For questions that involve concrete approximations, such as finding an approximate value of a number with some precision. For questions that belong to the mathematical area of Approximation Theory, use (approximation-theory).

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23
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0answers
258 views

Is there an integral for $\pi^4-\frac{2143}{22}$?

In Ramanujan's Notebooks, Vol 4, p.48 (and a related one in Quarterly Journal of Mathematics, XLV, 1914) there are various approximations, including the close (by just $10^{-7}$), $$\pi^4 \approx ...
9
votes
0answers
335 views

$\pi^4 + \pi^5 \approx e^6$ is anything special going on here?

Saw it in the news: $$(\pi^4 + \pi^5)^{\Large\frac16} \approx 2.71828180861$$ Is this just pigeon-hole? DISCUSSION: counterfeit $e$ using $\pi$'s Given enough integers and $\pi$'s we can ...
9
votes
0answers
193 views

Are there known pairs of simple numbers equal to huge precision, but not equal strictly?

Are there known pairs of numbers $a$ and $b$, which at first look at them seemed likely to be equal, and after checking up to $10^n$ decimal places appeared to agree, but suddenly for some $n$ they ...
8
votes
0answers
256 views

Approximation of the exponential

Let $c>1,k\in\mathbb{N}$. Let's consider two approximations of the exponential function : The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$ and the second one is ...
8
votes
0answers
433 views

Using Padé approximants for the quaternion exponential

On a lark, I wanted to know if one can use Padé approximants to compute the exponential $\exp(z)$ of a quaternion $z=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$. Since Mathematica has a package meant for ...
7
votes
0answers
215 views

How to evaluate the integral $e^{-(c\ln(\frac{1}{x}))^s} dx$?

Can anyone help me evaluate $$\int_{\alpha}^1 \exp{\left\{-\left(c\ln\left(\frac{1}{x}\right)\right)^s\right\}} dx$$, Where $0 \leq \alpha \leq 1$ and $s \in \mathbb{R}$. I tried changing ...
7
votes
0answers
329 views

Show that the function is constant

Let $S^n$ be an $n$-dimentional unit sphere. Consider $f: S^n \longrightarrow R_+$ even continuous function. Denote $$ ...
6
votes
0answers
89 views

Could this approximation be made simpler ? Solve $n!=a^n 10^k$

I need to find the smallest value of $n$ such that $$\frac{a^n}{n!}\leq 10^{-k}$$ in which $a$ and $k$ are given (these can be large numbers). I set the problem as : solve for $n$ the equation ...
5
votes
0answers
58 views

Given a vector $x\in \mathbb R^n$, how can we find $z\in \mathbb Z^n$ which is closest to a scalar multiple of $x$?

I am looking for how to find integer approximations to scalar multiples of real valued vectors. This is close to the problem of finding a best rational approximation to a real number, but kind of ...
5
votes
0answers
54 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
5
votes
0answers
322 views

Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
5
votes
0answers
143 views

Watson's Lemma Extension

We all know that Watson's Lemma is used to approximate the integral $$ F\left( s \right)=\int_0^\infty {{e^{ - st}}f\left( t \right)dt} $$ for large $s$. However, for arbitrary $s$, are there any ...
4
votes
0answers
107 views

Radial Basis Functions Interpolation

$ \let\oldcdot\cdot \renewcommand{\cdot}{\!\oldcdot\!} \newcommand{\e}{\varepsilon} \renewcommand{\p}{\varphi} \renewcommand{\p}{\varphi} \renewcommand{\vp}{\vec{\boldsymbol\p}(x)} ...
4
votes
0answers
31 views

Shaking a box of rocks (Optimal Packing)

My coworker was telling me that when he plants seeds on his farm, he puts them all in a large container on the tractor and after a period of just driving, the seeds are more densely packed than when ...
4
votes
0answers
35 views

Asymptotic behavior of many derivatives

To compute the residue of a pole of very high order $M$ at $z=0$, one needs to compute $\frac{d^M}{dz^M} g(z)$ Suppose that $g(z)$ is a reasonable but not trivial function, that itself may depend on ...
4
votes
0answers
170 views

Diophantine approximation with additional constraints

I am trying to compute best rational approximations to various transcendental numbers $c$, subject to the following constraints: $$\frac {i j} {2^k} = c + \epsilon, \space\space2^n \le i, j \lt ...
4
votes
0answers
215 views

Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Could you provide examples ...
4
votes
0answers
87 views

Can we use a sum of residues to develop an asymptotic expansion for this unknown function?

