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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6 views

Approximation technique of common probability distributions that can be convoluted and integrated fast

I am looking for a approximation technique of functions with two conditions: It is possible to perform a fast approximate convolution with the approximate functions. It is possible to numerically ...
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0answers
34 views

Lebesgue integration of $f(x)=\frac{1}{x}$ where $x\in[0,3]$

We have the function $f(x)=\infty$ if $x=0$ and $f(x)=\frac{1}{x}$ if $x$ otherwise. So, in this two values of function, I made simple approximation of $f(x)$ by the help of simple function : ...
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4answers
35 views

What is the approximation of trigonometric function by simple function

for $f(x)=\sin x$, $g(x)=\cos x$, $h(x)=\tan x$, What is the approximation of each function by using simple function?
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2answers
15 views

How is this approximation related?

I'm in the dark about the following approximation. Given that $r >> (l_1 + l_2)$ $\phi(P) \approx \dfrac{\lambda}{4\pi\epsilon_0}ln(\dfrac{r+l_2}{r-l_1}) \approx \dfrac{\lambda(l_1 + ...
-1
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0answers
22 views

Linear approximation of f(g(x)) [closed]

Suppose that the functions f(x) and g(x) are differentiable everywhere. Some values for these functions and their derivatives are given in the table. x f (x) f ′(x) ...
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1answer
27 views

Expansion in large and small limits

Let $$f(x) = \frac{1}{\log(\frac{x}{c})}$$ where $c$ is some constant number. Consider the variable $x$ in the large regime where $x \gg c$ and small regime where $x \ll c$. How would $f(x)$ depend on ...
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1answer
65 views

Comparison between lebesgue integral and riemann integral of $f(x)=x^2$ in $[0,2]$

If we have an example $f(x)=x^2$ let's say for $[0,2]$. In lebesgue integral, I already use a sequence of function $f_n(x)$ as approximation to $f(x)$ ($f_n(x)$ converges to $f(x)$) which is stated ...
2
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1answer
53 views

Numerical approximation, or is there a better way? $x=1+\ln(1+\ln(1+\ln(x)))$

Don't have enough reputation to post on this but is there any way of calculating the following? $x=1+\ln(1+\ln(1+\ln(x)))$
4
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1answer
45 views

The Cosine and an Enigmatic Parabola

In the interval $[-\frac{\pi}{2},\frac{\pi}{2}]$, the cosine function superficially resembles an inverted parabola of the form $-ax^2+1$: I wanted to know more and computed the $L^2$ norm ...
3
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2answers
56 views

Can one apply a WKB method to an inhomogeneous first order differential equation in order to find the asymptotic expansion of the solution?

Consider \begin{equation} \varepsilon \frac{dy}{dx} = Q(x)y + R(x) \end{equation} where $\varepsilon$ is a small parameter. Can one apply a WKB method to find an asymptotic expansion for the ...
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0answers
11 views

Probability - Characterizing goodness of moment matching method.

I have a question about how to characterize the goodness of approximating a distribution using its moments. Suppose I have a probability density function $p(x)$ (e.g., normal distribution), and I am ...
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0answers
15 views

Fitted function - Which is better to use?

So I have some data for program running time, that follows a power law relation aN^b. I log-log plotted the data and saw that it became a straight line, so I calculated the slope of this line to get ...
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1answer
46 views

How to write approximations of a sequence $x_n = {1/3^n}$

Write three approximations of the sequence ${x_n} = {1/ 3^n}$, using the following scheme - $P_0= 1, P_1 = 0.33332$ and $P_n = (6/5)P_{n-1} - (1/5)P_{n-2}$ for $n = 2, 3,\dots$ Further, make a ...
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0answers
63 views

Does a greedy task selection algorithm find a c-approximate solution?

A scheduling problem can be stated as: Given a set $\{(s_i,f_i)\}_{1\le i\le n}\}$ of tasks identified by their start and end times, choose the maximum size subset of non-overlapping tasks. ...
0
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0answers
17 views

Finding when $(x^3 + …)/(x^4 + …)$ reaches $10^{-n}$

A problem I’m working on throws up equations like: $$ \frac{1}{4k + 9} + 3\frac{k+1}{(4k + 9)^2} + \frac{2k+1}{(4k + 9)^3} + \frac{k}{(4k + 9)^4} $$ I need to know the value of $k$ for which this ...
0
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1answer
17 views

Error evalution for Newton-Raphson method

I'm supposed to approximate a solution of an equation using the Newton-Raphson method, knowing one real solution to this , namely $x \approx 0.61803$. $$x^4 + 3x - 2 = 0 $$ Therefore I start by ...
2
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0answers
17 views

How to interpolate multidimensional functions?

