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The function e (−1/x 2) is not analytic at x = 0: the Taylor series is identically 0, although the function is not. If f ( x ) is given by a convergent power series in an open disk centred at b in the complex plane (or an interval in the real line), it is said to be analytic in this region.
e aX e bX = e (a + b)X; e X e −X = I; Using the above results, we can easily verify the following claims. If X is symmetric then e X is also symmetric, and if X is skew-symmetric then e X is orthogonal. If X is Hermitian then e X is also Hermitian, and if X is skew-Hermitian then e X is unitary.
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 mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. Lagrange inversion is a special case of the inverse function theorem .
The approximation ( +) and its equivalent form + ( + ( +)) can be obtained by rearranging Stirling's extended formula and observing a coincidence between the resultant power series and the Taylor series expansion of the hyperbolic sine function.
An application for the above Taylor series expansion is to use Newton's method to reverse the computation. That is, if we have a value for the cumulative distribution function , Φ ( x ) {\textstyle \Phi (x)} , but do not know the x needed to obtain the Φ ( x ) {\textstyle \Phi (x)} , we can use Newton's method to find x, and use the Taylor ...
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. A simulation-based alternative to this approximation is the application of Monte Carlo simulations.
For given x, Padé approximants can be computed by Wynn's epsilon algorithm [2] and also other sequence transformations [3] from the partial sums = + + + + of the Taylor series of f, i.e., we have = ()!. f can also be a formal power series, and, hence, Padé approximants can also be applied to the summation of divergent series.