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
128 views

Counting Weak Compositions and Approximating Alternating Sum

I have the following problem: "Suppose you have a universe of $N$ distinct objects, and you observe $k$ of them, possibly with repetition. The order in which the objects are observed does not matter. ...
3
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1answer
224 views

Laplace's method with unknown exponent.

Given the integral: $$I = \int_0^a{e^{-\lambda g(x)}f(x)dx}$$ Where $g(x)$ and $f(x$) are both low order positive polynomials, and $\lambda \gg 1$, Laplace's method is commonly used to approximate ...
4
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1answer
561 views

Why does relative error give number of correct digits?

I learnt that if the relative error is 5*$10^{-s}$ then the number of correct digits the result has $s$. Why is this so? Can you illustrate with an example and/or a proof? Another way to put it ...
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0answers
60 views

Riemann Siegel formula modification?

$$Z(t)= \sum_{n=1}^{\lfloor\sqrt t/2\pi\rfloor }\frac{\cos(N(t)-t\log p_{n})}{\sqrt n}$$ here $N(t)$ is the smooth part of the zeros and $ p_{n} $ are the primes since $ p_{n} =n\log n $ then $ \log ...
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6answers
357 views

Approximating $\pi$ with least digits

Do you a digit efficient way to approximate $\pi$? I mean representing many digits of $\pi$ using only a few numeric digits and some sort of equation. Maybe mathematical operations also count as ...
3
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1answer
117 views

probability involving matching of discrete shapes on a square grid

Figure F exists on a regular square grid. T transforms F by any combination of horizontal or vertical reflection as well as rotation by 90 or 180 degrees. A larger background grid of X by Y contains ...
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2answers
95 views

Approximate (but as accurate as it can) location of sound

I quess that this is relatively easy question, but I have been struggling this for a two days now (basicly investigating different formulas) and couldn't find a solution. So, let's think this case: ...
2
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1answer
178 views

Polynomial Approximation of an Integral

I require a polynomial $p(x)$ such that $$\left|p(x) - \int_0^x \cos{(t^2)} dt\right| < \frac{1}{10!}$$ for all $x \in [-1, 1]$. I know that I should probably use the fact that if $$m\leq f^{n+1} ...
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0answers
66 views

Approximating number of nodes expanded by A* search

When searching over a graph expressed as a uniform, 8-connected grid using the A* algorithm, is there any way to give a rough approximate of the number nodes expanded? I appreciate this is a somewhat ...
0
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1answer
123 views

Approximation of function on interval

I'm looking for an accurate but as simple as possible approximation of $S(x,\lambda) = \frac{1}{(1-x) [x-(1-\lambda )]}\left((1+\lambda ) \left(\frac{x(1+\lambda)}{1-\lambda ...
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3answers
2k views

Smooth approximation of absolute value inequalities

Is there an analytic approximation to the inequality: $$\sum_{i=1}^{n} |x_i| \leq \delta ? $$ I would like to replace the above inequality with a smooth inequality that is "valid" in the sense that ...
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4answers
497 views

Approximate solution for the root of a non-linear function

I have been working with a system which involves computing the roots of functions that look like \begin{equation} e^t (g\cos(\omega t) + b) = c \end{equation} where $t$ is the independent variable ...
2
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1answer
1k views

Approximations for the partial sums of exponential series

Though the question here (Partial sums of exponential series - Stack Exchange) is similar, it is more specialized and I rather need a general approximation for an arbitrary partial sum. Essentially, ...
2
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3answers
784 views

How to calculate the area of bizarre shapes

I'm looking for an algorithm to calculate the area of various shapes (created out of basic shapes such as circles, rectangles, etc...). There are various possibilities such as the area of 2 circles, 1 ...
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0answers
71 views

How can I compare two approximants to a bivariate function?

An application I'm working on has required me to find simple approximations to a rather complicated bivariate function $g(x,y)$ that also takes a long time to evaluate on the computer. Through sheer ...
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1answer
502 views

Significant figures with a plus-minus

A question on my homework asks me to give the amount of significant figures of $2900±100$. Would this be one, two, or both?
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1answer
50 views

Approximation when $|a(t)|\ll b$

If $|a(t)|\ll b$ is it alright to take $a\left({a\cdot \dot{a} \over b^2}\right)$ as $0$? Would the following argument make sense? I know that we can take $\left({a\cdot a \over b^2}\right)$ as $0$ ...
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1answer
976 views

Approximating Coins Flips Problem

Approximate the probability of getting 500 heads out of a 1000 coin flip of unbiased coins to be within 5% of its true value (without the use of a calculator). I know that an exact probability ...
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4answers
338 views

What is the correct value of $\pi$

I have seen that: $\pi = 22/7$ $\pi = 3.14\ldots$ $\pi = 17 - \sqrt{192}$. $22/7 \gt \pi$ $22/7 \lt\pi$ My brain storming doubt was, is A = B? Is B = C? Is C= A = B? how D and E are correct or ...
3
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3answers
258 views

How to approximate $\sum_{k=1}^n k!$ using Stirling's formula?

How to find summation of the first $n$ factorials, $$1! + 2! + \cdots + n!$$ I know there's no direct formula, but how can it be estimated using Stirling's formula? Another question : Why can't ...
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1answer
150 views

errors, random and approximations

Hello good evening all! If a reading is reported as R = 200.045 + 0.001 or 200.045 - 0.001 Ohm. Does +0.001 or -0.001 Ohm represents a systematic or random error? Thanking you.
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2answers
609 views

Approximate $\int_a^b \frac{1}{\sqrt{2 \pi \sigma^2}}e^{-(x-\mu)^2/2 \sigma^2}\log(1+e^{-x}) \ \ dx $

I am trying to find an approximation to $$ I = \int_a^b \frac{1}{\sqrt{2 \pi \sigma^2}}e^{-(x-\mu)^2/2 \sigma^2}\log(1+e^{-x}) \ \ dx. $$ My attempt is as follows: $$ \begin{align} I &= \int_a^b ...
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1answer
235 views

calculate surface normal with random sampling of points

Given a surface in $R^3$ and a point P on the surface, I want to calculate the surface normal in this point, the vector that is perpendicular to the surface. However, I do not know the whole surface, ...
2
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1answer
67 views

Trying to rederive an exponential approximation

So I was reading a paper where the following approximation was made. Note that $p$ is small, $L$ is large, and $pL$ is $O(1)$: ...
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0answers
85 views

integral with bessel function represented as a series [duplicate]

Possible Duplicate: prove equality with integral and series This integral was my homework question with $p=2$ and $n=1$. I am wondering if one can get the general formula for p, or at least ...
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2answers
103 views

Limit of exponentials

Why is $n^n (n+m)^{-{\left(n+m\over 2\right)}}(n-m)^{-{\left(n-m\over 2\right)}}$ asymptotically equal to $\exp\left(-{m^2\over 2n}\right)$ as $n,m\to \infty$?
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1answer
435 views

Using binomial theorem find general formula for the coefficients

Using binomial thaorem (http://en.wikipedia.org/wiki/Binomial_theorem) find the general formula for the coefficients of the expantion: $$ ...
2
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1answer
805 views

approximation hypergeometric distribution with binomial

Let $X$ be $\rm{Hypergeometric}(2n,\ell,n)$ and $E(X)=\frac{1}{2} \ell=:\mu$. Is it possible and how to approximate the $q$-th central moment $E(X-\mu)^q$ of the hypergeometric distribution by the ...
2
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2answers
95 views

Queries regarding saddle point

I was reading this article in wikipedia related to saddle points. When I came across this line In one dimension, a saddle point is a point which is both a stationary point and a point of ...
3
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1answer
91 views

is the approximation of the sum true?

Someone commented under my question Calculation of the moments using Hypergeometric distribution that $$ \sum_{k=0}^l\frac{{l \choose k}{2n-l \choose n-k}(2k-l)^q}{{2n\choose n}}\sim \sum_{k=0}^l ...
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0answers
185 views

Orthogonal basis for waveform expansion

I have many signals where each signal has a different waveform f(x). One example of such a waveform could be this f(x) sampled at 11 x positions: I am looking for a basis, Bi, for a series ...
2
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1answer
160 views

Need help understanding Chebyshev approximation

I have a function $f(x)$ sampled at 11 $x$ positions: I want to approximate the function by a Chebyshev expansion: $$ \ f(x) \simeq \sum\limits_{i=0}^m c_i T_i(y) - \frac{1}{2}c_0,\qquad ...
5
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1answer
164 views

Why does $\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$

I read on Wikipedia that $$\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$$ to a high degree of accuracy. Why is this true? Is this pure coincidence or is there some mathematical ...
3
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1answer
99 views

A problem on $C^k$

Can anyone help me solve the following problem ? Let $a,k >0$, $a$ is real and $k$ is integer. Consider the set $S$ of all function $f\in C^k([0,a])$ such that 1) $f(0)=0$ and $f(a)=1$ 2) ...
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2answers
136 views

Approximation by $C^1$ path of a Lipschitz continuous path

I was wondering if the following equality holds: $$\inf\left\{\int_0^1 G(\gamma(t))|\gamma'(t)|dt, \gamma \in X \cap (\text{Lipschitz})\right\}\stackrel{??}{=}\inf\left\{ \int_0^1 ...
0
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1answer
25 views

desintegration constant

I have this system in maple and I can't get it work write: sist:=diff(x(t), t) =-k*x(t), x(0)=x0; Can someone help me get the desintegration constant k if the half-time is t1/2=700*10^6 Thanks
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1answer
84 views

Finding the asymptotic limit of an integral.