I'm learning about interpolation and I wanted to ask if there's a "good" method to interpolate multidimensional functions (when the dimension can be even a few thousands)? Is there a theoretic limit ...
3
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0answers
31 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) - ...
0
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0answers
38 views

How can we evaluate the troublesome integral?

How can we evalutate the integral $$ I=\int_0^{\infty}\Big[\arctan\big(\frac{\pi}{\ln 5-5-x-\ln x}\big)-\pi\Big]\frac{dx}{x+5} $$ analytically? I tried to evaluate it by parametrical integral, i.e. $$ ...
1
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1answer
28 views

Question on probability and approximation

Okay I think you are all familiar to YouTube videos and some facts are: to comment, like and dislike on a video you need a Google account. when someone views the video the view count of the video ...
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1answer
23 views

Binomial cumulative probability

Here is the question I need help on : Let $X$ be a binomial random variable with p = 0.5 and n = 100. Give $P(X \geq 60)$ rounded to two decimal places without using a calculator (by using ...
2
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0answers
94 views

Lower bound on a polynomial far from its zeros

Let $p(x) = \sum_{i=0}^{d}c_{i}x^{i} \in \mathbb{R}[x]$ and assume that all its zeros are real and in $[-1,1]$. I am interested in lower bounding the value of $|p(a)|$ in case $a \in [-1,1]$ is far ...
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0answers
6 views

Root finding: Distance b/w 2 objects = 0. (and other examples of finding roots?)

Can someone explain general uses of finding roots? I understand you can find roots to help manually graph a function, but there's gotta be more. For example, in video games, I recall something about ...
0
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1answer
43 views

Approximating a power of a root of unity to within $\delta$

I have an estimate of $\omega$, a root of unity. I'm really wondering how small the error (in the estimate), which I give as $\epsilon$, has to be, so that when I take my estimate of omega to the ...
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0answers
39 views

Univariate Polynomial Approximation

I'm working on an algorithm in which I need to approximate the behavior of a polynomial by computing its roots to some $\epsilon$ precision. The problem can be defined as follows: Let $f(x) = x^n + ...
2
votes
1answer
48 views

How to approximate this nasty exponential function with an integral?

What is the best way to approximate a function of the following form, $$ \text{exp}\left(-\int_{y}^{+\infty} f(x)\ dx \right)$$ Any approximation to this, does taylor series work? The reason I am ...
14
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2answers
154 views

A Mathematical Coincidence, or more?

According to the paper "Ten Problems in Experimental Mathematics", $$\int_0^\infty \cos(2x)\prod_{n=1}^\infty \cos\left(\frac{x}{n}\right)dx \quad = \quad \frac{\pi}{8}\color{blue}{-7.407 \times ...
1
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1answer
27 views

Angle measurement

Assume I want to compute one of the angles of a right triangle doing $n$ measurements of the sides with a ruler. In order to increase the precision I make several measurements. After that I compute ...
2
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2answers
69 views

Methods for calculating $\pi$ that use the sphere?

The area of the unit circle is $\pi$ and its circumference is $2\pi$. Consequently, many elementary methods for calculating and approximating $\pi$ use a geometric approach on the circle, such as ...
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0answers
15 views

MATLAB implementation Spline Fitting

Check the attached problem please. I am a beginner in spline fitting and have a few questions: 1) How to find the coefficients c[n]. Is it by DTFT? 2) I understand how to find the derivative but ...
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2answers
37 views

Approximation of numbers [closed]

How could we approximate an irrational number by rationals?? Could you give me some hints?? I don`t have any idea how we could approximate them by rationals...
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1answer
31 views

Finding a sequences so the image approximates integers

$x$ is fixed in $[0,1]$, $a_n,b_n,c_n$ are integers and not all of them are $0$ $y(a_n,b_n,c_n) = a_n x^3 + b_n x^2 + c_n x = Y_n$ Find an algorithm to go from $(a_n,b_n,c_n)$ to ...
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0answers
29 views

WKB approximation for multiple turning points

I'm working on a numerical program which approximates the eigenvalues of a Schrödinger equation by making use of the WKB approximation formulas. For example, if the Schrödinger equation is $$ y''(x) = ...
0
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1answer
38 views

Can we prove the Riemann-Lebesgue lemma by using the Weierstrass approximation theorem?

I'd like to prove the following version of the Riemann-Lebesgue lemma: Let $f: [0,1] \to \mathbb R$ be continuous. Then $$\int_0^1 f(x)\sin(nx) \, dx \xrightarrow{n \to \infty} 0$$ It's quite ...
0
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0answers
41 views

Are there ever cases where it's easy to get coefficients for the series representation for an integrand, but hard to approximate the integral?