In the course of solving a particular physical problem, I have derived a relationship between two unknown functions: $$ f(s) = \frac{s \sinh{\frac{\pi s}{2}}}{2 \pi i \beta} \int_{-c- i ...
4
votes
0answers
41 views

How to approximate derivative inside derivative

I am using a box-scheme for solving partial differential equation. The function is approximated with: $$ f_p=\psi\cdot\left(\theta\cdot f_{j}^{k+1}+(1-\theta)\cdot f_{j}^{k} \right) + (1-\psi)\cdot ...
4
votes
0answers
148 views

Approximate the integral $\int_0^\pi \sin(x^3)\mathrm{d}x$ with a standard pocket calculator

I came over the following integral $$ \int_0^\pi \sin(x^3) \mathrm{d}x $$ when a friend of mine tried to approximate it. The most obvious way is to use taylors formula, and then turn the integral ...
4
votes
0answers
102 views

Using formal power series to solve nasty equations

Consider a function $f:[0,\infty)\times \mathbb R\to\mathbb R$, and suppose that given some $a>0$, I would like to solve for $x\in\mathbb R$ satisfying \begin{align} f(\delta, x) = a. \end{align} ...
4
votes
0answers
127 views

Approximating two-variable function with a product of one-variable functions

Suppose I have an integrable non-const function $f(x,y)$ defined on $V$. I'd like to approximate it with a product $a(x)b(y)$. How can I find such $a(x)$ and $b(y)$ that $$\tag1 ...
4
votes
0answers
235 views

Low-rank approximation to the Graph Laplacian matrix of a regular grid.

As mentioned in the title, does anybody know any methods of efficient low-rank approximation $LL^T$ to the Graph Laplacian matrix $A$ corresponding to a square lattice? (except PCA)
3
votes
0answers
21 views

Estimating special values of the Riemann zeta function on the critical line

If $p,q$ are primes, is it necessarily true that $$\left|\zeta\left(\frac{1}{2} + i\frac{p}{q}\right)\right| > (p+q)^{–(p+q)} ?$$ (Here $\zeta$ is the Riemann zeta function.)
3
votes
0answers
45 views

How good is my approximation of this complicated sinc function? (plot included)

Part 1: The following function (for $N=256$) has the plot shown nexr $$ G(x) = \frac{1}{N}\text{exp} \bigg( j \frac{\pi}{2} \, x(N-1)\bigg) \frac{\sin (\frac{\pi N}{2} x)}{\sin (\frac{\pi}{2} x ...
3
votes
0answers
83 views

Solution for an ODE given only at discrete points

The problem I have: For each $n \in \mathbb N$ I have $$\begin{align} x_0^n & \in \mathbb R \\ h_n & \in \mathbb R \\ x_k^n & = x_0^n + k \cdot h_n \text{ for } k \in \{0,1,\ldots n\} \\ ...
3
votes
0answers
40 views

Function Looks Poisson-Like: But What's the Parameter $\lambda$?

(On pause) I have $$f\left(x\right)=-x\left( x\sqrt{4-x^2}-4\arccos\left(\frac{x}{2}\right) \right)\arccos\left(\frac{x^2+d^2-1}{2dx}\right)$$ which looks a bit like the continuous version of ...
3
votes
0answers
87 views

Taking a stationary phase approximation of a multidimensional integral

I'm looking for a way to take a stationary phase approximation of an integral of the following form: $$ \int_{-\infty}^\infty d\vec{q} \exp\left(2 \pi i N \left(S(q_{n+1}, \vec{q}, q_1) - ...
3
votes
0answers
48 views

Approximating this definite integral

I ran into the following integral in my research that I believe has no closed-form solution: $$ I = \int_{s_0}^{s_1} \frac{(\alpha_x s + \beta_x)^{\lambda_x}}{(\alpha_y s + \beta_y)^{\lambda_y}} ds ...
3
votes
0answers
16 views

Approximating the action of the U(N) exponential map

Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group: $$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$ I can approximate this to first order as: $$\tilde ...
3
votes
0answers
147 views

What domains of $\mathbb{R}^n$ have the property that $H^1(\Omega)=H^1_0(\Omega)$?

i wonder what are sufficient conditions on an unbounded domain of $R^n$ called $\Omega$ to get : $C_c^\infty (\Omega)$ dense in $H^1 (\Omega)$ ? where $C_c^\infty$ stands for the set of functions with ...
3
votes
0answers
51 views

Truncation error in Padè approximants

Suppose only the following data are known about a rational function $R(x)=P(x)/Q(x)$ (for $P,Q$ polynomials): (a) the degree of $P$ is $\leq m$ and the degree of $Q$ is $\leq n$; (b) the first $k$ ...
3
votes
0answers
403 views

Taylor Series of Integral

I'm trying to come up with the Taylor expansion of an integral expression. For simplicity, consider the toy integral $$ ...
3
votes
0answers
50 views

Why does the moment of approximation matter for the end result?

