Questions tagged [taylor-series]
Taylor series is a method to analyze functions as polynomials.
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Polynomial approximation for square root function with fast convergence and bounded coefficients
Let $\delta, \varepsilon \in (0,1)$. I am interested in a sequence $\{f_n\}$ of polynomial approximations of the square root function $x \to x^{1/2}$ on $[\delta,1]$, of the form
$$
f_n(x) = \sum_{i=0}...
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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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Do smooth cutoff functions analytically continue functions?
My goal is to prove (or disprove) that sufficiently smooth and quickly decaying cutoff functions being tacked on to a Taylor series correctly extend the radius of convergence to the analytic ...
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Norm of a Taylor approximation of a multivariate function
I have a function $f:\mathbb{R}^n\to\mathbb{R}^m$. My goal is to bound the first order Taylor approximation of $f$. Given $x,x'\in\mathbb{R}^n$ I have that
\begin{equation}
f(x)-f(x')\approx (x-x')^...
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Approximate expectation of a random variable that is the logarithm of a function of a binomial
I want the expectation of the following random variable: $\log\left(\frac{X}{k-X}+\alpha \right)$ with $X \sim Bin_{(k-1),p}$ and $\alpha > 0$, Therefore I derived the Taylor Series:
\begin{...
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Calculating the polynomials which are invariant under the action of a simple finite group
Let $G$ be a simple, finite group. In general, $G$ is not abelian.
Let $\rho$ be a representation of this group, where each $\rho(g)$ for $g\in G$ is a unitary, complex, $d$-dimensional matrix, $\rho(...
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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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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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What's an example of a function whose Taylor series converges to the wrong thing?
Can anyone provide an example of a real-valued function f with a convergent Taylor series that converges to a function that is not equal to f (not even locally)?
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A certain expectation of a function of independent gammas
Suppose that $Y_1...,Y_n$ are independant gamma random variables: $Y_i \sim \Gamma(\alpha_i,\beta_i)$, with density $f_i(t) = \frac{t^{\alpha_i-1} e^{-\frac{x}{\beta}}}{\Gamma(\alpha) \beta^{\alpha}}$....
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Hadamard lemma without integration
Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero.
By the product ...
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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
$$...
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Some multivariate Taylor series and corresponding smoothness balls
Suppose I have a multivariate function $f$ from $\mathbb{C}^d$ to $\mathbb{C}$ that accepts a Taylor expension of the form
$$f(\mathbf x) = \sum\limits_{\mathbf k \in \mathbb N^d} a_{\mathbf k} \...
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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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Are there any zeta functions with concurrent derivative shifts in multiple variables?
Expressions for rational zeta series have been obtained by considering the Taylor series of zeta functions. For instance, one has \begin{align}\zeta(s,x+y) &= \sum_{k=0}^{\infty} \frac{y^{k}}{k!} \...
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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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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.
$$ \dot M ...
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Estimation of parameters through multivariate Taylor expansion?
I do have a function $$f(t) = \prod\limits_{j=1}^{n} \left(1 + \sum\limits_{i=1}^{n} M_{i,j} t_i\right)^{-\alpha_{j}}$$ defined by parameters:
$M_{i,j} \in \mathbb{R}_{+}, \;\forall i \in 1,...,d,\; ...
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Fourier series of $\log(a +b\cos(x))$?
By numerical computation it seems like, if $a_0 < a_1$:
$$
\begin{multline}
\log({a_0}^2 + {a_1}^2 + 2 a_0 a_1 \cos(\omega t)) = \log({a_0}^2 + {a_1}^2) \\
+ \frac{a_0}{a_1}\cos(\omega t)
- \frac{...
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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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Are there any techniques that can be used in the case when a Neumann series doesn't converge?
Suppose we have a bounded linear operator $A = A(\gamma):H_1\to H_2$ where $H_1$ and $H_2$ are Hilbert spaces and $\gamma>0$ is some parameter, and we are interested in the solution to
$$
(I-A)x = ...
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Taylor expansion of determinant of Riemannian metric in normal coordinates up to higher order
Let $(M,g)$ be an $n$-dimensional Riemannian manifold. Let $p\in M$, and let $\{x^i\}_{i=1}^n$ be normal coordinates centered around $p$.
