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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 interchangeably.
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.
Recall from above that an n × n matrix exp(tA) amounts to a linear combination of the first n −1 powers of A by the Cayley–Hamilton theorem. For diagonalizable matrices, as illustrated above, e.g. in the 2 × 2 case, Sylvester's formula yields exp(tA) = B α exp(tα) + B β exp(tβ), where the B s are the Frobenius covariants of A.
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.
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 ...
Boltzmann's distribution is an exponential distribution. Boltzmann factor (vertical axis) as a function of temperature T for several energy differences ε i − ε j.. In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution [1]) is a probability distribution or probability measure that gives the probability that a system will be in a certain ...
and the second-order term does not affect the limit, yielding = (+). Euler's number can be defined as e = exp ( 1 ) {\displaystyle e=\exp(1)} . It follows from the preceding equations that exp ( x ) = e x {\displaystyle \exp(x)=e^{x}} when x is an integer (this results from the repeated-multiplication definition of the ...
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