Questions tagged [approximation]

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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68 views

Why this approximation for $\pi$ is so accurate?

Berggren and Borwein brothers in "Pi: A Source Book" showed a mysterious approximation for $\pi$ with astonishingly high accuracy: $$ \left(\frac{1}{10^5}\sum_{n={-\infty}}^\infty e^{-n^2/10^{10}}\...
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2answers
28 views

Saddle point approximation gives a null result

So I want to compute the following integral $$I=\int_0^1 x\sqrt{1-x}\exp \left(a^2x^2\right) dx$$ where $a>>1$. If we try to do a Saddle point approximation \begin{align} I&=\int_0^1 f(x)...
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27 views

What are some ways we measure irrationality?

I am wondering in what ways we may quantify an irrational number's approximability. This came up as I was reading about badly approximable numbers, which are those numbers $x$ such that $$\liminf_{q ...
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1answer
23 views

Different approximations of this function

Was told to post here. However, I have heard about this site, as well, but I am hesitant on posting on the internet, hence I made an account. Anyway, my question is: How can I approximate the sin(2) ≈...
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1answer
16 views

Limit of ratio of incomplete gamma function

In order to derive Sterling's approximation, I need to show that the following integral decays quicker than at least $\mathcal{O}(n^2)$: $\lim_{n\to\infty}\dfrac{\int_{2n}^\infty x^ne^{-x}dx}{\int_{0}^...
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2answers
68 views

Is $x \approx x$?

If I write $x \approx y$, does this mean (a) $x$ is sufficiently close to $y$ for some practical purpose, or (b) $x$ is sufficiently close to $y$ for some practical purpose, but is not equal to $y$? ...
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27 views

Maclaurin series approximate problem

Use the Maclaurin series to find an approximate value for the following integral: $$\int_{1}^{12} \sin(x^{4})dx$$ Need help with this question please
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15 views

Minimizing the distance to a subspace (orthogonal projection) if a norm is not induced by an inner product

Let $V = \mathbb{R}^n$ with the inner product $\langle\cdot,\cdot\rangle$ and $U \subset V$ a vector subspace. Then for $v\in V$ and $x \in U$ the inequality$$ ||v-x|| \leq ||v-u||$$ is satisfied for ...
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38 views

How can I approximate the following objective function?

I have the following objective function. I want to approximate the third term with norm-2 in a way that I could decompose my problem by $j$ (I need $\sum_{j \in J}$ out of the square root). I am ...
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1answer
37 views

Calculating Lambert W Function

I'm trying to evaluate Lambert W Function , I used the formula $$ W(z)e^{W(z)} = z \Rightarrow W(z) = \frac{z}{W(z)} $$ $$ W(z) \approx ln(z)-ln(ln(z)-ln(...)) $$ But the result is very bad If I used ...
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34 views

Traveling salesman with a twist: must visit each node twice and must “wait” between returning to each node?

Please be kind if this question violates the rules as I am still working on my Stack Exchange question-asking intuition. The motivation for this problem is the optimal deployment of automated ...
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1answer
55 views

How is 'Convergence' defined?

I've noticed that some papers, e.g. in theoretical computer science and numerical mathematics, provide pseudo-algorithms for their proposed methods. Often these pseudo-algorithms have instructions ...
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1answer
23 views

Determine second degree polynomial by least squares method

Determine the polynomial of the second degree that approximates by the method of least squares in the following function, at the interval [0,5, 1,5]. $f(x)=\frac{3}{\sqrt{x}}$ I´ve done least ...
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1answer
25 views

Approximation of Integral for T = 1K

Suppose we have the following integral: $$\int_{0}^{x_{\text{max}}}\frac{x}{1+e^x}dx,$$ where $x\equiv \frac{\Delta E}{k_BT}$. We are supposed to argue that it is justified to consider $x_{\text{max}}\...
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27 views

What is known about the approximation constant?

A badly approximable irrational is one whose continued fraction denominators are bounded; equivalently, if $\alpha$ is badly approximable then there is a $c(\alpha) > 0$ such that $$c(\alpha) = \...
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1answer
40 views

Show that $\Gamma(x) \sim \sqrt{2 \pi} e^{-x}x^{x-\frac12}$.

The gamma function is defined by $$\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} dt $$ where $x > 0$. Show that $\Gamma(x) \sim \sqrt{2 \pi} e^{-x}x^{x-\frac12}$. $\sim$ denotes that the ratio ...
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2answers
43 views

Numerical algorithm for finding the inverse of a function

Is there a numerical method to approximate the inverse of a function for a given interval? Thank you
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3answers
53 views

How do I calculate the limit of $\lim_{n\to \infty} (1-\frac{\theta^2}{2n^2})^{2(n+1)}$

I'm going through some physics problems about polarizers and one problem is about the case where $n+1$ polarizers are stacked up and I have to look at the case where $n \to \infty$. Now I came up for ...
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17 views

Numerical $n$-th order mixed partial derivative

To start with, this might be a naive question since I do not have a lot of experience with numerical analysis. Let $f(\boldsymbol{x})$ be a function in $M$ variables, i.e. $f:\mathbb{R}^M\to\mathbb{R}...
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2answers
65 views

Approximate solution to a transcendental equation in the limit of a variable

I have the following transcendental equation: $$2\cot{x}=\frac{kx}{hL}-\frac{hL}{kx}\tag1$$ I would like to inquire whether an approximate solution to $(1)$ can be developed in the limit $h\...
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59 views

How to find path for method of steepest descent

We have integral: $$\int_0^1\exp\left(n\left(\frac{itz}{\sqrt{a(1-a)n}}+a\ln(z)+(1-a)\ln(1-z)\right)\right)dz=\int_0^1\exp(nf(z))dz,$$ where $0<a<1$. We want to approximate this integral when $...
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1answer
22 views

What is the power series of $1/(D-h(x))$ if $h(x)\ll D$?

I have a problem, which I do not conceptually understand. I need to approximate an arbitrary function $$\frac{1}{D-h(x)}$$ where $h(x)$ is arbitrary, $h(x)\ll D$ and $D$ is a constant. Friends say ...
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1answer
16 views

How to find x's for representing a specific type of integral using a quadrature formula?

I have this integral which I want to approx with quadrature formula for some fixed $n$: $$ \int\limits_0^{+\infty} x e^{-x} f(x) dx \approx \sum\limits_{k = 0}^n A_k f(x_k) $$ I found info about how ...
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22 views

How can we speed up and improve circle-approximation using symmetries?

I am currently on the hunt for a way to approximate a half-circle function $$t \to f(t) = \sqrt{1-t^2}$$ By means of truncated power series (polynomials). I found several issues with a previous ...
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0answers
18 views

Deriving the Taylor expansion of a half-circle.

When answering a question a while back, I found a set of truncated circle shaped functions $x\to \sqrt{1-h_{n,k}(x)^2}$ which seem to be able to arbitrarily well approximate the monomials $x\to x^m$ ...
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0answers
26 views

Combination of normal distributions and binomial distributions

I am trying to create a functional approximative probability calculator based on a set of random number generators with unknown functions. However, I am really lacking concrete practical experience ...
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1answer
48 views

$\sum_{t=1}^x e^{-\frac{1}{t}} $ approximating $\log_e(\pi(e^x))\sim x$

Related to a previous question: Is $\ln(\pi(e^x)) \sim x?$ $\sum\limits_{t=1}^x e^{-\frac{1}{t}} $ approximates a modified prime counting function $\ln(\pi(e^x))\sim x$. This is similar I guess to $\...
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17 views

How do you find the Inverse of Incomplete Elliptic İntegral of Second Kind when modulus is large

So I tried to take the inverse of EllipticE when k modulus is large, in Mathematica, but the solution gives wrong answer. ...
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107 views

Solving a difficult integral

I have been stuck on the following integral for some time: $$ I = \int^{2 \pi}_0 \mathrm{d}\theta \frac{\cos \theta\left(x + \Delta\cos\theta\right)}{\sqrt{k + \left(x + \Delta\cos\theta\right)^2}} \...
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1answer
61 views

Is there a way to represent functions using circles, similar to how Taylor series work?

I was curious whether or not there is a method of representing a function as an infinite amount of circles multiplied or added or something? Similar to a Taylor series, such that the method would ...
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2answers
48 views

Why $\pi$ appears in Dirac? [duplicate]

"With hands", what is the reason why $\pi$ appears in this result : $\lim_{\epsilon\to 0}\frac{\epsilon}{\epsilon^2+x^2}=\pi\delta(x)$
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15 views

How the following Approximation is done to obtain the Inverse Laplace Transform?

The following highlighted part from a textbook is where I am stuck. The Laplace transform is independent of the derivations made prior to equation $(2.77)$. Please help me to understand how they have ...
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0answers
26 views

Second order Taylor approximation

$$\log E_t(X_{t+1}) = E_t(\log X_{t+1}) + \frac{1}{2}Var_t(\log X_{t+1}) $$ How can I prove it?I knwe that, $$g(x) = g(E(x))\ +\ g'(E[x])\ (x-E[x]) + \ g''(E[x])(x-E[x])^2$$ But I do not know how to ...
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4answers
49 views

Where does this relation come from? $n^2-1 \approx (n-1)2$ for $n-1 \ll 1$

I came across the relation in the title in a physics textbook and wondered how I get to it. $$n^2-1 \approx (n-1)2$$ for $$n-1\ll 1$$ Could anybody maybe help me out? Thanks!
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1answer
24 views

Approximation of an Integral for T >> T_D

In a lecture, we had the following integral: $$C_{\text{V, mol}} = 9R\left(\frac{T}{T_D} \right)^3 \int_{0}^{x_D}\frac{x^4e^x}{\left( e^x-1\right)^2}dx,$$ where $x_D \equiv \frac{\hbar\omega_D}{k_BT}...
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1answer
19 views

Working on an approximation of $\pi$. How do I demonstrate an inequality on the taylor expansion of arctan?

