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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.
In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function: [1]
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.
Following a proposal by William Kahan, it may thus be useful to have a dedicated routine, often called expm1, which computes e x − 1 directly, bypassing computation of e x. For example, one may use the Taylor series: e x − 1 = x + x 2 2 + x 3 6 + ⋯ + x n n ! + ⋯ . {\displaystyle e^{x}-1=x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+\cdots ...
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 ...
The theorem was proved by Lagrange [2] and generalized by Hans Heinrich Bürmann, [3] [4] [5] both in the late 18th century. There is a straightforward derivation using complex analysis and contour integration ; [ 6 ] the complex formal power series version is a consequence of knowing the formula for polynomials , so the theory of analytic ...
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.