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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 ...
To find a second-order approximation for the covariance of functions of two random variables (with the same function applied to both), one can proceed as follows.
Download as PDF; Printable version; ... for example, in solutions of the ... which converges more rapidly for all real values of x than a Taylor expansion, ...
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 .
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 ]
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.
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.
Download as PDF; Printable version; In other projects Wikidata item; Appearance. ... is the Taylor series expansion at the origin of the arctangent function: [1]