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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7
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0answers
128 views

Approximating intervals and squares by increasingly dense disjoint finite sets with special properties

Apologies for the length of the question. Consider interval $I=[0,1]$. For any $n \in \mathbb{N}$ we can always find two finite sets $S_{1n} \subset I$ and $S_{2n} \subset I$ such that: a) ...
0
votes
1answer
61 views

Estimate an error using method similar to Stirling's approximation?

In the application of WLLN, which is the polynomial approximation. For any function $F\in C([0,1])$ can be approximated by a polynomail $G$ so as to make $||F-G||=\max_{0\le x \le 1}F(x)-G(x)$ as ...
1
vote
2answers
81 views

Stirling's Approximation for binomial coefficient

In this proof, it is assumed that, for $k << n$, ${n \choose k} \approx \frac{n^k}{k!}$, given Stirling's approximation. How does Stirling's Approximation, in either form $\ln n! \approx ...
1
vote
0answers
35 views

Solving of numerical equation with integrals

Let's have equation $$ \cosh(2 \pi x) = \cos\left[\text{Re}\int \limits_{0}^{2 \pi}\sqrt{A - 2q\cos(2z)}dz \right]\times \cosh\left[ \text{Im}\int \limits_{0}^{2 \pi}\sqrt{A - 2q\cos(2z)}dz\right], $$ ...
0
votes
1answer
25 views

Approximating basic trigonometric functions to a integrable form

I wondered if there is a way of approximating trigonometric functions in terms of basic functions (possibly trigonometric functions) so that one can derive the indefinite integral of said function. ...
13
votes
9answers
377 views

How do I Approximate $\log{2}\approx 0.693$ without using the Maclaurin series?

How do I approximate the value $\log{2}\approx 0.693$ without using the Maclaurin series? The book gives the hint: consider $f(x)=e^x-e^{-x}-2x$.
6
votes
0answers
269 views

Response Surface Methodology using Moving Least Squares Method

I would like to obtain the response surface of a mathematical function for reliability-based design optimization (RBDO). To obtain a reliably response surface, I learned that moving least squares ...
0
votes
0answers
14 views

Sobolev space for classic function approximation?

Hi guys could be any convenience in using a sobolev space instead of square integrable space for function approximation? I know that sobolev space are mostly used for PDE, but i was wandering if ...
3
votes
0answers
35 views

Solution for 4th grade polinomial equation

I'm development a physics model that require a expression for elongation of a elastic material, $\lambda=\frac{L}{L_o}$ [where $L$ is the thickness of the material and $L_o \equiv L(\sigma = 0)$] as ...
1
vote
1answer
28 views

ODE with time-dependent frequency

Let's have equation $$ \tag 1 \frac{d^{2}y(t)}{dt^{2}} +\omega^{2}(t)y(t) = 0, \quad t \in (t_{\text{in}}, \infty) $$ Here $$ \omega^{2}(t) = A(t) - B(t)cos(2t), $$ and functions $A(t), B(t)$ have ...
2
votes
2answers
39 views

Approximation of $(x+cx^2)/(1+cx^2)$, when c is small

I'm reading a paper and can't wrap my head around the following approximation: $f(x) = \frac{x+cx^2}{1+cx^2}$ $,$ $0 \le x \le 1$ Assuming that $c$ is small, $(c << 1)$, the following ...
0
votes
1answer
10 views

Function for approximating the definite integral of a function using an r-degree polynomial

We have the Midpoint Rule which approximates the definite integral of a function $f(x)$ over $[a, b]$ using $n$ sub-intervals with width $\Delta x$ using a degree-0 polynomial $A$: ...
2
votes
1answer
53 views

How can I get $\alpha$ and $\beta$ numerically? (Euler constant)

Let $\gamma_n=\sum_{k=1}^{n}\frac{1}{k}-\ln(n)$ and $\gamma=\lim_{n}\gamma_n$ From the fact $\frac{1}{2(n+1)}\leq\gamma_n-\gamma\leq\frac{1}{2n}$, we have $\gamma_n-\gamma \sim \frac{1}{2n}$ By ...
5
votes
0answers
122 views

Finding closed-form approximations of the solutions of $f(x,y)=0$

Consider $$f(x,y)=\sum_{i=1}^n\dfrac{\sin(\omega_ix)}{\sin(\omega_iy)}r_i$$ where $n,\omega_i,r_i>0$ are known parameters. Restrict to domains where $ \sin(\omega_iy)\neq 0$, and by symmetry ...
1
vote
0answers
13 views

Least Square solution for approximating a sequence

Suppose I have a sequence of length N $a_1,...,a_N$ I want to approximate this sequence by $k^1,...,k^N$ where $k$ is my variable. What is the least square solution of this? is there a closed ...
10
votes
8answers
554 views

What are better approximations to $\pi$ as algebraic though irrational number?

