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Advanced. Specialized. Miscellaneous. v. t. e. In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point.
v. t. e. In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function.
The Volterra series is a model for non-linear behavior similar to the Taylor series.It differs from the Taylor series in its ability to capture "memory" effects. The Taylor series can be used for approximating the response of a nonlinear system to a given input if the output of the system depends strictly on the input at that particular time.
Universal Taylor series. A universal Taylor series is a formal power series , such that for every continuous function on , if , then there exists an increasing sequence of positive integers such that. In other words, the set of partial sums of is dense (in sup-norm) in , the set of continuous functions on that is zero at origin.
z is a complex variable. The radius of convergence r is a nonnegative real number or such that the series converges if. and diverges if. Some may prefer an alternative definition, as existence is obvious: On the boundary, that is, where | z − a | = r, the behavior of the power series may be complicated, and the series may converge for some ...
Identity theorem. In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions f and g analytic on a domain D (open and connected subset of or ), if f = g on some , where has an accumulation point in D, then f = g on D. [1] Thus an analytic function is completely determined ...
Mercator series. Polynomial approximation to logarithm with n=1, 2, 3, and 10 in the interval (0,2). In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm : In summation notation , The series converges to the natural logarithm (shifted by 1) whenever .
Brook Taylor. Brook Taylor FRS (18 August 1685 – 29 December 1731) was an English mathematician and barrister best known for several results in mathematical analysis. Taylor's most famous developments are Taylor's theorem and the Taylor series, essential in the infinitesimal approach of functions in specific points.