Search results
Results from the WOW.Com Content Network
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 ...
Print/export Download as PDF; Printable version; In other projects Wikidata item; ... and consider a Taylor expansion of the score 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.
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 finite difference coefficients for a given stencil are fixed by the choice of node points. The coefficients may be calculated by taking the derivative of the Lagrange polynomial interpolating between the node points, [3] by computing the Taylor expansion around each node point and solving a linear system, [4] or by enforcing that the stencil is exact for monomials up to the degree of the ...
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]
Given a twice continuously differentiable function of one real variable, Taylor's theorem for the case = states that = + ′ () + where is the remainder term. The linear approximation is obtained by dropping the remainder: f ( x ) ≈ f ( a ) + f ′ ( a ) ( x − a ) . {\displaystyle f(x)\approx f(a)+f'(a)(x-a).}
These follow by Taylor expansion about w = 1, with the help of the basic properties discussed above. The radii of convergence will be dependent on the value of z and on the G-function that is expanded. The expansions can be regarded as generalizations of similar theorems for Bessel, hypergeometric and confluent hypergeometric functions.