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).

learn more… | top users | synonyms

0
votes
2answers
34 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
29 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
62 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: $$ ...
3
votes
0answers
75 views
+300

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
49 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
30 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
21 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
36 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
62 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
59 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
69 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
33 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
17 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
51 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
29 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
65 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 ...
0
votes
0answers
22 views

Minimum vertex cover of two edge disjoint perfect graphs

How well can the minimum vertex cover of the union of a perfect graph and bipartite graph (the two graphs are edge disjoint but not vertex disjoint) be approximated?
2
votes
1answer
38 views

How to find solutions for this nonlinear equation?

I want to find an analytical solution $x$ as a function of parameters $(e,u,r,t)\in\mathbb{R}^4$ that satisfies the following condition: ...
0
votes
2answers
44 views

Find to how many digits is the value 355/113, an accurate approximation to $3.1415929204$

Find to how many digits is the value 355/113, an accurate approximation to $3.1415929204$ What i did was i computed using calculator value of 355/113 which came out to be $3.14159265$ Now i see up ...
1
vote
1answer
19 views

Approximate first order derivative without solving the best fitted polynomial

For example, I have $K$ points of the form $(x_k,y_k,f(x_k,y_k))$ for $k=1, ..., K$ near $0$. The distances between the points and $0$ are not the same. Is there an approximation for first order ...
0
votes
0answers
13 views

Universal polynomial approximation algorythm

I would like to ask, is there any universal algorythm to fill this matrix for any n value? $\textbf{A} = \matrix{n & \sum x_i & \sum x_i^2 & \cdots & \sum x_i^n \cr \sum ...
3
votes
4answers
98 views

Why does $\sqrt{3x} \left(\dfrac {x}{2} \right)^x$ approximate $x!$ pretty well?

I was just messing around and trying out things in the desmos calculator and found that $\sqrt{3x} \left(\dfrac {x}{2} \right)^x$ is pretty close to $x!$ most of the time, here is a graph. Why does ...
1
vote
1answer
57 views

Is there a “greater than about” symbol?

To indicate approximate equality, one can use ≃, ≅, ~, ♎, or ≒. I need to indicate an approximate inequality. Specifically, I know A is greater than a quantity of approximately B. Is there a way to ...
1
vote
1answer
54 views

Question about normal approximation on roulette

Problem: At roulette, you bet a dollar on red $30$ times in a row. Each time you win a dollar with prob. $18/38$ and lose a dollar with prob. $20/38$. Find approximately the probability that after ...
0
votes
1answer
38 views

What are some applications of “separable” spaces?

A separable space is a space that contains a countable dense subset. For example, the space of continuous functions $C[a,b]$ is separable. Are there some practical applications arising out of this ...
0
votes
0answers
14 views

What is the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims?

A company has a one-year group life policy that divides its employees into two classes as follows: Class, Probability of Death, Benefit, Number in Class, A, 0.01, ...
0
votes
1answer
62 views

Finding the closest function describing a “magnetic line” (given magnetic readings)

I'm collecting data from a smartphone magnetometer while I move a magnet along a straight line (a slider). I am collecting the values of the magnetic field strength along the three axes. I would like ...
0
votes
1answer
30 views

How to approach to fitting curve?

I'd like to approximate fitting curve some kind of curves like below. (1, 3.5), (2, 4.3), (3, 7.2), (4, 8) which is having 4 points. and I heard that this solver is PINV() of matlab function. But ...
0
votes
2answers
59 views

How to find a equation from approximate curve? [closed]

I want to know a way to find equation from a curve. for example, if I have 4 point (1, 3.5), (2, 4.3), (3, 7.2), (4, 8) then how to find a good approximation equation ? What if I got above curve ...
0
votes
5answers
185 views

best approximation of $\sqrt{2}$

The approximation \begin{align} \sqrt{2} &\approx \frac{1}{8} \operatorname{csch}\left(\frac{3\pi}{2}\right) \operatorname{sech}^3(\pi) \, \left[2+3 \, ...
1
vote
1answer
26 views

How is the approximation justified and how to improve it?

In an attempt to find the solution to the equation $Mx=e^x$ with $M$ being a large real number and the solution $w \gt 1$ I was asked to justify why $\ln M$ is a reasonable approximation to ...
5
votes
3answers
806 views

How to find an approximation to $1 - \left( \frac{13999}{14000}\right )^{14000}$?

