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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1answer
313 views

Approximations. Newton's method - composite Simpson's rule

Can you help me please to solve this problems and if you can give me some helpful information. Thanks!
0
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1answer
142 views

Approximating piecewise linear function

I'm trying to derive an analytic approximation to the following piecewise linear function: $$ f(x) = \left\{ \begin{eqnarray} \frac{x}{x_s} & & \text{if} & x < x_s \\ \frac{1-x}{1-...
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0answers
62 views

Approximating $C^2$ functions with compactly supported $C^2$ functions

Let $C^2$ be the space of twice-continuously differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$ and $C^2_K$ be the subset of functions in $C^2$ with compact support (that are zero outside ...
0
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1answer
756 views

Approximation to the square root

I was reading an article that approximated a square root operator as follows $\sqrt{1+x+y} \cong \sqrt{1+x} + \frac{1}{2}y + O(xy,y^2) $ At first glance that looks like a Taylor series expansion, ...
5
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1answer
210 views

Subdifferential boundary conditions: Testing with $L^2$ or $H^{1/2}$ functions

My question was essentially this: Does it make a difference if I test subdifferential boundary conditions with functions from $L^2(\Gamma)$ or $H^{1/2}(\Gamma)$? In the following, I will phrase the ...
3
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3answers
778 views

Approximate $|x|$ with a smooth function

I am trying to get the derivative of $|x|$, and I want that derivative function, say $g(x)$, to be a function of x. So it really needs the |x| to be smooth (ex. $x^2$); I am wondering what is the ...
23
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6answers
1k views

Why is $e^\pi - \pi$ so close to $20$?

$e^\pi-\pi\approx 19.99909998$ Why is this so close to $20$?
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1answer
16 views

approximation for formula — relation to exp function

Let $n>0$ be a natural number. I am looking for an M(n) of the following function: $$ f(n) = (1 - n^{-1/4})^n < M(n) $$ I know that if $n$ goes to infinity $f(n)$ goes to $0$. Now, I wonder if ...
2
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1answer
190 views

Existence of a simultaneous rational approximation of real numbers in (0,1)

I have a simple question the rational approximation of real vectors. Dirichlet's simultaneous approximation theorem states: Given any $d$ real numbers $\alpha_1,\ldots,\alpha_d$ and for every ...
0
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1answer
29 views

Approximating an interval [duplicate]

Corollary C is derived from $E_(x) = \frac{f''(x)}{2} \cdot (x-a)^2$ I'm having serious issues understanding this problem and I'm just not getting it right. I'm preparing for a test so this isn't ...
3
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1answer
344 views

Error formula for linearization

Can anyone shed some light on this formula? I can't find any information on it. It has three corollaries that I also need to understand:
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0answers
358 views

Best Fit Curve and Function for 4D Data?

I have experimental data of 4 dimensions, and I want to computer-generate an approximated function from that. If it helps to clarify, I'm testing a pendulum's period based on variable length, starting ...
3
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1answer
102 views

Approximating hypergeometric function F(1,1+a,2+a,z) for z->1

in my studies a normalization constant for a pmf includes the hypergeometric function ${}_2F_1(1,1+a,2+a,z)$ The parameters are in the range $0.99<z<1$ and $0<a<5$. I have tried some ...
1
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0answers
42 views

Approximation of eigenvalue

I'm having trouble with the following problem: Let $A\in M_n(\mathbb{C})$ be a Hermitian matrix, $\mu\in\mathbb{C}$, $\epsilon>0$ and $x\in\mathbb{C}^n$ such that $\mid\mid x\mid\mid=1$ and $\...
1
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3answers
41 views

An approximation: 2*number of events = twice the probability

My statistical mechanics textbook uses an approximation to derive a well-known result. The approximation is: Suppose for an event, the probability of an outcome is P. For n events, the probability of ...
6
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2answers
353 views

Find the integral part of $\sum_{i=2}^{10000}\frac1{\sqrt{i}}$

$$A = \frac1{\sqrt{2}}+\frac1{\sqrt{3}}+\cdots+\frac{1}{\sqrt{10000}}$$ Find $\lfloor A\rfloor$ where $\lfloor x\rfloor$ is the greatest integer less than, or equal to $x$ I got stuck on this, so ...
2
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1answer
78 views

Approximation of a factorial

In books I can find a lot of approximations of factorial, which doesn't require the a lot of multiplications, like for example Stirling's approximation. Very popular is also the Euler's solution which ...
0
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1answer
60 views

Forcing coefficient of best fit quadratic

So I have a question regarding best-fit polynomials. Currently, using matrices and then Gaussian elimination, I come up with f(x) = Ax^2 + Bx + C and use the derivative of that fit to come up with the ...
0
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0answers
55 views

