# Questions tagged [taylor-series]

Taylor series is a method to analyze functions as polynomials.

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### Antiderivatives via Taylor series and the FT of Calculus

If $f$ is a real function on an interval $[a,b]$ such that $f$ is computationally tractable on $[a,b]$: you can calculate $f(x)$ to $n$ bits of precision using an algorithm which is polynomial in $n$ ...
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### Exponential-like function equivalent for the Dixonian Elliptics

Is there some exponential-like function that acts as partner for the intricate Dixonian Elliptics, in a similar way that the Exponential function acts as a partner for the trigonometric functions ?
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### Expressing a vector valued function in terms of its derivatives

Consider a function $$f:\mathbb{R}^n\rightarrow\mathbb{R}^m$$ given by $m$ functions $f_i:\mathbb{R}^n\rightarrow \mathbb{R}$ that we can assume to be polynomials in $x_1,\dots,x_n$. Does there ...
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### Weaker version of the Borel lemma for vector-valued functions

Borel's lemma for Frechét-spaces $V$ says: (i) For every $(v_j)_{j \in \mathbb{N}} \in V^\mathbb{N}$ there exists a smooth $f: \mathbb{R} \to V$ such that $$f^{(j)}(0) = v_j.$$ For general locally ...
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### Analyzing the decay rate of Taylor series coefficients when high-order derivatives are intractable

This could be a soft question. I am trying to show that the $n$-th Taylor series coefficient of a function is $O(n^{-5/2})$. However, because the function is a function composition of another function ...
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I do have a function $$f(t) = \prod\limits_{j=1}^{n} \left(1 + \sum\limits_{i=1}^{n} M_{i,j} t_i\right)^{-\alpha_{j}}$$ defined by parameters: \$M_{i,j} \in \mathbb{R}_{+}, \;\forall i \in 1,...,d,\; ...