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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56 views

Three almost-integers of the form $ce^{H_a+H_b}\approx 2^k\pm1$

The approximation $$H_n\approx log(2n+1)$$ http://math.stackexchange.com/a/1602945/134791 suggests that the harmonic number for composite odd numbers might be close to the sum of the harmonic ...
2
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0answers
64 views

Integral of a Gaussian with Trigonometric functions Involved

I am having a difficult time evaluating an integral unlike any integral I have seen before. To get right into things here is the integral: $$\frac{A}{\sigma_o\sqrt{2\pi}}\int_{-\infty}^\infty ...
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0answers
27 views

Cubic curve as approximation of Euler spirals?

I was reading the wiki article about Euler spirals and I reached this passage: Unaware of the solution of the geometry by Leonhard Euler, Rankine cited the cubic curve (a polynomial curve of ...
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1answer
30 views

How many “seeks” are there in these binary sequences?

Consider a set of $k \geq 1$ random, IID binary sequences of length $n$, denoted $S_i,\;i = 1\ldots k$, and a "master sequence", also of length $n$, and denoted $S_M$ (see figure for $k = 4$).   ...
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1answer
83 views

Can this integral be evaluated/approximated?

I've been trying to evaluate this integral without much success: $\displaystyle \int_{-\infty}^\infty dx\, e^{iax} \frac{1- e^{-c\sinh^2 bx}}{\sinh^2 bx}$ I've tried contour integration. There are no ...
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111 views

How to verfy if the approximations of the complex error function have no poles?

I found an article published few days ago in arXiv:1601.01261 that shows a very simple Matlab code for computation of the complex error function (aka the Faddeeva function) defined as \begin{equation} ...
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23 views

Galerkin and Ritz method

I know Galerkin method gives us the best approximation. But i couldn't understand the difference between this method and Ritz method for approximation. Do not give us the same results? What is the ...
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75 views

Series for Stieltjes constants from $\gamma= \sum_{n=1}^\infty \left(\frac{2}{n}-\sum_{j=n(n-1)+1}^{n(n+1)} \frac{1}{j}\right)$

Euler's constant has the following representations (Euler-Mascheroni constant expression, further simplification, http://math.stackexchange.com/a/129808/134791, Question on Macys formula for ...
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1answer
317 views

Why is $e$ close to $H_8$, closer to $H_8\left(1+\frac{1}{80^2}\right)$ and even closer to $\gamma+log\left(\frac{17}{2}\right) +\frac{1}{10^3}$?

The eighth harmonic number happens to be close to $e$. $$e\approx2.71(8)$$ $$H_8=\sum_{k=1}^8 \frac{1}{k}=\frac{761}{280}\approx2.71(7)$$ This leads to the almost-integer ...
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14 views

looking for approximation/expansion of $f(t)=a(t)/\sqrt{b(t) + \epsilon(t)}$ with $\epsilon(t) << b(t)$

I have the following function $ f(t) = \frac{a(t)}{\sqrt{b(t) + \epsilon(t)}} $ defined for $t\geq 0$. I know that $a(t) > 0$, $b(t) > 0$, $\epsilon(t) \geq 0$ and $\epsilon(t) << b(t) $ ...
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45 views

How can I use Stirlings inequality to prove this inequality?

Let $p,k$ be natural numbers with $p\ge k$, show that $$ \frac{(p-k)!}{(p+k)!} \le \frac{1}{p^{2k}} \left(\frac{e}{2} \right)^{2k^2/p}. $$ The text where I come across this says to use Stirling's ...
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252 views

Rational series representation of $e^\pi$

This question is related to Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$? by Tito Piezas III. Andrew Fraker (2014) found an almost-integer which is equivalent to the following ...
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2answers
34 views

Approximation of $2$nd Derivative Up to $O(h^4)$

Investigate if it is possible to obtain 4th order accuracy using 5 points for a 2nd derivative approximation, i.e. is it possible to determine a, b, c, d, e in $$y''(0) = ...
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23 views

Aproximation (Proof) [duplicate]

Can you help me? If $f$ and $f'$ are continuous at $[a,b]$ (where $a,b\mathbb{\in R}$), then $\forall\epsilon>0$ exists a polynomial $p$ such that $\left\Vert f-p\right\Vert _{\infty}\leq\epsilon$ ...
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65 views

integral involving error function (erf)

Does anybody know if a closed form of this integral exist? $\int \mbox{erf}(x) \ln(\mbox{erf}(x)) \Bbb dx$ where erf is so called error function. In case there is no closed form solution. Is it ...
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0answers
56 views

Why is $\sum{\frac{1}{(j/2)!}(\xi)^j}\approx\sum{\frac{1}{j!}(\xi)^{2j}}$?

Why is $\sum{\frac{1}{(j/2)!}(\xi)^j}\approx\sum{\frac{1}{j!}(\xi)^{2j}}$? This equation is used for solving the Schrödinger equation of the harmonic oscillator at one point in Griffiths, ...
0
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2answers
49 views

Prove that if $2a^2 - b^2 = \pm1$ then $\frac ba \approx \sqrt2$ [closed]

Prove that if $2a^2 - b^2 = \pm1$ then $\frac ba\approx\sqrt2 $ (a,b) (1,1),(2,3),(5,7),(12,17),(29,41),(70,99)....
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2answers
59 views

How to estimate ln(1.1) using quadratic approximation?

So the general idea for quadratic approximation is assuming there a function $Q(x)$ we want to estimate near $a$: $Q_a(a) = f(a)$ $Q_a'(a) = f '(a)$ $Q_a''(a) = f ''(a)$ But then how do you ...
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1answer
13 views

Derive a second order difference approximation

Derive a second order difference approximation to $y(a)$ using the values $$y(a + h/2), \space y(a + h) \space,\space y(a + 2h)$$ Verify the order of your approximation. Have no idea how to tackle ...
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0answers
32 views

Approximately identical distributions

Suppose we have two random variables $x,y$ in $\mathbb{R}^n$. Assume that for some scalar $\epsilon>0$, for any set $S\subset\mathbb{R}^n$ there exists sets $S_1\supset S,S_2\supset S$ such that ...
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1answer
497 views

Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$?

This was inspired by this post. Let $q = e^{2\pi\,i\tau}$. Then, $$x := \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q} - 24 + 276q - 2048q^2 + 11202q^3 - 49152q^4+ \cdots\tag1$$ ...
0
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1answer
26 views

Need help in understanding appoximation for heavy hitters problem

I am reading a paper and do not understand the following "We allow the space used by a solution to grow as $1/ \epsilon $, so as $ \epsilon ↓ 0$ the space blows up..." I do not understand the ...
2
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1answer
102 views

Error Bounded Cubic B-Splines with fewest segments

I have some odd constraints in my project. Suppose we want to use Cubic BSplines to approximate a set of Points. There is two Constraints: error value should be an input to the algorithm (error ...
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1answer
21 views

Asymptotic approximation to radial wave function

This seemingly easy analysis is driving me up the wall. $$\frac{\text{d}^{2}u}{\text{d}\rho^{2}}=\left[1-\frac{\rho_{0}}{\rho}+\frac{l(l+1)}{\rho^{2}}\right]u$$ why is it for $$\rho\rightarrow0,$$ ...
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0answers
31 views

Softmax for Continuous Functions?

The softmax $\log \sum_{i=1}^n \exp(f_i)$ of vector $f$ is a smooth upper bound on $\max_i f_i$. However, the same cannot be said of $\log \int_{X} \exp(f(x))dx$ in relation to $\max_{x \in X} f(x)$ ...
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2answers
31 views

Approximation of $\frac{x}{\sqrt{x^2+R^2}}$

How do you prove this statement? If $x\gg R$ then $$\frac{x}{\sqrt{x^2+R^2}}\cong 1-\frac{1}{2}\left(\frac{R}{x}\right)^2$$ I have no ideas even how to start.
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1answer
43 views

Method for finding square roots quickly (manual)

I was recently studying AC circuits and there I need to use Pythagoras theorem a lot.So I was looking for a method with which square roots can be calculated very fast,manually up to 1 decimal ...
2
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1answer
63 views

Find a>1 s.t. $a^x = x$ has a unique solution

What $a$ makes $\{x\mid a^x = x\}$ a singleton? $$(1.4444)^x - x \le 0 \tag 1$$ has real solutions. $$(1.4447)^x - x \le 0 \tag 2$$ has no real solutions. I guess $1.4444 < a < 1.4447$ I ...
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0answers
29 views

Asymptotic behaviour of generalized binomial coefficient $\frac{an(an-1)…(an-n+1)}{n!}$

Let $a\in(0,1)$. What is the asymptotic behaviour of $\frac{an(an-1)...(an-n+1)}{n!}$ as $n\rightarrow\infty$? It looks like it might be possible to express this in terms of gamma functions and use ...
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1answer
53 views

Order of the error for the Trapezoidal and Simpson's method of numerical integration

What are the order of the error for the Trapezoidal and Simpson's method of numerical integration? What is the definition of order of the error of a quadrature formula? Is it true that order of ...
3
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1answer
28 views

Approximation formula on a surface

[Beginning calculus question.] We can get an approximation to the value of a function of two variables, I think, by saying $$ f(a+\Delta x , b+ \Delta y) \approx f(a,b) + f_x(a,b)\Delta x ...
2
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1answer
38 views

Approximating logarithmic series

I'm trying to solve the below recurrence $$T(n) = 2 T(n/2) + n/\log n$$ I referenced some online articles they all are approximating $n \sum_{i=0}^{h-1} \frac{1}{\log n - i}$ to $n ...
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2answers
73 views

Approximating a Harmonic Sum

The infinite sum $\sum_{n=1}^{\infty}\frac{1}{n}$ diverges. However, it is possible to find bounds from some $n$ to another integer $n$. Wolfram alpha is able to give a decimal approximation of the ...
2
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2answers
214 views

