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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.
Examples of functions that are not entire include the square root, the logarithm, the trigonometric function tangent, and its inverse, arctan. For these functions the Taylor series do not converge if x is far from b. That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor ...
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 .
In particular, the Taylor expansion holds in the form = + (), = = ()! (), where the remainder term R k is complex analytic. Methods of complex analysis provide some powerful results regarding Taylor expansions.
The Mittag-Leffler function can be used to interpolate continuously between a Gaussian and a Lorentzian function. In mathematics, the Mittag-Leffler functions are a family of special functions.
In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size.. This is used for defining the exponential of a matrix, which is involved in the closed-form solution of systems of linear differential equations.
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 interpolation is done using the quadratic Taylor expansion of the Difference-of-Gaussian scale-space function, (,,) with the candidate keypoint as the origin. This Taylor expansion is given by: This Taylor expansion is given by: