For questions related to approximations
9
votes
6answers
2k views
Simple numerical methods for calculating the digits of Pi
Are there any simple methods for calculating the digits of Pi? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that ...
5
votes
3answers
343 views
$\lim_{n\to\infty} f(2^n)$ for some very slowly increasing function $f(n)$
I should be able to answer this myself, but feel insecure anyway. I want to know, whether a function f(n) is bounded if n goes to infinity (and if it's bounded, the limit). Heuristically it appears ...
1
vote
0answers
269 views
An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function
I think, here, I found
$$
P_\color{red}x(\color{blue}s)=\sum_{p<\color{red}x} \frac{1}{p^{\color{blue}s}} =\sum_{\color{green}n=1}^{\infty}\frac{ \mu (\color{green}n)}{\color{green}n}
...
35
votes
5answers
2k views
Find the average of $\sin^{100} (x)$ in 5 minutes?
I read this quote attributed to VI Arnold.
"Who can't calculate the average value of the one hundredth power of the sine function within five minutes, doesn't understand mathematics - even if he ...
3
votes
4answers
2k views
Why is $22/7$ a better approximation for $\pi$ than $3.14$?
This seems counterintuitive, but $22/7$ is closer to $\pi$ than $3.14=314/100$ which has a significantly greater denominator.
Why is $22/7$ a better approximation for $\pi$ than $3.14$?
This has ...
20
votes
9answers
2k views
What is the purpose of Stirling's approximation to a factorial?
Stirling approximation to a factorial is
$$
n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n.
$$
I wonder what benefit can be got from it?
From computational perspective (I admit I don't ...
2
votes
4answers
627 views
Approximating $\pi$ using Monte Carlo integration
I need to estimate $\pi$ using the following integration:
$$\int_{0}^{1} \!\sqrt{1-x^2} \ dx$$
using monte carlo
Any help would be greatly appreciated, please note that I'm a student trying to ...
53
votes
3answers
2k views
Is there an integral that proves $\pi > 333/106$?
The following integral,
$$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$
is clearly positive, which proves that $\pi < 22/7$.
Is there a similar integral which proves ...
21
votes
5answers
1k views
How do you calculate the decimal expansion of an irrational number?
Just curious, how do you calculate an irrational number? Take $\pi$ for example. Computers have calculated $\pi$ to the millionth digit and beyond. What formula/method do they use to figure this out? ...
12
votes
5answers
874 views
Approximation of $e^{-x}$
Is there a method to mentally evaluate $e^{-x}$ for $x>0$? Just to have an idea when computing probabilities or anything that is an exponential function of some parameters.
7
votes
6answers
264 views
Approximating $\pi$ with least digits
Do you a digit efficient way to approximate $\pi$? I mean representing many digits of $\pi$ using only a few numeric digits and some sort of equation. Maybe mathematical operations also count as ...
6
votes
1answer
288 views
Tensor products of functions generate dense subspace?
Let $X$ and $Y$ be two spaces in certain category, $F(\cdot)$ a functor associating each space with a function space (with certain topology). Assume that for any $f\in F(X)$ and $g\in F(Y)$, $f\otimes ...
5
votes
2answers
868 views
How can I calculate non-integer exponents?
I can calculate the result of $x^y$ provided that $y \in\mathbb{N}, x \neq 0$ using a simple recursive function:
$$
f(x,y) =
\begin {cases}
1 & y = 0 \\
(x)f(x, y-1) & y > 0
\end ...
3
votes
2answers
3k views
Relation between Simpson's Rule, Trapezoid Rule and Midpoint Rule
I am studying numerical approximation and verifying $S_{2n} = \frac{1}{3}\left(T_n +2 M_n\right)$. ($S_n$ refers to Simpson's Rule approximation, $T_n$ refers to Trapezoid Rule approximation and $M_n$ ...
0
votes
1answer
69 views
How to show that a measurable function on $R^d$ can be approximated by step functions?
In Stein's Book: Real Analysis, Theorem 4.3 says that any measurable function $f$ on $R^d$ can be approximated by step functions. But the proof provided there only show that when $f=\chi_E$, with ...
0
votes
1answer
21 views
Approximation related to resonance
Can someone help me with this problem.
We have $$x(t)=N \sin (w_{0} t)+\frac{w_0}{w_1}e^{\frac{-t}{T}}\sin (w_{1}t)$$
and $w_1=(1+\frac{\delta_1}{N^2})w_0$ for some $|\delta_1|\leq 1$.
I need to ...
0
votes
1answer
99 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 ...
16
votes
5answers
692 views
How best to explain the $\sqrt{2\pi n}$ term in Stirling's?
I recently showed my Algorithms class how to bound $\ln n! = \sum \ln n$ by integrals, thereby obtaining the simple factorial approximation
$$ e \left(\frac{n}{e}\right)^{n} \leq n! \leq ...
10
votes
3answers
556 views
Sine Approximation of Bhaskara
An Indian mathematician, Bhaskara I, gave the following amazing of the sine (I checked the graph and some values, and the approximation is truly impressive.)
$$\sin x \approx \frac{{16x\left( {\pi - ...
17
votes
1answer
362 views
A series problem by Knuth
I came across the following problem, known as Knuth's Series which originally was an American Mathematical Monthly problem.
Prove that $$\sum_{n=1}^\infty ...
8
votes
1answer
229 views
Why is $10\frac{\exp(\pi)-\log 3}{\log 2}$ almost an integer?
I read that
$$10\frac{\exp(\pi)-\log 3}{\log 2} =318.000000033252\dots \approx 318$$
Is this simply a coincidence or can this somehow be explained?
7
votes
2answers
533 views
Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$?
The $\Gamma (x)$ function has just one minimum for $x>0$ . This result uses some
properties of the gamma function:
$\Gamma ^{\prime \prime }(x)>0$ and $\Gamma (x)>0$ for all $x>0$
$\Gamma (1)=\Gamma ...
6
votes
3answers
363 views
How is it that this shape can converge to what looks like a triangle but has a different perimeter?
I had this strange notion some time ago, and I recently wrote a blog post about it, as a mere curiosity. I don't really consider it a "serious" mathematical question; but out of interest, I wondered ...
21
votes
3answers
843 views
The right “weigh” to do integrals
Back in the day, before approximation methods like splines became vogue in my line of work, one way of computing the area under an empirically drawn curve was to painstakingly sketch it on a piece of ...
12
votes
1answer
198 views
Why is integer approximation of a function interesting?
I have recently learnt the following result:
Let $g \in \mathbb{R}[x]$ be a polynomial with $g(0) = 0$. Then, for any $\varepsilon > 0$, the set of positive integers $n$, such that $g(n)$ is ...
9
votes
2answers
402 views
Complex Zeros of $z^2e^z-z$
Can anyone give me a hint on showing (in a relatively elegant way, as I know the answer from WolframAlpha), that the complex valued function $z^2e^z-z$ has at most 2 roots with norm less than 2? ...
4
votes
1answer
148 views
Evaluating a limit involving binomial coefficients.
If $N_c=\lfloor \frac{1}{2}n\log n+cn\rfloor$ for some integer $n$ and real constant $c$, then how would one go about showing the following identity where $k$ is a fixed integer:
$$\lim_{n\rightarrow ...
4
votes
1answer
190 views
Approximating Lambert W for input below 0
As a small part of a much bigger project, I need to be able to approximate the numerical output of the Lambert W function. I have found decent approximations (good up to at least 4 decimal places), ...
4
votes
3answers
1k views
Find formula from values
Is there any "algorithm" or steps to follow to get a formula from a table of values.
Example:
Using this values:
...
3
votes
2answers
314 views
How to bound the truncation error for a Taylor polynomial approximation of $\tan(x)$
I am playing with Taylor series! I want to go beyond the basic text book examples ($\sin(x)$, $\cos(x)$, $\exp(x)$, $\ln(x)$, etc.) and try something different to improve my understanding. So I ...
3
votes
1answer
381 views
Approximation for Lambert W function near zero
I am looking for a good approximation for the $W_0$ branch of the Lambert $W$ function. I am looking for values $0 < x < e$ only, so I expect something simpler than the general Taylor expansion. ...
2
votes
1answer
440 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, ...
10
votes
2answers
725 views
Euler's Approximation of pi.
I recently stumbled across the formula:
$$\pi=20\arctan\frac{1}{7}+8\arctan\frac{3}{79}$$
developed by Euler, for approximating pi. I evaluated it to several thousand decimal places and up to that ...
6
votes
3answers
2k views
How to justify small angle approximation for cosine
Everyone knows the picture that explains instantly the small angle approximation to the sine function (as defined by the parametrisation of the unit circle): "what's the length of that arc?" "See how ...
5
votes
5answers
229 views
What exactly is “approximation”?
There are a lot of great "approximations" that exist in the mathematical field:$$\dfrac{22}{7} \approx \pi$$
$$e \approx \left(1 + \dfrac{1}{n}\right)^n$$But the fact that I have yet to know what ...
5
votes
3answers
1k views
How to simplify or calculate a formula with very big factorials
I'm facing a practical problem where I've calculated a formula that, with the help of some programming, can bring me to my final answer. However, the numbers involved are so big that it takes ages to ...
4
votes
1answer
312 views
Hermite Interpolation of $e^x$. Strange behaviour when increasing the number of derivatives at interpolating points.
I am trying to understand Hermite Interpolation. Here is my pedagogical example.
I want to approximate $f(x)=e^x$ on the domain $[-1,1]$ using Hermite interpolation.
I choose the Chebyshev zeros ...
3
votes
2answers
107 views
Bounds on $ \sum\limits_{n=0}^{\infty }{\frac{a..\left( a+n-1 \right)}{\left( a+b \right)…\left( a+b+n-1 \right)}\frac{{{z}^{n}}}{n!}}$
I have a confluent hypergeometric function as $ _{1}{{F}_{1}}\left( a,a+b,z \right)$ where $z<0$ and $a,b>0$ and integer.
I am interested to find the bounds on the value it can take or an ...
3
votes
2answers
348 views
Modern formula for calculating Riemann Zeta Function [duplicate]
Possible Duplicate:
How to evaluate Riemann Zeta function
I have an amateur interest in the Zeta Function. I have read Edward's book on the topic, which is perhaps a little dated. I would ...
3
votes
1answer
465 views
With Euler's method for differential equations, is it possible to take the limit as $h \to 0$ and get an exact approximation?
I was recently watching a tutorial on Euler's method for approximating differential equations, and the whole time I was thinking "why can't you just take the limit of the step size $h$ as it goes to ...
3
votes
2answers
227 views
Asymptotic number of unlabeled graphs
A rather tight lower bound $c(n)$ of the number of unlabeled digraphs of order $n$ (loops allowed) seems to be
$$c(n) = 2^{n^2}/n!$$
because there are $2^{n^2}$ labeled graphs, almost all of them ...
2
votes
3answers
253 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 ...
2
votes
2answers
141 views
approximation formula for the integral
Get an approximation formula for the following integral:
$$
\sum_{k=1}^n \left( \frac{1}{35} \right)^{k-1}\int_0^{\frac{\pi}{2}}\cos^{2(n-k)+1}(y) \cdot \sin^{2(k-1)}(y) \, dy
$$
2
votes
1answer
230 views
Solving an integral with Laplace method
I'm trying to approximate the sum $$\sum_{\alpha=1}^{\mu} \Big(1-\frac{(\alpha(2 \mu-\alpha))^2 \gamma_1 \gamma_2}{2n^2 \mu^4}\Big)^{\frac{\lambda}{2}}$$ with an integral ...
1
vote
2answers
60 views
Ei[x] Approximation
I'm working with the function $F(x)=e^{-k(x+1)}\int_1^x\frac{N^2}{t(N-t)}e^{kt}dt$.
Breaking it down into into single fractions helps a little, yielding: $F(x)=Ne^{-k(x+1)} \int_1^x [\frac{1}{t} ...
1
vote
2answers
107 views
Continued Fractions Approximation
I have come across continued fractions approximation but I am unsure what the steps are.
For example How would you express the following rational function in continued-fraction form:
$${x^2+3x+2 ...
1
vote
2answers
164 views
Triple Recursion Relation Coefficients
I am reading Atkinson's "An Introduction to Numerical Analysis" and I am trying to understand how a certain equation was reached. It is called the "Triple Recursion Relation" for an orthogonal family ...
0
votes
0answers
13 views
Can a measurable function in $R^d$ be approximated by step function? [duplicate]
Consider the question here: How to show that a measurable function on $R^d$ can be approximated by step function?
I have work on it for a while, now I can show that (by essentially apply Egorov's ...
0
votes
0answers
106 views
Please help me find this limit and inverse Laplace transform
I need help solving this (I suggest something hereafter but I am not sure if it's ok):
I would like to find an approximate solution of the function $\bar{f}(s)$ defined in the Laplace space. At long ...
0
votes
1answer
69 views
Quadratic approximation of a cost function with a Taylor expansion
See also http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-832-underactuated-robotics-spring-2009/readings/MIT6_832s09_read_ch12.pdf, page 92.
Given an instantaneous cost ...
