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That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for analytic functions ...
The Taylor series of f converges uniformly to the zero function T f (x) = 0, which is analytic with all coefficients equal to zero. The function f is unequal to this Taylor series, and hence non-analytic. For any order k ∈ N and radius r > 0 there exists M k,r > 0 satisfying the remainder bound above.
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A function is analytic if and only if for every in its domain, its Taylor series about converges to the function in some neighborhood of . This is stronger than merely being infinitely differentiable at x 0 {\displaystyle x_{0}} , and therefore having a well-defined Taylor series; the Fabius function provides an example of a function that is ...
Multipole expansions are useful because, similar to Taylor series, oftentimes only the first few terms are needed to provide a good approximation of the original function. The function being expanded may be real - or complex -valued and is defined either on R 3 {\displaystyle \mathbb {R} ^{3}} , or less often on R n {\displaystyle \mathbb {R ...
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In calculus, the power rule is used to differentiate functions of the form () =, whenever is a real number.Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule.
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