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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 ...
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 k {\textstyle k} of the Taylor series of the function.
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 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 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 mathematics, the Euler numbers are a sequence E n of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion = + = =!, where is the hyperbolic cosine function.
Data source: Fiserv. Fiscal years end Dec. 31. This steady growth underscores the popularity of its service offerings and also demonstrates superb operating leverage as the business's net income ...
A nonlinear operator = (,, | |) is elliptic if its linearization is; i.e. the first-order Taylor expansion with respect to u and its derivatives about any point is an elliptic operator. Example 1 The negative of the Laplacian in R d given by − Δ u = − ∑ i = 1 d ∂ i 2 u {\displaystyle -\Delta u=-\sum _{i=1}^{d}\partial _{i}^{2}u} is a ...