WHY I'M ASKING THIS I'm working on a faster way to approximate integrals of series. So I'd like to know if this could be useful. THE QUESTION If we suppose that we can get a formula for the ...
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0answers
17 views

integration with pade approximant

given the function $$ \int _{0}^{\infty}\sqrt{x}exp(-x) $$ can we use Pade approximants to integrate this i mean let bhe te rational approxsiamtions of $ \sqrt{x}= \frac{A(x)}{B(x)} $ and $ ...
6
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2answers
145 views

Approximation of minimum among many binomials

We choose $k$ numbers independently from the binomial distribution $B(n,1/2)$, where we can think of $n$ as large. What is the expectation of the minimum of the $k$ numbers? Is there a good way to ...
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0answers
42 views

A question about $f(x)\equiv C$

Let $f(x)$ is Continuous function on $[0,\pi]$,and for $n=1,2,.....,$ the function $f(x)$ has the following property:$$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0.(n=1,2,......)$$ Proof: $f(x)\equiv C$(C is ...
0
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2answers
16 views

Use of small approximations to get the following formula

I cannot see how the following formula has been found $\displaystyle \cos(\theta + \frac{v\epsilon^2 \Omega}{L}+O(\epsilon^4))-\cos(\theta) = -\epsilon^2\frac{\Omega v}{L}\sin(\theta)+O(\epsilon^4)$ ...
0
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1answer
28 views

Approximating Pirrational Numbers

A while back I wrote this question on PPCG.SE about the numbers I termed Pirrational numbers. They are defined as follows: Let $P_i$ be the $i$th Pirrational number for some $i \in \mathbb{N}_0$ ...
0
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2answers
39 views

When $a\ll b$, how to approximate $f = \int_0^a \sqrt{b^2+x^2}/\sqrt{a^2-x^2} \, \, dx$?

Suppose $a\ll b$. How do I then approximate $$\int_0^a \frac{\sqrt{b^2+x^2}}{\sqrt{a^2-x^2}}dx$$ ? I think that maybe Taylor approximation may help, but I am not sure how to proceed. My physics ...
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0answers
14 views

How do they calculate players' chances of winning in 9-handed Hold 'em or Omaha poker with hidden information?

So Omaha poker is a card game where each player is dealt 4 private cards, and then 5 community cards are dealt in the middle, and each player makes the best possible 5 card poker hand by using 2 cards ...
0
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1answer
42 views

Linear Approximation with log

The problem: Find the linear approximation of the function $$ f(x,y) = \ln(e+x+y) $$ at point $(0, 0)$. Use it to approximate the value of the function at $(0.1, 0.2)$ What I have so far: I found ...
0
votes
1answer
50 views

Calculating originally arc approximated by cubic bezier curve

I have an cubic bezier curve, which is representing an arc by an approximation. The approximation was calculated with the kappa constant: $$ \\k = \frac43*(\sqrt{2}-1) $$ This means, that the ...
1
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0answers
22 views

Approximate Solution to Backwards Recurrence of Dynamic Game

Suppose we keep tossing a fair dice until we reach some cumulative sum greater than or equal to $N$. Then, let $S_k$ be the expected value of the final sum, given that the current sum is $k$. We have ...
2
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0answers
34 views

How do we know that the first few digits of an approximation for $\pi$ are correct?

For Gregory–Leibniz series, wikipedia has - "after 500,000 terms, it produces only five correct decimal digits of π.". But how do you know that those five decimal values are correct when you reach ...
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0answers
28 views

Bounds on constant for Stirling approximation

Stirling's approximation says that $$n!\sim\sqrt{2\pi n}\left(\dfrac{n}{e}\right)^n.$$ What is known about constants $c_1$ and $c_2$ such that $$c_1\sqrt{n}\left(\dfrac{n}{e}\right)^n\le n!\le ...
2
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2answers
50 views

Estimating the behavior for large $n$

I want to find how these coefficients increase/decrease as $n$ increases: $$ C_n = \frac{1}{n!} \left[(n+\alpha)^{n-\alpha-\frac{1}{2}}\right]$$ with $\alpha=\frac{1}{br-1}$ and $0\leq b,r \leq 1$. ...
2
votes
3answers
46 views

Is there an approximation to the natural log function at large values?

At small values close to $x=1$, you can use taylor expansion for $\ln x$: $$ \ln x = (x-1) - \frac{1}{2}(x-1)^2 + ....$$ Is there any valid expansion or approximation for large values (or at ...
3
votes
2answers
55 views

Correcting Error in the Leibniz $\pi$ formula… why does it work?

You are probably familiar with the Leibniz $\pi$ formula: $$ 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots = \frac{\pi}{4} $$ For a CS homework assignment I had to write a ...