Approximation of $\pi$ using Brahmagupta's Identity

Brahmagupta, an ancient Indian Mathematician, gave an pretty efficient algorithm for finding integer solutions to the famous Pell's Equation, far before Fermat propounded this before the European ...
2
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1answer
35 views

Finding $w_1,w_2,b$ such that $w_2 \sigma(w_1 x + b) \approx x$

Let $\sigma(z) = 1/(1+e^{-z})$. How can I find $w_1, w_2, b$ such that $w_2 \sigma(w_1 x + b) \approx x$ for $x \in [0,1]$? The hint provided was to rewrite $x = \frac{1}{2}+\Delta$, assume $w_1$ is ...
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1answer
51 views

Probability that sum of two random variables is $1$

Let $X$ and $Y$ be two independent random variables with density functions $f_x(x) = x\exp\left(-\frac{1}{2}x^2\right) \mathbb{}$ where $x \in \mathbb{R}^+$ and $f_y(y) = \frac{1}{2}$ with $y \in ...
2
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0answers
35 views

Meaure-theoretic induction: Why dyadic approximation?

In measure-theoretic induction proofs we always use the dyadic approximation of a non-negative measurable function $Y$ as $$Y_n = \sum_{k=0}^{n2^n-1} k/2^n 1\left(\frac{k}{2^n} \leq Y < ...
1
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1answer
46 views

How are polynomials graphs approximated?

Say I have the data: $x=[ 1, 2, 3.3, 4, 5.5, 8, 9, 10.2, 11, 45 ]$ $y=[ 9,27,64,91,164,330,462,540,630,10218]$ The data is subjective though. How would one approximate a valid polynomial for this ...
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4answers
637 views

How can I calculate or at least approximate the sum?

As a part of a complexity analysis of the algorithm, I have to calculate the following sum: $$n^{1/2} + n^{3/4} + n^{7/8} + ...$$ where in total I have $k$ elements to sum: ...
0
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1answer
29 views

error bound for polynomial interpolation with derivative matching

We all know the following formula for the maximum error (evenly spaced) polynomial interpolation: $|f(x) - p_n(x)| \leq \frac{h^{n+1}}{4(n+1)} \max_{x\in [a,b]} f^{(n+1)}(x)$ where $p_n(x)$ is the ...
2
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1answer
53 views

Ways to generate triangle wave function.

I recently when searching for parameters on a unit cube in $\mathbb{R}^9$ (we all have our more or less peculiar hobbies, don't we?) found a practical reason to implement a triangle wave function ...
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0answers
30 views

Normalization in least-p'th minimax algorithm

In the book "Practical Optimization: Algorithms and Engineering Applications", the least-$p$th minimax algorithm is presented, for approximation of the minimax optimizer (Alg. 8.1): $Loss_x(k)$ = ...
2
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3answers
80 views

Bounds on Gaussian infinite sum

What are some good upper and lower bounds on the following sum? $$S=\sum_{n=-\infty}^{+\infty}\dfrac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{1}{2}\left(\frac{n}{\sigma}\right)^2}$$ I am looking for ...
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3answers
484 views

After 6n roll of dice, what is the probability each face was rolled exactly n times?

This is closely related to the question "If you toss an even number of coins, what is the probability of 50% head and 50% tail?", but for dice with 6 possible results instead of coins (with 2 possible ...
2
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1answer
68 views

Exponential series approximation

I have a series of the following form: \begin{align} \sum_{k=2}^\infty \left( 1 - e^{-ns^{k-1}} \right)^k \end{align} where $0<s<1$. I would like to compute an approximation of this series, for ...
2
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3answers
66 views

Approximation by using Taylor Polynomials - why?

Could anyone tell me why would I want to approximate a function $f$ by using its Taylor expansion (is it the same as saying approximation by Taylor polynomials?), if I have the exact formula of the ...
1
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1answer
51 views

(Ab)using the factorial and gamma functions

I have a product of the following form: $$ P = (N-\alpha)(N-2\alpha)\cdots(N-k\alpha) $$ where $k$ is an integer between $1$ and $N-1$, and $\alpha$ is a real number in $(0,1]$. Clearly, for ...
0
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1answer
26 views

Minimizing sums of values versus minimizing cubes of sums.

I am attempting to find the best path from start to finish from a set of points. Say that one path has costs $x_1,x_2,...,x_n$ and $y_1,y_2,...,y_n$ associated with it. I am attempting to find the ...
1
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2answers
40 views

Complex Equation Formula

Can someone show me how the following two expressions are equivalent: $$\Gamma = \frac{i X - R_c}{i X + R_c} = -e^{-i 2 \mathrm{tan}^{-1} (\frac{X}{R_c})}$$ I'm working through a calculation and I ...
2
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0answers
31 views

Approximation of integrable function by polynomials

Assume $f\in \mathscr{R}(\alpha)$ on $[a,b]$, and prove that there are polynomials $P_n$ such that $$\lim \limits_{n\to \infty}\int_{a}^{b}|f-P_n|^2d\alpha=0.$$ Proof: Let $\varepsilon>0$ be given ...