Using Jacobi field, one can show that the metric $g$ has the ...
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Hadamard-like product on infinitely differentiable functions
Has the following operation $*$ on formal power series $f,g$ been studied before?
$$[X^n](f*g) = n! \cdot [X^n]f \cdot [X^n]g$$
where $n$ is a nonnegative integer? This is the typical Hadamard ...
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Bounding higher derivatives of $f(x) = 1/(1+x^2)^r$
Let $r\in \lbrack 0,\infty)$. Define $f(x) = 1/(1+x^2)^r$. It would seem to be the case that $$|f^{(k)}(x)|\leq \frac{2r \cdot (2r+1) \dotsb (2r + k-1)}{(1+x^2)^{r + k/2}}$$ for all even $k\geq 0$. ...
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power series of the reciprocal... does a recursive formula exist for the coefficients [closed]
Let $f(x)=\sum _{n=0}^{\infty } b_nx^n$ and $\frac{1}{f(x)}=\sum _{n=0}^{\infty } d_nx^n$. Then the coefficients of the reciprocal of $f(x)$ can be written down. The first few terms are:
$d_0 = \frac{...
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Radius of convergence of multivariate Taylor series
Consider the function $f$ on $\mathbb{R}^{l}$ given by \begin{eqnarray}\left(x_{1},...,x_{l}\right)\mapsto\left(\sum_{i=1}^{l}\frac{1}{\left(1+x_{i}\right)^{k_{i}}}-\left(l-1\right)\right)^{-1} \end{...
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Taylor $k$-differentiability of a real function at a point
I am interested in the standard name for the following weak form of $k$-differentiability.
Definition. A function $f:\mathbb R\to\mathbb R$ is called Taylor $k$-differentiable at a point $x_0$ if ...
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Taylor expension of a simple integral [closed]
I'm trying to derive some weights expression for a boosting algorithm on a L2-ISE loss function, and i have trouble with the taylor expension.
Suppose that $f$ and $g$ are two densities from $\...
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Why is $ \frac{\pi^2}{12}=\ln(2)$ not true?
This question may sound ridiculous at first sight, but let me please show you all how I arrived at the aforementioned 'identity'.
Let us begin with (one of the many) equalities established by Euler:
...
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Complexity of calculating $f^{(n)}(0)$/extracting a coefficient of a functions taylor-series
Many combinatorial problems can be solved using generating functions.
In such a case, we obtain a function $f(x)$, which (for usual) has a taylor-expansion:
$$
f(x) = \sum_{n\ge 0 } a_n x^n
$$
So ...
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Taylor approximation of $f(q) = \left(1 + q \dfrac{w_s}{w_0}\right)^{\alpha}$
I am trying to prove equations (3) given in this paper
http://users.cecs.anu.edu.au/~thush/publications/vtc_final.pdf.
The authors use taylor series to approximate function
$f(q) = \left(1 + q \...
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Taylor's theorem for a composition with $\min:\mathbb R^2\to\mathbb R$ and differentiability Lebesgue almost everywhere
Let
$f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1$$
$g:=\ln f$ (and assume $g'$ is Lipschitz continuous)
$n\in\mathbb N$, $$s(x,y):=\sum_{i=1}^n\left(g(y_i)-g(x_i)\right)$$ and $$h(...
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Taylor-like expansion for a holomorphic function in non-simply-connected domain
Suppose $f$ is a holomorphic function in a simply connected open set $U$, and we know it's Taylor expansion at a point $p\in U$. We can then find a holomorphic map $g$ of $U$ to the unit disc which ...
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Show that the absolute value of this function is twice differentiable except on a set of Lebesgue measure $0$
Let
$f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1\tag1$$
$g:=\ln f$ and assume that $g'=\frac{f'}f$ is Lipschitz continuous (note that this implies that $f'(x)\xrightarrow{|x|\to\...
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Covergence of fractional Taylor series
Let $f(x)$ be a function that is continuous and infinitely smooth on entire $\mathbb R$. Let's consider Taylor-Maclaurin series for this function:
$$f(x) = \sum_{0}^{∞}\frac{f^n(x_0)(x-x_0)^n}{n!}$$
...