Matching First and Second Derivatives: Taylor Series

I have $$ f(x) = \sqrt{\ln\left(a\cosh^{2}(mx)(1+bx^{2})\right)} $$ If I expand this as a series I should get something of the form $$ \sqrt{\ln a}+gx^{2}+\mathcal{O}(x^{4}) $$ but I'm having ...
0
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0answers
57 views

Approximation for the logarithm of a summatory

I would like to find an approximation for: $$ \log \left(\sum_{i=1}^{N} a_i\exp(-b_i^2)\exp(-c_i^2)\right) $$ with $$ a_i = \frac{1}{\sqrt{(e^2 + e_i^2)(g^2 + g_i^2)}} \\ b_i = \frac{b-d_i}{2(e^2 + ...
1
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0answers
61 views

Does a complex polynomial can be approximated by a Euler-type exponential function?

Can one uniformly approximate a complex polynomial with a Euler-type exponential function? I mean: let $E\subset\Bbb C$ be a compact set with the connected complement $\Omega:= \bar {\Bbb C}\setminus ...
0
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1answer
240 views

MatLab: estimate number of iterations

I want automatically estimate iterations number in matlab. Suppose we have for(int i = 1; i < N; i++). It's clear that for-loop prodices ...
1
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0answers
23 views

Copying the Curvature of One Function onto Another: Approximation

I have a polar function $$ r(\theta)=\left(r+\epsilon\right)\cos(\theta)-\sqrt{r^{2}-\left(r+\epsilon\right)^{2}\sin^{2}(\theta)} $$ Is it possible to methodically conjure another polar function ...
2
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0answers
36 views

Using Polars to Approximate a Cartesian line: Approximating an Integral

I have the equation of the lower semicircle of radius $r$ centred at a distance $a+r$ above the x-axis $$ f(x)=r+a-\sqrt{r^{2}-x^{2}} $$ which I can approximate (for small $x$) as $$ f(x)\approx a+\...
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2answers
28 views

Divide $n$ seats by a list of $\mathbf{w}$ weights proportionally.

I have $n$ number of seats and I have list of weights $\mathbf{w} \in \mathbb{R}^{k}$ which is a probability distribution with $k$ possible values, $\sum_{i=1}^k{w_i}=1$. I want to divide the $n$ ...
0
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2answers
46 views

How to use $t(29/\sqrt{2})<0$ where $t(x)=x^2-41x+420$ to prove that $41/29<\sqrt{2}<42/29$??

So I was investigating different ways to approximate $\sqrt{2}$. Here's my latest: $$Let:t(x)=x^2-41x+420$$ then the roots of $t(x)$ are $20$ and $21$. I showed that then $t(x)=(x-20)(x-21)$ and ...
4
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2answers
184 views

Evaluate $\sum_0^\infty \frac{1}{n^n}$

Courtesy of this xkcd comic I now know that $$ \sum_{n=1}^\infty \frac{1}{n^n} \approx \ln^e(3) $$ Echoing the views of the comic itself, if I ever find myself taking $\ln^e(x)$ then something has ...
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0answers
48 views

How to evaluate the derivate of a hypergeometric function w.r.t. one of its parameters?

I have to numerically evaluate the derivative of the hypergeometric function w.r.t. its first and second parameters $\large\frac{\partial}{\partial a}{_2F_1}\left(a , b ,c;z\right)$ and $\large\...
0
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1answer
40 views

Approximations to the Roots of a Function

I want to find approximations to the root of a function in two variables using the Newton-Raphson method. I can use the method on a function in a single variable but I'm lost as to how you can use it ...
3
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1answer
855 views

Weierstrass Approximation Theorem for $\Bbb C$

The Weierstrass approximation theorem states that any continuous function $ f : I \rightarrow \Bbb R $ on a closed, bounded, connected subset $ I \subseteq \Bbb R $ can be uniformly approximated by ...
31
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4answers
2k views

Geometric explanation of $\sqrt 2 + \sqrt 3 \approx \pi$

Just curious, is there a geometry picture explanation to show that $\sqrt 2 + \sqrt 3 $ is close to $ \pi $?
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4answers
139 views

Approximation to $\sqrt{\cos(\theta)}$?

I have this formula, (it is just the law of cosines angle formula): $$ d = \sqrt{a^2 + b^2 - 2ab \ cos(\theta)} $$ Here is my issue. I am wondering if there is a way to 'extract' the $cos$ term. My ...
1
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2answers
49 views

Approximating 'big' ratio with 'small' ratio

Given a ratio $ \frac{m}{n}, p \in N, q \in N $ where either $m$ or $n$ (or both) is a very big number, how can we find a ratio $ \frac{p}{q}, p \in N, q \in N $ which estimates $ \frac{m}{n} $ up to $...
3
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1answer
557 views

Why is Simpson's rule exact for cubics?

