Search results
Results From The WOW.Com Content Network
t. e. In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point.
v. t. e. In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function.
Taylor expansions for the moments of functions of random variables. 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.
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 .
Arctangent series. In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function: [1] This series converges in the complex disk except for (where ).
Analytic function of a matrix. 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.
In mathematics, linearization is finding the linear approximation to a function at a given point. 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 ...
Itô's lemma. In mathematics, Itô's lemma or Itô's formula (also called the Itô–Doeblin formula, especially in the French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule.