Questions tagged [taylor-series]
Taylor series is a method to analyze functions as polynomials.
144 questions
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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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A problem on monotonicity rule for the ratio of two Maclaurin power series
In the paper [1] below, a monotonicity rule for the ratio of two Maclaurin power series was presented as follow.
Let $a_k$ and $b_k$ for $k\in\{0\}\cup\mathbb{N}$ be real numbers and
the power series ...
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Lower bound for variance of ratio of dependent random variables
I'm trying to find a lower bound on $\text{Var}(X / Y)$ for dependent random variables $X, Y \in [0, 1]$ with $X \leq Y$.
More specifically, $X$ and $Y$ are defined as follows:
Let $h, n \in \mathbb{N}...
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Showing bound $\|\nabla_t \dot{\tilde{x}}_Y(h, 0)\| \le L \|Y\|_{2, \infty}$ for smooth homotopies of geodesics
This question pertains to Lemma 3.5 of this article. Let $M$ be a smooth Riemannian manifold and $\gamma$ some geodesic with respect to the Levi-Civita connection $\nabla$. For any $C^2$ vector field $...
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Is a mixture of real analytic functions again analytic?
Let $$h : \mathbb{R}^2 \to \mathbb{R}^+.$$
Suppose that for each $x$, $h(x, y)$ is a real analytic function of $y$.
Let $\mu(dx)$ be a finite measure on $\mathbb{R}$, and for each $y$, suppose that
$$...
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Approximation Rates for Multivariate Taylor Series
Let $k,n,m$ be positive integers and suppose that $f$ is $C^{k}(\mathbb{R}^n,\mathbb{R}^m)$ functions. For any given $\epsilon>0$ and $x_0\in \mathbb{R}^n$, are there known sharp approximation ...
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Estimation of parameters through multivariate Taylor expansion?
I do have a function $$f(t) = \prod\limits_{j=1}^{n} \left(1 + \sum\limits_{i=1}^{n} M_{i,j} t_i\right)^{-\alpha_{j}}$$ defined by parameters:
$M_{i,j} \in \mathbb{R}_{+}, \;\forall i \in 1,...,d,\; ...
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Hadamard-like product on infinitely differentiable functions
Has the following operation $*$ on formal power series $f,g$ been studied before?
$$[X^n](f*g) = n! \cdot [X^n]f \cdot [X^n]g$$
where $n$ is a nonnegative integer? This is the typical Hadamard ...
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Radius of convergence of multivariate Taylor series
Consider the function $f$ on $\mathbb{R}^{l}$ given by \begin{eqnarray}\left(x_{1},...,x_{l}\right)\mapsto\left(\sum_{i=1}^{l}\frac{1}{\left(1+x_{i}\right)^{k_{i}}}-\left(l-1\right)\right)^{-1} \end{...
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Asymptotics to Taylor expansions?
I posted a question on MSE about approximating Taylor series but Despite a bounty I did not receive any answers or comments.
Maybe you guys can help.
https://math.stackexchange.com/questions/1440931/...
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Generating a series representation for the inverse of the operator $f(f)$
I am considering the following problem:
Suppose you are given a function $u: C \rightarrow C$, find a function $g$ such that $g(g) = u$ (Let's assume that such a function exists). And by "find", I ...
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Properties and name of some polynomials
I have encountered in a problem some polynomials given by $P_k(x) = \prod_{j=0}^{k-2} (kx-j)$. I need to understand if these polynomials are known, and if they have certain special properties, as ...
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Estimate the scale of the power series with Poisson pdf/pmf-like terms
I would like to have an estimate for the series
$$P(t) = \sum\limits_{k = 0}^\infty (e^{-t}\frac{t^k}{k!})^m,$$
where $e$ is the base of natural logarithm, $k!$ is the factorial of the integer $k$, $t$...
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Applying the ideas of power series to certain convolutions - which identities transfer?
Let's suppose I'm working with some set of functions $f_k(n)$. $f_1(n)$ is essentially the root of my functions, and could be nearly anything, and then $f_k(n) = (f_1(n) * f_{k-1}(n))$ for some ...
