of a fixed number of events. When k = 1 {\displaystyle k=1} , the distribution simplifies to the exponential distribution. The Erlang distribution...

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln ⁡ Γ ( z ) = Γ ′ ( z ) Γ ( z )...