I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to ...
0
votes
0answers
99 views

Integration with Log of error function (erf)

Can anybody help me evaluating the closed-form or an approximate form of $H(x) = \int P(x) \ln(P(x)) \Bbb dx$ where $P(x) = \frac{C(x)}{v\int C(x) \Bbb dx}$ and $C(x) = {\frac ...
5
votes
2answers
149 views

PI aproximation with the $ {x\over y^2 } \approx \pi $ pattern

there are commands in mathematica and maple for finding Rational integer forms that approximate a floating point or a decimal number ... Mathematica function Rationalize Maple function ...
1
vote
2answers
51 views

Error in approximating e via a finite sum

I need some help with a homework problem. I have to find an upper bound for the error in approximating $e$ by the series $$\sum_{k=0}^{n} \frac{1}{k!}$$ I thought about using Taylor's theorem with ...
0
votes
1answer
38 views

Approximation of $\frac{e^{-\lambda |D|}(\lambda |D|)^k}{k!}$ when $|D|$ is very small

EDITED: Sorry. I made a mistake, I think I want to ask why this approximation is valid when $|D|$ is very small, rather than how to find its limit when $|D| \to 0^+$; also, I made a typo, it should be ...
1
vote
1answer
30 views

Exponential regression plus a constant

I've been looking for a convenient way to fit an expression of the form $y=a\cdot b^x + c$ to a set of points in the (x; y) plane. The technique I'm looking for should be sufficiently "light" from a ...
3
votes
1answer
58 views

Estimating $\sum_1^\infty e^{-k^2}$

While reviewing the integral test, I remembered a formula for bounding sums the test applies to (non-negative, monotone decreasing $f(x)$): $$\sum_{N+1}^M f(n) \leq \int_N^M f(x)dx$$ Specifically, ...
5
votes
1answer
90 views

Inverse of $\frac{1-e^{-x}}{x}$ on $(0,1)$

I am trying to invert (or to estimate the inverse of) $$y=\frac{1-e^{-x}}{x}$$ for $y\in(0,1)$. The function 'looks' monotonically decreasing between $x=0$ and $x=\infty$, but I have not been able to ...
0
votes
0answers
70 views

Perturbation Method for a Partial Differential Equation

Find the two term asymptotic expansion of the solution $u(x,t)$ of the weakly anisotropic diffusion equation $$\frac{∂u}{∂t} = \frac{∂}{∂x}\left[D\left(ε \, \frac{∂u}{∂x}\right) \, ...
1
vote
2answers
50 views

Implicit solution of ODE to explicit or approximate explicit function

Working with the following ODE and implicit solution but need an explicit solution for J: The ODE with$J_c$ and $G$ as constants is: $$-\frac{1}{J^2}\frac{dJ}{dt} = G(J-J_c)$$ The implicit solution ...
-1
votes
3answers
102 views

Golden Ratio Approximation

$$\sqrt{1000}-30.0047 \approx \varphi $$ $$[(\sqrt{1000}-30.0047)^2-(\sqrt{1000}-30.0047)]^{5050.3535}\approx \varphi $$ Simplifying Above expression we get ...
2
votes
0answers
15 views

Problems with Exchange Procedure in Remez Algorithm

So first off: *** This code is not being used in production software. It is a personal project of mine, trying to understand approximation theory and advanced curve fitting. In other words, I'm ...
1
vote
1answer
20 views

Diophantine approximation with integer vectors

I would like to determine whether or not there exists ${\beta > 0}$ and ${\gamma \geq 2 }$ such that ${ \forall (m_{1},m_{2}) \in \mathbb{Z}^{2} \setminus (0,0) }$, one has the inequality $$ ...
2
votes
0answers
55 views

Asymptotic Approximation

After analyzing performance of a cooperative system, I get the following expression for the system outage probability: $P = 1 - \frac{{{e^{ - 2\mu /{\beta _M}}}}}{{\Gamma \left( {{\alpha _3}} ...
0
votes
0answers
23 views

upper-band of the Integral expression

Consider below integral expression $$\int_{0}^{\infty}g(y)[\int_{a}^{\infty}(1-e^{-(k+y)x})f(x)dx ]dy \ \ \ \ (1)$$ Where, we know: $$f(x)>0\ ,\ \ a\leq x \leq \infty$$ $$\ k>0$$ $$g(y)>0\ ...
0
votes
1answer
38 views

Can Stirling's approximation be used to obtain lower and upper bound for $\pi(x)$?

The Willan's formula is given as follows (taken from Ribenboim's Little book of bigger primes): $$ \pi(x)=\sum_{j=2}^{x}f(j) \text{ where } ...
1
vote
0answers
24 views

Trapezoidal Rule yielding the exact value of the integral

It is clear that if a function $f(x)$ is linear over the domain $a \leq x \leq b$, then one application of the trapezoidal rule, over the same domain, will yield the exact value of ...
0
votes
1answer
255 views

For small values of $x$, the approximation $\sin(x)\approx x$ is often used. Estimate the error using this formula with the aid of Taylor's Theorem.

