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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.
For example, the exponential function is the function which is equal to its own derivative everywhere, and assumes the value 1 at the origin. However, one may equally well define an analytic function by its Taylor series. Taylor series are used to define functions and "operators" in diverse areas of mathematics. In particular, this is true in ...
This is the form of the remainder term mentioned after the actual statement of Taylor's theorem with remainder in the mean value form. The Lagrange form of the remainder is found by choosing G ( t ) = ( x − t ) k + 1 {\displaystyle G(t)=(x-t)^{k+1}} and the Cauchy form by choosing G ( t ) = t − a {\displaystyle G(t)=t-a} .
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 integral representation of the Mittag-Leffler function is (Section 6 of [2]) , =, >, >, where the contour starts and ends at and circles around the singularities and branch points of the integrand.
For example, the definite integral over the positive real axis of any function g(x) that can be written as a product G 1 (cx γ)·G 2 (dx δ) of two G-functions with rational γ/δ equals just another G-function, and generalizations of integral transforms like the Hankel transform and the Laplace transform and their inverses result when ...
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 .
When g is applied to a random variable such as the mean, the delta method would tend to work better as the sample size increases, since it would help reduce the variance, and thus the taylor approximation would be applied to a smaller range of the function g at the point of interest.