I can't understand: Why is Simpson's rule exact for cubic polynomials?
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5answers
9k views

Approximating Logs and Antilogs by hand

I have read through questions like Calculate logarithms by hand and and a section of the Feynman Lecture series which talks about calculation of logarithms. I have recognized neither of them useful ...
2
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0answers
44 views

Time Discretization

I wonder why we work with constant discretization in Time Discretization of numerical approximation for numerical scheme if we take not necessarily constant Discretization Is the numerical scheme (...
1
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0answers
75 views

Counterexample to smooth approximation of sobolev function on closure of set without $C^1$ boundary

I'm working through the following problem, and I just need a hint to finish it I think. Consider the set $\Omega = B(0,1) \backslash \left\{x\in \mathbb{R}^N : x_N = 0 \right\}$. We are given the ...
0
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1answer
67 views

Absolute error computation

Given is a function $f(x)=4x^{2}$, which we want to evaluate for $x\in \left [ 1,2 \right ]$, $\widetilde{x}\in \left [ 1,2 \right ]$ is the approximation of $x$. What can be the value of the ...
0
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1answer
50 views

What is a good approximation of $(1-p)(1-q)$ as $(1-x)^2$, for $p,q \in (0,1)$?

I'm doing some scientific modeling, and I want to use $(1-x)^2$ to approximate $(1-p)(1-q)$, with $p, q \in (0,1)$. $p$ and $q$ are probabilities, and are not near zero. My intuition is that since I'...
2
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2answers
1k views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
4
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1answer
482 views

Approximation of indicator function of an open set by continuous functions

Let $(X,\mathcal{T})$ be a locally compact separable Hausdorff space and $A \in \mathcal{T}$ open. Does there exist a sequence $(f_n)_{n \in \mathbb{N}}$ of (bounded) continuous functions such that $$...
3
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2answers
917 views

Approximating Trig Functions with Polynomials

I was thinking about the graphs of different trig functions and noticed that most of them are of a similar shape to some different types of polynomials. For example: Higher degree polynomials create ...
2
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4answers
78 views

Approximating $\ln{\frac{x-y}{x+y}}$

For $x>>y$, $$\ln{\frac{x-y}{x+y}} = \ln{\left[ x\left( 1-y/x \right) \right]} -\ln{\left[ x\left( 1+y/x \right) \right]} \approx -2\frac{y}{x}$$ However, the following does not work: $$\ln{\...
0
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1answer
52 views

A power approximation function

I am trying to construct a function that would approximate $a^b$ using Maclaurin series. Here are my reasoning: Since $$a^b=e^{b\ln a}$$ and $$e^x=\sum^{\infty}_{k=0} \frac{x^k}{k!}$$ it should ...
0
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1answer
32 views

Show $\int_{M}^{\infty}\left|\frac{e^{-x^a}\cos(Cx)}{x^{b+1}}\right|dx<\frac{1}{aM^{a+b}e^{M^a}}$?

is there a way to show the inequality \begin{equation}\int_{M}^{\infty}\left|\frac{e^{-x^a}\cos(Cx)}{x^{b+1}}\right|dx<\frac{1}{aM^{a+b}e^{M^a}}\end{equation} for positive constants $M$ and $C$ ...
3
votes
0answers
487 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 $$ I(\epsilon)=P\int_{-1}^{\epsilon^2}\frac{\epsilon}{x\sqrt{(\epsilon^2-x)(1-x)}}...
2
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1answer
191 views

Stone-Weierstrass theorem with $p(1/x)$

I am trying to prove that for a continuous $f\colon[1,\infty)\rightarrow\mathbb R$ and $f(x)\to a$ as $x\to\infty$ it could be approximated by $g(x)=p(1/x)$ where $p$ is a polynomial.
1
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0answers
564 views

Sharpness of the upper bound $(1-x)^n \leq 1 + \frac{nx}{2}$

Here is a known inequality: $$(1-x)^n\leq 1+\frac{nx}{2}\qquad \text{for} \, \frac 1n\leq x\leq 1 $$ I am wondering if there is a better upper bound than this? Thank you.
9
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0answers
219 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 ...
1
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0answers
60 views

Approximating an IVP

I wish to solve the IVP: \begin{align} x(0) =& -1 \\ x' =& 1 + x^2 - t^3 \end{align} With a fourth order taylor series method, I solved the ODE on the interval [0, 2] and then made the ...