I am trying to wrap my head around the reason why the moment of approximation matters for the end result of my analysis. As an example, let's take an equation for which we can still find the full ...
3
votes
0answers
93 views

Representation for a function that, when added/multiplied/composed with another function of the same form, yields a new function of the same form

I apologize for the possibly unclear wording of the title. I'm not well versed in math terminology. I'm after a concrete representation of a function, eg $y(x) = Ax^p$ (where $A$ and $p$ are ...
3
votes
0answers
135 views

Useful approximation of the pdf

Good day to everyone. In my research work I came out with a function, which looks like this (it is the pdf of some random variable): $$f(x,\rho,\psi)=\frac{2}{\pi }+\sqrt{\frac{2}{\pi }} ...
3
votes
0answers
78 views

How to solve a distance problem inside of a picture?

sorry for my bad english. I have the following problem: In the picture you can see 4 different positions. Every position is known to me (longitude, latitude with screen-x and screen-y). Now i want ...
3
votes
0answers
64 views

Root calculation by hand (division-like algorithm)

I remember from my highschool days a division-like algorithm for calculating square, even cubic roots. I know the continued fraction method, some series and Newton's method. I have checked similar ...
3
votes
0answers
176 views

Runge-Kutta Error Analysis

Could anyone explain to me how to reduce the error propagated by using Runge-Kutta of order 4? Or can anyone give me a nice reference to it.
3
votes
0answers
33 views

Characterize a large class of shapes using a finite number of parameters

I am doing some numerical computations searching for an optimal shape for a certain functional. In my particular case, the shape $\Omega$ is a 2 dimensional star shaped domain by the origin, which ...
3
votes
0answers
82 views

Question about linearization

Given a data matrix $D\in\mathbb{R}^{N \times N}$ Can one construct another matrix $M$ that for all permutation matrices $Q^A$,$Q^B$, if $[\sum_i\sum_j (Q^A_{ij}D_{ij})]^2 \geq [\sum_i\sum_j ...
3
votes
0answers
402 views

An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function

I think, here, I found $$ P_\color{red}x(\color{blue}s)=\sum_{p<\color{red}x} \frac{1}{p^{\color{blue}s}} =\sum_{\color{green}n=1}^{\infty}\frac{ \mu (\color{green}n)}{\color{green}n} ...
3
votes
0answers
61 views

When can the commutator of two matrices be neglected in series expansions?

Under what conditions can the higher order commutators in the Baker–Campbell–Hausdorff formula be neglected when the commutators does not vanish exactly and there is no small parameter in the ...
3
votes
0answers
280 views

Minimizing maximum absolute column sum norm of the residual between a matrix and its $k$-rank approximation

Let $X \in \mathbb{R}^{m\times n}$ be a matrix with rank $r$. How can we find the optimal $\tilde{X} \in \mathbb{R}^{m\times n}$ whose rank is $k$ where $k\leq r$ and the reconstruction error in ...
3
votes
0answers
82 views

Is that observation really a property of the log of coefficients of continued fractions (example: cf(log(3)/log(2))

I'm again looking at the problem of approximation of perfect powers of 2 to that of 3 (I assume $\small q_N = 2^S / 3^N \gt 1 $) and specifically using the continued fraction representation of $\small ...
3
votes
0answers
262 views

Approximating a system of differential equations as a Bézier curve

I am looking for a general transform to approximate the solution to an n-dimensional system of differential equations and initial conditions as a cubic or quadratic Bézier curve. Sorry if my ...
3
votes
0answers
248 views

Approximation of a real number as a linear combination of two reals with coprime integral coefficients

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, give me an algorithm that will provide coprime integers $a$ ...
2
votes
0answers
50 views

Approximate an integral with Bessel functions

Given $r,a,\lambda\in\mathbb{R}$, $r<a$, how can I find an approximate solution for the following definite integral? $$ \int_0^\infty J_0 (\lambda r)J_1(\lambda a)\frac{1}{\sqrt{n+\lambda^2 ...
2
votes
0answers
58 views

Is there a closed-form solution (even approximated) to this inequality?

I have the following function: $f(x, \theta) = (1-\theta)(x+1)^{-\theta}\left[ \frac{2-2\theta}{1- 2\theta} (N^{1-2\theta} - (x+1)^{1-2\theta}) - (x+1)^{-\theta}(N^{1-\theta} - (x+1)^{1-\theta}) ...
2
votes
0answers
74 views

Why the quadrature formula is exact one not an approximation?

I am reading this material on the algorithm of calculating the centroid of a polyhedron. I am confused by the last step of the deduction: The three coordinates of the centroid can be obtained: ...