I want to find an approximation to the expression $$ 1 - \left( \frac{13999}{14000}\right )^{14000} $$ I tried by taking logarithm $$ \ln P = \ln\left(1 - ...
2
votes
5answers
118 views

Approximating $\sqrt{1+\frac{1}{n}}$ by $1+\frac{1}{2n}$

I was wondering how to approximate $\sqrt{1+\frac{1}{n}}$ by $1+\frac{1}{2n}$ without using Laurent Series. The reason why I ask was because using this approximation, we can show that the sequence ...
0
votes
1answer
10 views

Comparing a function and its estimate

What are some clever ways of comparing (visually) a function with its estimate? For regions where the function does not cross zero, plotting the ratio of the functions and plotting the relative error ...
0
votes
0answers
12 views

Approximate CDF of arbitrarily aggregated random variable

I would like to know if my solution for the following is mathematically correct in general: I have a random variable $Z$ that is an arbitrary function of two other rvs $X$ and $Y$, so: $Z = f_{arb}(X, ...
0
votes
1answer
13 views

Which function is suitable to approximate a convex piecewise linear function

Im trying to fit a convex piecewise linear function into a smooth function. However I have no idea which kind of function is suitable? Can anyone give me some examples of function that is suitable ...
-1
votes
1answer
19 views

Numerical integration of functions over computable Cauchy sequences

I'm interested in exact real arithmetic (and by extension constructive analysis). A nice representation of real numbers is via Cauchy Sequences. The basic idea being that you have a function which, ...
1
vote
1answer
54 views

Use stirlings approximation to prove inequality.

I have come across this statement in a text on finite elements. I can give you the reference if that will be useful. The text mentions that the inequality follows from Stirling's formula. I can't ...
9
votes
2answers
83 views

Existence of a function

I need some help: I am thinking about this problem. Any advice would be appreciated. Let's fix $\epsilon>0$. Does there exists some $f\in C^0([0,\pi])$ such that: ...
0
votes
0answers
35 views

What functions stem from Fourier Series with rational-only coefficients?

Given the Fourier series $$f(z) = \sum_{k=-\infty}^\infty c_k e^{ikz}$$ but with $c_k\in(\mathbb Q+ i\mathbb Q)$ instead of $\mathbb C$ (or even purely real), are the functions obtained this way in ...
4
votes
2answers
66 views

Iterative calculation of $\log x$

Suppose one is given an initial approximation of $\log x$, $y_0$, so that: $$y_0 = \log x + \epsilon \approx \log x$$ Here, all that is known about $x$ is that $x>1$. Is there a general method of ...
2
votes
2answers
37 views

How to choose the extent of approximation for this equation?

Given the curve $$y={1 \over{x^2-1}}-{1\over {x^2}}$$ Find reasonable approximations for the intersections of this curve with the straight line $y=Ax$ (a) when $A$ is a very small positive number ...
0
votes
1answer
33 views

functions that jump from zero to value k and then decay back to zero

I am looking for functions that are zero for all $x< a$ and then from a suddenly jump to a specified value $K$ and then drop back to zero either quickly or slowly and perhaps even not necessarily ...
2
votes
3answers
154 views

Is $e^{e^{2}}$ a relatively good approximation for $1000\phi$? [closed]

Yesterday night, I found that $e^{e^{2}} \sim 1000\phi$, where $\phi$ is the golden ratio. I believe that it is correct to four decimal places. Would it be considered a relatively good ...
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}) ...
1
vote
2answers
15 views

Approximation of derivative of discrete function

I have a function of which I only know the value of at some discrete points. Now I want to calculate the derivative of this function. The approximation of taking the difference of two consecutive ...
0
votes
2answers
37 views

Runge Phenomena and Taylor Expansion

From The Weierstrass Approximation Theorem Vs The Runge's Phenomenon: We contrast this to polynomial interpolation: this is a specific method for generating a sequence of polynomials that ...
0
votes
1answer
30 views

Fourier expansion of absolute value of a periodic function

For an arbitrary periodic function p(x), whose period and Fourier expansion might have been known in advance, how can we get the Fourier expansion/coefficients of |p(x)| from them? Or, if possible, ...
4
votes
2answers
44 views

Why can the elements of $L^\infty$ be approximated in $L^p$ by $C^1$-functions?

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $f\in L^\infty(\Omega)$. From which theorem does the existence of $(f_k)_{k\in\mathbb N}\subseteq C^1(\overline\Omega)$ with ...