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Expansion of an integral
I have an integral of the form
$$ I=\int_0^{\infty}{dx}\ln \bigg(1+\exp(-\frac{f(x)}{a})\bigg) $$ where $a$ is a positive constant and $f(x)$ is a regular and positive function such that $I$ is finite ...
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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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One-Sided Analyticity Condition Guarantees Analytic Function?
Let $f \ \colon \ [0,\infty) \to \mathbb{R}$ be a function satisfying:
$f$ is differentiable infinitely many times in $(0,\infty)$, and has a right-derivative of any order at $0$.
$f$ satifsfies the ...
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Compute inverse series for implicit equation $b=-\log(1-e^{-x})/x$
In financial mathematics, the inverse series of: $$b(x) = -\frac{\log(1-e^{-x})}{x}$$ is needed in order to perform fast calculation on swaptions for G2++ calibration model. (see this post for ...
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Generalization of Carleman coefficients to multivariable functions - Carleman tensor?
Recently I learned about a matrix called
Carleman matrix. It is a matrix used to represent function iteration with matrix multiplying.
Carleman linearization is a technique used to embed a finite
...
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Functional Taylor expansion for differential entropy
Consider an continuous distribution $F$ with density $f$. The (differential) Shannon entropy of $f$ is
$h(f)=-\int f(x)\log f(x) dx$.
In the literature of differential entropy estimation, ...
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Mean value theorem in terms of Wirtinger calculus?
The mean value theorem for vector-valued function in the real domain $f: \mathcal{R}^n \rightarrow \mathcal{R}^d$ can be expressed as
\begin{equation}
f(x)-f(y)=\int_{0}^{1}\nabla f(x(\tau))d \tau \...
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Asymptotic growth of the of Taylor coefficients of the inverse of a function
Let $f(x)=\sum_{n\geq 1} c_n\cdot x^n$ be a function given by a power series. Further there is some $\alpha >1$ such that for all $n$, $c_n = \Theta(1/n^{\alpha})$. What can one say about the ...
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Is there a bound for Lipschitz constant in terms of second differences?
It is easy to show that if $f\colon[0,1]\to\mathbb R$ and $|f|\leq A$ and $|f''|\leq B$ then~$|f'|\leq 4A+B$. Indeed, by Taylor formula with remainder $f(x)=f(c)+(x-c)f'(c)+\frac12(x-c)^2f''(d)$ where ...
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Properties of signomial Functions in one variable
I am interested in functions of the form: \sum_{j=1}^\infty a_j x^{p_j} where p_j can be any non-negative real number. Wikipedia has informed me that this is a subset of the signomial functions, but ...
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A curious series related to the asymptotic behavior of the tetration
The tetration is denoted $^n a$, where $a$ is called the base and $n$ is called the height, and is defined for $n\in\mathbb N\cup\{-1,\,0\}$ by the recurrence
$$
{^{-1} a} = 0, \quad {^{n+1} a} = a^{\...
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A polynomial and its reciprocal expansion [closed]
Suppose $f(x)=\prod_{k=1}^n(x-a_k)$ where all $a_k>0$.
Expand the function $\frac1f$ at $\infty$ so that
$$\frac1{f(x)}=\frac{b_n}{x^n}+\frac{b_{n+1}}{x^{n+1}}+\cdots.$$
Does it follow that each $...
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Taylor expansion of cumulant generating function
For the characteristic function $\mathbf E e^{i t X}$ of a random variable $X$ with $n+1$ finite moments, there is the well known and easy to prove bound on the remainder of the Taylor series
$$\left\...
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Determining the rate of spread of geodesics when the sectional curvature is zero
I have posted this question in mathSE a few weeks ago (and proposed a bounty) but so far got no response.
In the book Riemanian geometry (by do-Carmo), the following result is proved (Corollary 2.9 ...
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A closed form of a summation or the taylor series expansion of some function with a closed form?
Let $Z_N = \displaystyle{\sum_{k+j\leq N}} \frac{N!N^{k+j}}{N^{N+1}}\frac{u^kv^j}{k!j!}\binom{N-j}{N-j-k}$ where $u$ and $v$ are two unknowns.
My question is: Is there a closed-form for $Z_N$ or is $...
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