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Higher order Approximation of Lie groups [closed]
Maybe the following is trivial or folklore, but I can't find any concrete proof of
the theorem, that higher order derivatives of Lie groups don't give any new information
above what is coded in its ...
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Multi-variate Picard-Lindelöf? Convergence of analytic PDEs (w/ commutative partial derivatives & value at a base point)
I am looking for a theorem to give existence and uniqueness of solutions to PDEs of the following form.
Find an analytic $u : \mathbb{R}^n \to \mathbb{R}^m$ satisfying the equations $\partial_{x_i}u =...
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Estimating the bound of the integral over whole $\mathbb{R}$ of the Taylor remainder term?
Let $f : \mathbb{R} \to \mathbb{R}$ be a smooth function which has a smooth inverse and satisfies the estimate
\begin{equation}
\lvert f(x) \rvert \leq \lvert x \rvert.
\end{equation}
Also, let $d\mu$ ...
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A holomorphic function in the open unit disk satisfying certain properties
Does there exist a function which is holomorphic in $|z|<1,$ continuous in $|z|\leq1$ and such that the series $\sum |a_n|$ is divergent, where $a_n$'s coefficients in the Taylor series expansion ...
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Expected value of Taylor series with central moments of binomial variate
I want to understand this entry, but do not understand how the $\mathcal{O}\left(\frac{1}{n^2}\right)$ in the accepted answer comes into play.
I reproduce the question here: We have $x \sim \mathrm{...
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Integration algorithm and analytic property
This question is the continuation of the previous one.
In the article about the integration of analytical polynomial - time computable function $f(x)$ with the Taylor series $$f(x) = \sum_{n=0}^{\...
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A polynomial and its reciprocal expansion [closed]
Suppose $f(x)=\prod_{k=1}^n(x-a_k)$ where all $a_k>0$.
Expand the function $\frac1f$ at $\infty$ so that
$$\frac1{f(x)}=\frac{b_n}{x^n}+\frac{b_{n+1}}{x^{n+1}}+\cdots.$$
Does it follow that each $...
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Inequality of Partial Taylor Series
Hi,
For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds:
$$\sum_{k=0}^{N} \frac{x^k}...
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Small phase approximation
Does anyone known how to prove that if $|\phi_k (r)| \ll 1$ for all $r$ and all $k=1,...,n\,$, the following equation
$$ S=\left|\int_0^\infty A(r)e^{-i[\phi_0(r)+\sum_{k=1}^n \phi_k(r)]} dr \right|^2 ...
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Simplification on the estimation on error of the ratio of 2 random variables
Let $Z=\dfrac{X}{Y}$ the ratio of 2 random variables.
Distribution of $Z=\dfrac{X}{Y}$
Consider the case of two independent normal variables $X$ and $Y$ with strictly positive means and variances $\...
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Some multivariate Taylor series and corresponding smoothness balls
Suppose I have a multivariate function $f$ from $\mathbb{C}^d$ to $\mathbb{C}$ that accepts a Taylor expension of the form
$$f(\mathbf x) = \sum\limits_{\mathbf k \in \mathbb N^d} a_{\mathbf k} \...
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On proving the absence of limit cycles in a dynamical system
I'm studying this classical paper in nonlinear dynamics in biophysics and I want to understand it properly. I'm stuck at this two-variable problem which I've struggled with for too long now.
$$ \dot M ...
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Taylor expension of a simple integral [closed]
I'm trying to derive some weights expression for a boosting algorithm on a L2-ISE loss function, and i have trouble with the taylor expension.
Suppose that $f$ and $g$ are two densities from $\...
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Discrete Taylor's Formula in n dimensions [closed]
I am searching for discrete form of Taylor's formula in n dimensions. Please share the appropriate resources.
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Uniformization and constructive analytic continuation of Taylor-Maclaurin series
Context. In their paper, "Uniformization and Constructive Analytic Continuation of Taylor Series", Costin and Dunne present a constructive method to greatly increase the accuracy of a ...
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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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Do we have tetration uniqueness by $ A = \inf \sum_n a_n^2 $?