For small values of $x$, the approximation $\sin(x)\approx x$ is often used. Estimate the error using this formula with the aid of Taylor's Theorem. For what range of values of $x$ will this ...
0
votes
2answers
41 views

Binomial expansions question

In a physics book the autor make the following expansions, given the fact that $z>>d$ (much greater). However I didn't understand how he manage to get the final expression. ...
2
votes
1answer
42 views

Error estimation for the Wallis product

From the Wallis product we know $$\prod_{k=1}^{\infty} \left(\frac{2k}{2k-1} \cdot \frac{2k}{2k+1}\right) = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot ...
4
votes
1answer
70 views

Yet another Gamma function approximation

I know I have asked a similar question a couple of days, ago, but I still have a problem. I need a upper bound for: $$ ...
6
votes
0answers
151 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)} ...
2
votes
0answers
57 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 ...
1
vote
1answer
42 views

Lower and upper bound of the Stirling's approximation

Perhaps everybody has heard of the Stirling's approximation, namely: $$ \Gamma(z)\approx\sqrt{\frac{2\pi}{z}}\left(\frac{z}{e}\right)^z $$ Thus (the very basic example): $$ ...
1
vote
0answers
31 views

Please check this perturbation solution of polynomial root and truncation order.

I have a quintic polynomial where the coefficients depends on a parameter $c$, i.e. $$ a_0(c)+a_1(c)x+a_2(c)x^2+a_3(c)x^3+a_4(c)x^4+x^5 $$ I know that the roots of the polynomial are real and ...
0
votes
1answer
113 views

Coin Change Problem with Fixed Coins

Problem: Given $n$ coin denominations, with $c_1<c_2<c_3<\cdots<c_{n}$ being positive integer numbers, and a number $X$, we want to know whether the number $X$ can be changed by $N$ coins. ...
0
votes
2answers
71 views

Approximation of the Gamma function

I am having trouble obtaining a lower bound for the following formula: $$ \ln\frac{\Gamma\left(\frac{x}{3}\right)}{\Gamma\left(\frac{x}{4}+1\right)\Gamma\left(\frac{x}{12}+1\right)}. $$ I tried using ...
1
vote
2answers
31 views

Simple question, what is meant by 'as $x \to \infty$ the number of squares $\leq x$ is $\sqrt{x} + O(1)$?

For $x \to \infty$: the number of squares $n^2 \leq x$ is $\sqrt{x} + O(1)$. From here (page 6). More specifically, do they mean that... I'm confused now. I'm really not sure what they mean ...
0
votes
0answers
65 views

How to I approximate $I = \int_{-1}^{1} \sqrt{1-x^2}\cos(x)dx$ s.t. the error is bounded?

Edit: Because the original question was pretty trivial, I want to ask the same question but with:$I = \int_{-1}^{1} \sqrt{1-x^2}\cos(x)dx$. How to I approximate $I = \int_{-1}^{1} ...
1
vote
2answers
74 views

Series Approximation How to evaluate $1/3+1/3(1/3)^3+1/5(1/3)^5+…$?

How to evaluate $$\frac13+\frac13(\frac13)^3+\frac15(\frac13)^5+...$$? I faced this particular sum in the website www.toppr.com .And it is given under the heading "Problems on Approximation"...but I ...
3
votes
1answer
35 views

$\sum_{n \leq x} \frac{1}{n} = \int_{1}^x \frac{dt}{t} + O(1)$ help deriving it

On page 5 of: Probabilistic Number Theory by Dr.J¨orn Steuding, there's $\sum_{n=2}^{[x]} \frac{1}{n} \lt \int_{1}^{[x]} \frac{dt}{t} \lt \sum_{n=1}^{[x] - 1}$ Therefore integration yields: ...
1
vote
0answers
26 views

How to compare experimental data with teorethical prediction

I would like to know, what is the method to approximate experimental data to teorethical one. I have heard about polynomial regression. After calculating particular matrices and solving set of ...
1
vote
0answers
57 views

Is there an analytic approximation to integral of this form?

Started working on trying to find an analytical approximation to this integral and not getting very far. Any assistance or direction is greatly appreciated! Thanks Vince $$\int_{0}^{t} ...
0
votes
1answer
80 views

Approximating fractions

I have a fraction $\dfrac{a}{b}$ where $a$ and $b$ are both two large integers with $30$ digits each. I wish to approximate this fraction with a new fraction $\dfrac{c}{d}$ where $c$ and $d$ are both ...
2
votes
1answer
84 views

Upper bounding a tricky sum

For a problem in probability, I'm trying to find an upper bound for $$ \sum_{d=0}^k\binom{k}{d}\gamma^d(1-\gamma)^{k-d}\left(1-p^d(1-p)^{k-d}\right)^m$$ which will help me analyze what values of ...