# 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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### 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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### 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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### 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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### Taylor expansion theorem for Gateaux differentiable functions

I am having a hard time studying Gateaux derivatives (see https://en.wikipedia.org/wiki/Gateaux_derivative), it seems that every author mentions the concept but only as a cliffhanger to study Fréchet ...

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### Jet at a singular point or a submanifold

Let $M$ be a smooth manifold, $p\in M$ and $f\in C^\infty(M\setminus\{p\})$.
We will say that $f$ has a power-law singularity at $p$ of order $\eta$ if for every smooth immersion $\gamma:(-1,1)\to M$ ...

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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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### Other expansion for positive Taylor expansion

I was thinking of the following problem. Let $f$ be a Taylor expansion and $a_k$ the associated coefficients,
$$\forall x\in\mathbb{R},~f(x)\triangleq\sum_{k=0}^\infty a_kx^k.$$
Let suppose that we ...

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### Is there a restriction on the structure of the set of points where all derivatives of a $C^\infty$ real function are 0? [duplicate]

Let $f$ be an infinitely differentiable real function and let $Z(f)$ denote the set of points on which all derivatives of $f$ vanish. It is not hard to describe an $f$ such that $Z(f)$ is any ...

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### How to calculate the weights for Discrete Laplacian Operator?

I am following this paper step by step and want to build an isotropic Laplacian kernel. As shown in the following figure, I can understand until using Taylor to expand the 2D discrete Laplacian ...

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### Taylor expansion of Stieltjes Transform

I'm trying to derive a very basic result stated in several books on random matrix theory (e.g. Terry Tao's book and Potters & Bouchaud's book).
Given a symmetric matrix $A \in \mathbb{R}^{N \times ...

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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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### 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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### Is this a valid method of extending convergence of the Maclaurin series for $\frac{x}{x+1}$ from $|x|<1$ to $\Re(x)>-1$?

I originally asked this question on Math StackExchange a few months ago and no answers or even comments have yet been posted, so I'm asking this question again here on Math OverFlow.
This Math ...

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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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### What or where is the series expansion of the function $\ln\bigl(\frac{\tan x}{x}-1\bigr)$ or $\ln(\tan x-x)$ around $x=0$?

It is known that
\begin{equation*}
\tan x=\sum_{k=1}^{\infty}\frac{2^{2k}\bigl(2^{2k}-1\bigr)}{(2k)!}|B_{2k}|x^{2k-1}, \quad |x|<\frac{\pi}{2}
\end{equation*}
and
\begin{equation*}
\ln\tan x=\ln x+\...

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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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### $f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $?

While talking about tetration with my friend the following idea (re)occured.
$$f(f(z)) = z ,\quad f(\exp(z)) = \exp(f(z)) \tag{A}\label{A}$$
or variations of it like the weaker
$$f(f(f(f(z)))) = z ,\...

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### Fourier series of an arbitrary function of a cosine function

Is there a general expression for the Fourier series of the function $f(a\cos(\omega t))$ in terms of the derivatives of $f$? Obviously, the function can be expressed as a Maclaurin series $f(0)+af'(0)...

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### Computing $_2F_2(a,a,a+1,a+1,z)$ (hypergeometric function)

Trying to implement the derivative of the gamma incomplete function, I encountered the hypergeometric function $_2F_2(a,a,a+1,a+1; z=-x)$, where $x$ would always be a positive real (and thus $z$ a ...

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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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### Convex minorants to convex functions, given partial Taylor expansion and smoothness estimate

Let $V$ denote a strictly convex function (in arbitrary dimension) whose Hessian is $L$-Lipschitz. Given only this knowledge, and the values of $\left\{ V \left( x \right), \nabla V \left( x \right), \...

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### What is the meaning of big-O of a random variable?

I encountered this problem in a book "Pattern Recognition and Machine Learning" by Christopher M. Bishop. I excerpt it below:
screenshot of the book
In the excerpt, the big-O notation $O(\xi^...

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### Family of functions with prescribed derivatives

Suppose $f: \mathbb C \times (-1,1) \to \mathbb C$ is a smooth function that satisfies $f(0,t)=1$ for all $t\in (-1,1)$. Assume that for any $k\in \mathbb N$, any $z \in \mathbb C$ and any $t \in (-1,...

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### Exponential taylor series for multiple variables with linear constraints for coefficients

I'm trying to simplify the sum
$$
\sum_{\vec x \in (\mathbb{N}_0)^n: M\vec x = \vec b} \prod_i \frac{(a_i)^{x_i}}{x_i!},
$$
where $M$ is a $\mathbb{N}_0$-valued $m\times n$ matrix, $\vec b$ is $\...

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### Probabilistic Taylor theorem for concave functions

This paper proves a probabilistic version of Taylor's theorem
\begin{equation*}
\mathbb{E}g(X) = \sum_{k=0}^{n-1} \frac{g^{(k)}(0)}{k!} \mathbb{E}X^k + \frac{\mathbb{E}X^n}{n!} \mathbb{E} g^{(n)}(X_{(...

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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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### 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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### Taylor series with less than differentiability

I have a function $f^0\colon (0;\infty) \to \mathbb R$ with the property that the following limit exists and is finite
$$
F^1 := \lim_{x\to \infty} x \cdot f^0(x)
$$
Then I consider $f^1(x) := x \cdot ...

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### The Basel problem revisited?

In the Basel problem, the $sinc$ function is considered at the Wikipedia page.
Let me try to make an alternative function definition:
$$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \...

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### Estimating the growth of the Taylor coefficients given the growth of the function at the boundary

Let $f(z)=\sum a_nz^n$ be a Taylor series that converges for $|z|<1$ and satisfies
$$
|f(z)|\le \frac{1}{(1-|z|)^{k}}
$$
for some fixed $k>0$.
Question: What can I deduce about the growth of the ...

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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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### 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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### 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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### 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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### 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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### Taylor series on a Riemannian manifold

I need some help for the following problem.
Let $M$ a riemannian manifold and $f$ a smooth differential function, then consider the following integral $$\int_M \Gamma(x,y)(f(y)-f(x))dV_y$$
where $dV_y$...

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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,\; ...