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  2. Taylor expansions for the moments of functions of random ...

    en.wikipedia.org/wiki/Taylor_expansions_for_the...

    In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite.

  3. Taylor's theorem - Wikipedia

    en.wikipedia.org/wiki/Taylor's_theorem

    Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, [2] although an earlier version of the result was already mentioned in 1671 by James Gregory. [ 3 ] Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis .

  4. Grassmann number - Wikipedia

    en.wikipedia.org/wiki/Grassmann_number

    The usage of the term "Grassmann variables" is historic; they are not variables, per se; they are better understood as the basis elements of a unital algebra. The terminology comes from the fact that a primary use is to define integrals, and that the variable of integration is Grassmann-valued, and thus, by abuse of language, is called a ...

  5. First-order second-moment method - Wikipedia

    en.wikipedia.org/wiki/First-order_second-moment...

    For the second-order approximations of the third central moment as well as for the derivation of all higher-order approximations see Appendix D of Ref. [3] Taking into account the quadratic terms of the Taylor series and the third moments of the input variables is referred to as second-order third-moment method. [4]

  6. Taylor series - Wikipedia

    en.wikipedia.org/wiki/Taylor_series

    That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for analytic functions ...

  7. Path-ordering - Wikipedia

    en.wikipedia.org/wiki/Path-ordering

    If an operator is not simply expressed as a product, but as a function of another operator, we must first perform a Taylor expansion of this function. This is the case of the Wilson loop, which is defined as a path-ordered exponential to guarantee that the Wilson loop encodes the holonomy of the gauge connection.

  8. Translation operator (quantum mechanics) - Wikipedia

    en.wikipedia.org/wiki/Translation_operator...

    [2] A nice way to double-check that these relations are correct is to do a Taylor expansion of the translation operator acting on a position-space wavefunction. Expanding the exponential to all orders, the translation operator generates exactly the full Taylor expansion of a test function: = ^ () = ⁡ (^) = (=!

  9. Shift operator - Wikipedia

    en.wikipedia.org/wiki/Shift_operator

    ⁠ The shift operator acting on functions of a real variable is a unitary operator on ⁠ (). In both cases, the (left) shift operator satisfies the following commutation relation with the Fourier transform: F T t = M t F , {\displaystyle {\mathcal {F}}T^{t}=M^{t}{\mathcal {F}},} where M t is the multiplication operator by exp( itx ) .