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  2. List of representations of e - Wikipedia

    en.wikipedia.org/wiki/List_of_representations_of_e

    Since e is an irrational number (see proof that e is irrational), it cannot be represented as the quotient of two integers, but it can be represented as a continued fraction. Using calculus, e may also be represented as an infinite series, infinite product, or other types of limit of a sequence.

  3. Taylor series - Wikipedia

    en.wikipedia.org/wiki/Taylor_series

    The function e (−1/x 2) is not analytic at x = 0: the Taylor series is identically 0, although the function is not. If f ( x ) is given by a convergent power series in an open disk centred at b in the complex plane (or an interval in the real line), it is said to be analytic in this region.

  4. Exponentiation - Wikipedia

    en.wikipedia.org/wiki/Exponentiation

    The definition of e x as the exponential function allows defining b x for every positive real numbers b, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm ln(x) is the inverse of the exponential function e x means that one has = ⁡ (⁡) = ⁡ for every b > 0.

  5. Exponential function - Wikipedia

    en.wikipedia.org/wiki/Exponential_function

    In this setting, e 0 = 1, and e x is invertible with inverse e −x for any x in B. If xy = yx, then e x + y = e x e y, but this identity can fail for noncommuting x and y. Some alternative definitions lead to the same function. For instance, e x can be defined as (+).

  6. Characterizations of the exponential function - Wikipedia

    en.wikipedia.org/wiki/Characterizations_of_the...

    The six most common definitions of the exponential function ⁡ = for real values are as follows.. Product limit. Define by the limit: = (+).; Power series. Define e x as the value of the infinite series = =! = + +! +! +! + (Here n! denotes the factorial of n.

  7. E-function - Wikipedia

    en.wikipedia.org/wiki/E-function

    The exponential function is an E-function, in its case c n = 1 for all of the n. If λ is an algebraic number then the Bessel function J λ is an E-function. The sum or product of two E-functions is an E-function. In particular E-functions form a ring. If a is an algebraic number and f(x) is an E-function then f(ax) will be an E-function.

  8. Bernoulli number - Wikipedia

    en.wikipedia.org/wiki/Bernoulli_number

    In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...

  9. Matrix exponential - Wikipedia

    en.wikipedia.org/wiki/Matrix_exponential

    For any real numbers (scalars) x and y we know that the exponential function satisfies e x+y = e x e y. The same is true for commuting matrices. If matrices X and Y commute (meaning that XY = YX), then, + =. However, for matrices that do not commute the above equality does not necessarily hold.

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