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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.
A function cannot be written as a Taylor series centred at a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable x; see Laurent series. For example, f (x) = e −1/x 2 can be written as a Laurent series.
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 .
Typical examples of analytic functions are The following elementary functions: All polynomials: if a polynomial has degree n, any terms of degree larger than n in its Taylor series expansion must immediately vanish to 0, and so this series
The Taylor expansion would be: + where / denotes the partial derivative of f k with respect to the i-th variable, evaluated at the mean value of all components of vector x. Or in matrix notation , f ≈ f 0 + J x {\displaystyle \mathrm {f} \approx \mathrm {f} ^{0}+\mathrm {J} \mathrm {x} \,} where J is the Jacobian matrix .
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.
The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. [1]
The implicit function theorem of more than two real variables deals with the continuity and differentiability of the function, as follows. [4] Let ϕ(x 1, x 2, …, x n) be a continuous function with continuous first order partial derivatives, and let ϕ evaluated at a point (a, b) = (a 1, a 2, …, a n, b) be zero: