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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5
votes
1answer
166 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
votes
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) ...
5
votes
2answers
140 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
votes
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
1
vote
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
votes
1answer
250 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
votes
1answer
96 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
votes
1answer
241 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
votes
1answer
48 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
votes
3answers
121 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 ...
1
vote
2answers
506 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 ...
0
votes
1answer
80 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
votes
1answer
41 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, $t$ ...
1
vote
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
vote
1answer
111 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 ...
23
votes
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
votes
1answer
280 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
votes
1answer
414 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 ...
1
vote
2answers
943 views

Help with Chebyshev Economization of $\exp(x)$?

This may be a stupid question, so I apologize in advance if it is. This is a very common example of Chebyshev Economization, but I still do not understand how the coefficients are found. I want to ...
5
votes
1answer
168 views

$C^1$ approximation of a continuous curve.

Suppose I have two points $\alpha,\beta \in \Bbb{R}^n$. Define $$ X=\{\gamma \in C^1([0,1] , \Bbb{R}^n),\ \gamma(0)=\alpha,\gamma(1)=\beta ,0 <|\gamma'|<K\}$$ parametrized curves joining ...
2
votes
2answers
134 views

Estimate the area restricted by $f(x) = \log(x+1)/x \, , f(x-1), \, y=0$, and $y=a$.

I need to estimate the area between the functions $f(x) = \log(x+1)/x \, , f(x-1), \, y=0$, and $y=a$. where $a>1$. Now I have tried quite a few ways to do this, but nothing comes to mind. I ...
0
votes
0answers
49 views

Steiner Tree Approximation

My question is about a subtlety regarding the $2$-approximation for the Metric Steiner Tree problem. The classical Metric Steiner tree problem: Given a metric space on $n$ points and a subset $S$ ...
3
votes
1answer
112 views

Approximating the logarithm of a Laplace transform

Suppose $X$ is a random variable on $\mathbb R_+$ with finite mean, i.e. $\mathbb E X <+\infty$. Let $F_X(t)$ be its c.d.f. and $\mathcal{L}_X(\cdot)$ its Laplace transform, i.e. ...
1
vote
0answers
77 views

Describe growth of $\epsilon n$

For all $\epsilon$ we have that $f(n)\le \epsilon n$ where n is a natural number. What can we say about the growth of $f(n)$? Clearly $f(n)=O(n)$, can we say anything sharper?
0
votes
1answer
115 views

asymptotic limit of $\int_0^{\infty}\left(1-\frac{t^2}{2(2k+3)}+\frac{t^4}{2\cdot 4\cdot(2k+3)\cdot(2k+5)}\right)^qdt$

Help me please with the following integral. I've asked this question before Asymptotic limit of the integral with polynomial, but it turns out that it was incorrect question. I should get an ...
1
vote
1answer
121 views

Algorithms for finding closed form approximations for integrals (with no closed form solutions)

It is well known that many integrals have no closed form solutions, normally what you would do is solve them numerically. My question is if there are algorithms that give you good closed form ...
2
votes
1answer
77 views

asymptotic limit at the integral

I would like to get an asymptotic limit at the following integral: for $p\ge 2, n \in N$, $t \ge 0$ $$ \int_{0}^{\frac 12 \sqrt{(n+1)!}}\left(1-\frac{t^2}{2^2(n+1)!}\right)^p \mathrm{d} t $$ I think ...
0
votes
1answer
811 views

Newton's Method, and approximating parameters for Bézier curves.

I've been wanting, for quite a while now, to polish up some source code I wrote for approximating arbitrary Bézier curves to given series of points. I managed to accomplish quite a bit, but I hit a ...
0
votes
1answer
269 views

Approximate solution for an exponential equation

Trying to solve this question: Probability of ball ownership I got at an expression for the solution, P: $$\frac{P}{M} = (1 - \frac{1}{M+N})^{N(1-\frac{P}{M})+M}$$ Where M, N are parameters. The ...
1
vote
1answer
520 views

Approximating the logarithm of sum

I would like to approximate $$ \ln(\sum_{k=0}^n(n-2k)^p) $$ Here $p\geq 2$
1
vote
0answers
132 views

Extending Hermite polynomial interpolation

Working with the definition of Hermite polynomials $x_0,\ldots,x_n$ are distinct in $[a, b]$, $f''(x)$ is continuous on [a, b], then $$H_{2n+1}(x)=\sum_{j=0}^{n} [f(x_j)H_{n,j}(x)] +\sum_{j=0}^{n} ...
5
votes
3answers
4k views

Approximating log of factorial

I'm wondering if people had a recommendation for approximating $\log(n!)$. I've been using Stirlings formula, $ (n + \frac{1}{2})\log(n) - n + \frac{1}{2}\log(2\pi) $ but it is not so great for ...
4
votes
1answer
103 views

Proof about binomial coefficient

I today see a approximated equation, when $n \ll u $: $$\log {u \choose n} \approx n \Big(\log \frac{u}{n} + 1.44\Big)$$ I would like to know how to prove it.
6
votes
2answers
440 views

Elementary bound of binomial coefficient

I'm working my way through an Erdős paper from the sixties and some of the elementary bounds he claims seem to be just beyond my reach. The expression looks horrendous but maybe there is a clever ...
3
votes
0answers
84 views

Numerically estimate $a^b$ [duplicate]

Possible Duplicate: How can I calculate non-integer exponents? What is the most efficient way to estimate $a^b$ ($a > 0$) numerically? My goal is not to use built-in math functions (like ...
5
votes
1answer
2k views

Approximation of a bounded measurable function with step functions?

I'm having trouble judging whether this statement is correct: For an arbitrary bounded measurable function $f$ defined on $[0,1]$, $\exists{}\ $a sequence of step functions $\{\phi_n\}$, such that ...
2
votes
2answers
224 views

How many significant figures are needed in base 2?

$x \in \mathbb{R}$ $2^{500}<x<2^{501} $ How many significant figures are needed in base 2, to know in high approximation whether $2^x$ is integer?
1
vote
0answers
293 views

Calculation of a 'double' sum

Let $n \in N$ and $q\geq 2$. I am trying to calculate the following sum: $$ \sum_{i=0}^{\sqrt n/2}\sum_{j= i \sqrt n }^{(i+1)\sqrt n}\frac{(-1)^q2^q(\frac{n}{2}-j)^q}{(n-j)!j!} $$ Any help will be ...
4
votes
1answer
214 views

Evaluating a limit involving binomial coefficients.

If $N_c=\lfloor \frac{1}{2}n\log n+cn\rfloor$ for some integer $n$ and real constant $c$, then how would one go about showing the following identity where $k$ is a fixed integer: $$\lim_{n\rightarrow ...
1
vote
1answer
104 views

Limit of a sum for which the upper limit is also in the argument of the sum - Taylor series of $e^x$

A book I'm reading claims that $\frac{1}{2}(k-1)!\sum \limits_{j=0}^{k-3} \frac{k^j}{j!} \sim (\pi / 8)^{1/2}k^{k-\frac{1}{2}}$ as $k \to \infty$. I can get most of the expression to work out nicely ...
1
vote
1answer
76 views

Bound for sum with geometric progression

Let $n_i$, $i=1,\ldots,m+1$ be nonnegative natural numbers, sum of which $\sum_{i=1}^{m+1}n_i=N$. I woul like to find an upper bound for the following$$ \sum_{i=1}^{m+1}\frac{\sqrt n_i}{2^{i-1}}$$
10
votes
4answers
3k views

Find bound for sum of square roots

Let $a_1,...,a_n$ be real numbers, such that $a_1+...+a_n=A$. What can we say about $\sqrt{a_1}+...+\sqrt{a_n}$? I would like to bound from above thus sum in terms of $A$.
2
votes
1answer
271 views

Best and most efficient way to numerically compute $e$?

There are many well-known methods for efficiently numerically computing $\pi$, such as Chudnovsky's Method or perhaps Gauss-Legendre's algorithm. I was wondering what the best method for computing $e$ ...
4
votes
2answers
383 views

computation of the sum

I am having trouble to compute the following sum: $$ \sum_{k=0}^n(n-2k)^p \frac{{n \choose k}{2m-n \choose m-k}}{{2m \choose m}} $$ Here $p\geq 2$. To simplify the question, we can even assume that ...
4
votes
1answer
1k views

Method for estimating the $n^{th}$ derivative?

When using numerical analysis, I often find that I am required to estimate a derivative (e.g. when using Newton Iteration for finding roots). To estimate the first derivative of a function $f(x)$ at ...
1
vote
1answer
238 views

Approximation of Integration by parts

I'm trying to approximate a integral of the form: $$\int_V{g({\bf x})f({\bf x})} \; d^3x$$ Where the functions $f(\bf{x})$ and $g(\bf{x})$ are positive functions, but where only $f(\bf{x})$ is known ...
2
votes
1answer
317 views

General bound on a polynomial's root with the largest norm

Is there a general bound on a polynomial's root with the largest norm? When Rouche's theorem is used, it still seems that the polynomial's root with the largest norm still needs to be found if we ...
1
vote
0answers
85 views

Approximation or calculation of the probability of getting “clumps” when sampling from a uniform distribution

Suppose that there are $n$ independent samples $X_1,X_2,...,X_n$ sampled from the uniform distribution on $[0,1]$ with the pdf $f(x)=1$. Is there a good way to calculate or approximate the ...
1
vote
2answers
458 views

Smoothing of absolute value and sign functions for numerical integration

I'm doing Numerical integration of ODEs. for a special system that has an always positive coordinate s and a conjugated momentum ...
0
votes
0answers
294 views

Comparing norms of a vector

Let $a$ be a vector in $\mathbb R^m$, such that $\sum_{i=1}^{m}a_i=0.$ I would like to compare $\sqrt{2m(2m−1)}\|a\|_{\infty}$ and $\sqrt{2m}\|a\|_2$, in the case when the vector $a$ satisfies the ...