I'm having trouble finding the asymptotic of the integral $$ \int^{1}_{0} \ln^\lambda \frac{1}{x} dx$$ as $\lambda \rightarrow + \infty$. Can anyone help? Thank you!
3
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1answer
240 views

Remainder term for Gauss-Laguerre quadrature

I need to calculate a quadrature rule with maximum degree of accuracy that looks like this: $$ \int_0^\infty e^{-x}f(x)dx = \sum_{i=0}^n A_if(x_i) + R_n(f) $$ where $n=2$. For $R_n(f)$ I have this ...
0
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1answer
92 views

Expansions of Hermite functions

I am wondering if someone knows good references. I am looking for expansions of Hermite functions, which gives connections between rates of decay and smoothness of coefficients. Thank you for your ...
6
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1answer
238 views

Integral with Bessel function

Let $n$ be half an odd integer, say $n=k+1/2, k \in \mathbb{N}$. Let $q\geq 1$. I would like to calculate (or approximate) the following integral: $$ \int_0^{\infty}\left(\sqrt{\frac{\pi}{2}}\cdot ...
0
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1answer
47 views

Finding coefficients of a polynomial

So, I have to approximate $f(x) = cos(x)$ using a second-degree polynomial. $$ P(x)=\sum_{i=0}^2 c_i \pi_i(x) $$ $\pi_i$ is the Laguerre polynomial. My professor instructed me that I can use the ...
2
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3answers
119 views

Big-$\mathcal{O}$ bounding of sums of logarithmic functions

I am reading a text which states that $$\sum \limits_{n \leq X} \left(\log X - \log n \right) = \mathcal{O}(X)$$ I can't quite see why this is true, though I can certainly believe it. Could anyone ...
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2answers
494 views

Finding the second-degree polynomial that is the best approximation for cos(x)

So, I need to find the second-degree polynomial that is the best approximation for $f(x) = cos(x)$ in $L^2_w[a, b]$, where $w(x) = e^{-x}$, $a=0$, $b=\infty$. "Best approximation" for f is a function ...
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1answer
79 views

Convincing Proof of Measure Theory Approximation

There is a standard means of approximating a bounded nonnegative function from below in a measure theoretic setting, which is $$f_n=2^{-n}\lfloor{2^nf}\rfloor\wedge ...
0
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1answer
40 views

Find value of a variable from measured data

I have a measurement from which I want to deduct the value of a physical size (velocity). The theoretical equation is $$ A\frac{(b+vt)^2}{(c-vt)^2} $$ Where $A$, $b$ and $c$ are all known sizes, $t$ ...
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0answers
125 views

Worst-case error related to Cramer-Rao bound

I would like to understand the relation (if any) between the Cramer-Rao Lower Bound of estimation theory and the following simple definition of "reconstruction accuracy" which doesn't use any ...
1
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1answer
110 views

Is there a typo in Calculus:Early Transcedentals?

I just finished doing my homework on Local Linear Approximations in 3-space (Ch.13.4). In one of the problems the answer I got is different from the answer key. Problem 39. We have a function ...
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5answers
4k views

How do you calculate the decimal expansion of an irrational number?

Just curious, how do you calculate an irrational number? Take $\pi$ for example. Computers have calculated $\pi$ to the millionth digit and beyond. What formula/method do they use to figure this out? ...
11
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1answer
279 views

Why is $10\frac{\exp(\pi)-\log 3}{\log 2}$ almost an integer?

I read that $$10\frac{\exp(\pi)-\log 3}{\log 2} =318.000000033252\dots \approx 318$$ Is this simply a coincidence or can this somehow be explained?
3
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1answer
387 views

Surprising approximation of weighted sum of binomial coefficients

The following sum appeared in connection to the problem addition of angular momentum in physics: $$ \frac{1}{2^{n+3}}\sum_{k=0}^n \left(\frac{n-2k-1}{\sqrt{k+1}}+\frac{n-2k+1}{\sqrt{n-k+1}}\right)^2 ...