I'm struggling with the following inequality. In particular, I don't understand how to demonstrate that this term is larger than the left hand side for all $n \in \mathbb{N}$. It seems to be ...
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1answer
52 views

Approximation of a distribution from $\mathcal{D}'$ by functions from $\mathcal{D}$

I need to prove that for any generalized function ( they are also called distributions) $f \in \mathcal{D}'$ there exists generalized functions $f_n \in \mathcal{D} $, given by normal functions from $...
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0answers
18 views

Can you identify this unknown sequence related to approximating circle area using unit squares?

I am interested in finding out whether the following sequence has already been "discovered" or if it is worth submitting to OEIS (Online Encyclopedia of Integer Sequences), since I can't find it on ...
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2answers
29 views

Finding the interval of a function that can be approximated to $x^3$

$f(x) = \ln(1+x^2) \cdot \arctan(x)$ Using the approximation $ f(x) = x^3 $ Find the interval centered at zero where this approximation is accurate to within $0.00001$, or $10^{-5}$ So I know that ...
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0answers
25 views

approximate solution for ODE, error estimation with Picard-Lindelöf's Theorem

Let be $$y'= \frac{t}{y^2+1} $$ $y(0)=0 $ I want to determine an approximate solution $ \tilde{y} $ from the starting point $ \phi_0 =0 $ so that the error is : $ \lVert \tilde{y} -y \rVert_{ C[-1,...
4
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1answer
376 views

Is this like the Birthday Problem? Poisson Halloween Party

Suppose that there are n guests at a Halloween party, and that each is wearing one of 200 possible costumes available at local store, uniformly at random and independently of all other guests. Using ...
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3answers
37 views

How to tell / quantify how much a number is close to some simple integer ratio?

It's easy for us to tell that 0.49999 is only 0.00001 away from being expressed as a simple ratio: 1/2. However, it may not be as obvious that 0.142858 is also at most only 0.00001 away from being ...
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0answers
16 views

Poissonization use for sample size

(Cross-post from Stats Stack Exchange) I'm interested in using the Poissonization trick to solve the following problem, which I made up: Suppose I have a categorical random variable $X$ taking ...
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1answer
10 views

Applying polynomial time approximation scheme (PTAS) on an algorithm

I am trying to understand how we can apply PTAS on an algorithm. I think that we apply PTAS on the running time equation of the algorithm. We use the term (1-ϵ) and (1+ ϵ) in the running time of the ...
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0answers
6 views

Approximation of convex hull by projecting points to four planes and then taking their supeposition? How does it work?

Anyone understand more generally the idea used to approximate convex hull of a tree top shape in this paper: At first, the crown points were projected onto four vertical planes through the ...
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0answers
36 views

Approximation for the second order derivative: $\frac{d^2y}{dx^2}\approx\frac{\Delta y}{\Delta x^2}$. Why?

My physics professor made the following approximation: $$\frac{d^2y}{dx^2}\approx\frac{\Delta y}{\Delta x^2}$$ How can you fundament this? I get how you can do $\displaystyle\frac{dy}{dx}\approx\frac{...
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1answer
39 views

Set Cover Problem: How to calculate the denominator?

I am trying to understand the set cover problem. I found the algorithm at: Set Cover which also contains the attached example: Somebody please guide me, how we calculate the denominator |S-C| and how ...
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0answers
13 views

Measuring great circle distance like length in cylindrical coordinate?

picture Is there a formula to calculate "great circle distance" like length in cylindrical coordinate? As in the yellow line between 2 points in the attached picture. I think I can interpolate and ...
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0answers
50 views

Integral in dimension 7

I want to compute the volume of region bounded by the inequalities : $x_2x_4x_6\geq x_1x_3x_5x_7, x_1x_4x_5\geq x_2x_3x_6x_7, x_3x_4x_7\geq x_1x_2x_5x_6, x_1x_2x_3\geq x_4x_5x_6x_7, x_2x_5x_7\geq ...
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0answers
10 views

Find the shortest curve that crosses given discs

Let $D_{n}$ be a set of discs. Where $n=1,2,...,n'$. Each disc is defined as follows $D_{n}=\{(x,y)\in\mathbb{R}^{2}:(x-a_{n})^2+(y-b_{n})^2\le C^2\}$ Where $\{a_{n}\}$ and $\{b_{n}\}$ are some sets ...

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