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
29 views

Is $e^{-r/2}$ equivalent to $r^{-(l+1)}$ in the radial solution of Laplace equation?

When we solve the Laplace equation for Hydrogen Wave Equation at large r, we obtain the expression below to account for the behavior of the wave at very very large $r$ $$R=e^{-(r/2)}$$ At very small ...
0
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2answers
29 views

Approximating a sum of reciprocals

What is a good approximation for the function: $$S_{N,k} = \sum_{i=N}^\infty {\frac{1}{i^k}}$$ when $k$ is a given constant (2, 3 or 4) and $N$ is large? $S_{N,k}$ is a decreasing function of $N$; ...
0
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1answer
36 views

When is right to kill $r^l$ and/or $r^{(-l-1)}$?

When we solve the Laplace equation in spherical polar coordinate, we get the radial part whose solution is: $$R=Ar^l+Br^{-(l+1)}$$ Now, some solutions keep this two terms, but when we derive the ...
1
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1answer
78 views

Simpler derivation to $\pi$ [closed]

I'm an amateur in mathematics, being in 9th grade. I have been trying to derive $\pi$. During this I reached a limit to find the value of $\pi$. $$\lim_{x \to 0} \frac{180\sin x}{x}$$ Where $x$ is in ...
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0answers
16 views

Continuous/interpolating alternative to order of magnitude?

Define $\operatorname{magnitude}\left(x\right) = 10 ^ { \lfloor \log_{10} x \rfloor }$ and $\operatorname{magnitude'}\left(x\right) = 10 ^ { \lfloor \log_{10} x \rceil }$ Currently I'm using this ...
1
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1answer
84 views

Which function to kill: Sine or Cos?

I got an equation which was a solution to some familiar Differential Equation I am solving, the solution takes the form of: $$V=Ce^{-ix}$$ but $$Ce^{-ix}=A\cos(x)+B\sin(x)$$ so ...
3
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1answer
118 views

Calculating ${(0.9)}^{\left(0.6\right)}$ with an approximation of ${10}^{\left(-4\right)}$

I'm having extreme difficulties understanding how to use Lagrange theorem to find an approximation. So far for my series I have: $$(1+(-x))^\frac{3}{5}= ...
0
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0answers
38 views

Sum Calculation: $\sum_{n=1}^\infty \left(1- \frac{\cosh^{-1} n}{\log 2x}\right)$

I was investigating the asymptotic properties of the $\cosh$ functions and how they all strongly relate to $e^x$ In my studying, I found out that $\cosh x\sim \frac{e^x}{2}$ By that definition, that ...
0
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0answers
45 views

What approximations for the Gamma function's inverse appear to work 'best'?

So I was wondering how we approximate the inverse of the Gamma function, where I tried a few methods: Lagrange inversion theorem: $$\Gamma^{-1}(z)=a+\sum_{n=1}^{\infty}\lim_{w\to ...
0
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1answer
267 views

Is there an integral or series for $\frac{\pi}{3}-1-\frac{1}{15\sqrt{2}}$?

The approximation $$\pi\approx\frac{22}{7}=3+\frac{1}{7}$$ suggests that the closest integer to $\frac{1}{\left(\pi-3\right)}$ is $7$. However, $$ \frac{1}{\left(\pi-3\right)^2}\approx49.879 $$ is ...
4
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1answer
44 views

QR(pivot) vs SVD for low rank approximation

Define the low rank problem as finding the approximation of matrix A, B: where we want to minimize rank(B) and we want the 2 norm of the residu of A-B to be less than epsilon. Could someone help me ...
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2answers
42 views

Error when approximating $\int_0^{1}(x^{2}+x)dx$ with midpoint rule.

Task is to define the exact error when approximating $\int_0^{1}(x^{2}+x)dx$ with midpoint rule using n subintervals. I know the error term is $E(f)=\frac{1}{24}(b-a)f^{''}(\varepsilon)h^{2}$ but im ...
0
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2answers
50 views

Bound for $\log { \binom{n}{i}}$?

(1) Are there better (smaller; tighter) bounds for $\log { \binom{n}{i}}$, than $O(n \log n )$? (2) Under what conditions $O(i \log n)$ is a good bound? Clearly this bound should be in a way that it ...
0
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1answer
47 views

Using Lagrange Remainder to find the approximation of $\sqrt(8)

I'm looking for an approximation of $\sqrt 8$ with an approximation of $10^{-4}$. It was given that $\sqrt 8 =3\sqrt\frac{8}{9}$ so I set up a general series for $ \sqrt{1+x}^\frac{1}{2} $ around zero ...
0
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1answer
24 views

Use linear scales for both the X- and Y- axes?

I am trying to understand this question: Use linear scales for both the X- and Y- axes. In which region, ƒ2(x) is a good approximation to ƒ1(x)? My equations are: ...
0
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0answers
27 views

Approximating a Riccati Differential Equation/Differential equation of two variables

I am currently really struggling to solve/approximate this Riccati equation $x'(t, v) = cv(1-x(t)) + \beta x(t)(1-x(t)) - \gamma x(t)$ st. $0 < c, \gamma, x_{0} < 1$ $ 0 \leq v \leq 1$ $ 0 ...
3
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2answers
77 views

Closed form or approximation of $\sum\limits_{i=0}^{n-1}\sum\limits_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)} (i + 2x)(j +2x)$

During the solution of my programming problem I ended up with the following double sum: $$\sum_{i=0}^{n-1}\sum_{j=i + 1}^{n-1} \frac{i + j + 2}{(i + 1)(j+1)}\cdot (i + 2x)(j +2x)$$ where $x$ is some ...
1
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0answers
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
63 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
25 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 ...
2
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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 $k = 4$).   ...
1
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1answer
82 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 ...
0
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0answers
110 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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0answers
22 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 ...
1
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0answers
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 ...
2
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1answer
313 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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0answers
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) $ ...
4
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0answers
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 ...
6
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0answers
250 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 ...
2
votes
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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0answers
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 $\left\Vert f-p\right\Vert _{\infty}\leq\epsilon$ ...
0
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0answers
61 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 ...
1
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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)....
1
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2answers
57 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 ...
0
votes
1answer
12 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 ...
0
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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 ...
14
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1answer
494 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
votes
1answer
54 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 ...
1
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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 $$\rho\rightarrow0,$$ ...
1
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0answers
28 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.
0
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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
votes
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 ...
1
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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 ...
1
vote
1answer
48 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
votes
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
votes
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 ...
1
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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 ...