Let $f$ be a real analytic (on at least $|x|<2$) and real solution of the functional equation $f(0) = 1,f(x+1) = \exp(f(x))$. For the existence of such $f$, see here.
Then
$$
f(x) = \sum_n a_n x^n ;...
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Could the patterns in the roots of the Riemann-Liouville differintegral of $(s-1)\,L(s, \chi_{q,j})$ be explained?
This question aims to extend this question to (automorphic) Dirichlet L-functions.
Take: $$f(s,q,j) = (s-1)\,L(s, \chi_{q,j})$$
with $q$ the modulus and $j$ the index of a character $\chi$. A fast way ...
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Linear approximation of multivariate function of bounded second order partial derivatives
I have a question about linear approximation in the multivariate case.\
Let $f:B^d_r\to \mathbb{R}$ be a real-valued $C^2$-function defined on the $d$-dimensional ball of radius $r$ centered at the ...
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Finding a square integrable dominating function for function class
problem statement
For any $x \in R^d$, consider the function $$\phi(x,a) = \min_{1\leq j \leq k} \|x-a_j\|^2,$$
where $a = (a_1, \dots, a_k) \in R^{kd}$ and $\| \cdot \|$ is the $L_p$ norm for any $p ...
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Taylor approximation of $f(q) = \left(1 + q \dfrac{w_s}{w_0}\right)^{\alpha}$
I am trying to prove equations (3) given in this paper
http://users.cecs.anu.edu.au/~thush/publications/vtc_final.pdf.
The authors use taylor series to approximate function
$f(q) = \left(1 + q \...
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Taylor series expansion of quantile function
Suppose $Y$ and $Z$ two random variables, $\lambda \in \mathbb{R} $.
We note $F^{-1}_{Y}(\alpha)$ the quantile function of the variable $Y$ at the quantile level $\alpha \in (0,1)$.
Do you have any ...
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Series expansion with remaining $log n$
Hi,
I'm studying the asymptotic behavior $(n \rightarrow \infty)$ of the following formula, where $k$ is a given constant.
$$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2}$$
I'm trying to do a ...
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Calculating entropy of adjacency matrix using eigenvalue decomposition?
How to calculate entropy using the eigenvalues when the eigenvalues are negative?
Is there a simple relation between the entropy of a matrix and its characteristic polynomial?
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Jet spaces for maps with constraints
Lets be in the category $\mathbf{M}$ of smooth finite dimensional manifolds with smooth maps:
Suppose we have the set of all smooth maps $Hom_\mathbf{M}(R^n,M)$ from $R^n$ to a smooth manifold $M$. ...
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show this inequality with $\frac{d^i}{dx^i}\left(1-\left(\frac{-x}{\ln(1-x)}\right)^{1/K}\right) \Bigg|_{x=0}>0, ~~~\forall i\in N^{+}$
I am trying to solve this Komal problem 661:
Let $K$ be a fixed positive integer. Let $(a_{0},a_{1},\cdots )$ be the sequence of real numbers that satisfies $a_{0}=-1$ and
$$\sum_{i_{0},i_{1},\cdots,...
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A question about approximation of Real analytic functions
Define $B$ to be the set of functions $f:[0,1]\rightarrow \mathbb{R}$
for which there exists a dense set $C\subset [0,1]$ of computables numbers and an algorithm $F$ such that for any $x_0\in C,$ in ...
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Taylor series of a complex function that is not holomorphic
I want to create Taylor series of a complex function that has complex conjugate in it. Obviously I cannot do a total derivative but derivations over real and imag parts exist.
Bonus question: Can I ...
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About the uniqueness of the Taylor polynomial
I'm in trouble understanding this theorem:
For a given function, differentiable n times at a given point $x_0$, there exist a unique polynomial $P_n$ (of degree $\le$n) such that
$$\forall \; k=0,...,...
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How to express a quadratic polynomial exactly as a power series [closed]
I claim, for $\operatorname{artanh}(\rho) = \frac{1}{2} \ln\left(\frac{1+\rho}{1-\rho}\right)$, i.e., the inverse hyperbolic tangent function, the following holds approximately under assumptions given ...
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