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).

learn more… | top users | synonyms

0
votes
0answers
7 views

Picking out a single term from a sum that depends on Associated Legendre Polynomials through integration

Although this question is rooted in physics, the question is a mathematical one, please read the entire question before voting. I have defined a state for some $m'$ \begin{equation} \left|\Psi_{m'}\...
1
vote
1answer
15 views

Pade approximant of infinite order

The Pade approximant states that you can approximate a function $f(x)$ by a rational function $R(x)$ of a given order. My question is, if the order of $R(x)$ goes to infinity, does $R(x)$ approach $f(...
0
votes
3answers
55 views

Proof that $w_n=e^{\sqrt{n}} - e^{\sqrt{n-1}} \approx \dfrac{e^{\sqrt{n}}}{2\sqrt{n}}$

According to a article, author wrote: $w_n=e^{\sqrt{n}} - e^{\sqrt{n-1}} \approx \dfrac{e^{\sqrt{n}}}{2\sqrt{n}}$ for $n>1$. Can you Explain for me this?
0
votes
0answers
25 views

How to obtain this expansion?

My goal is to find an expansion in powers of 1/ρ (and its first 2 or 3 terms) of the quantity \begin{equation} F(\rho,\mu,\nu)=(2n+1)E_n(\rho)E_n(\mu)E_n(\nu)I_n(\rho),\quad \rho \ge h_2 \end{equation}...
0
votes
1answer
25 views

Finding the Intersection of Normal CDF and y=x and Plot the Relationship Between $\sigma$ and Intersection Conditional on Different $\mu$

I'm trying to investigate the intersections of the Normal CDF and the $y=x$ line for $\mu \in [0, 1]$ and $ \sigma \in (0, 1] $ when $ x \in [0, 1] $ and plot the corresponding relationship between $\...
1
vote
1answer
26 views

Probability of being in a circle, given normal

Let's assume a bivariate normal distribution with center $\mu$ and covariance matrix $\Sigma$. Let a circle $C$ be given as $C=\{x\in\mathbb{R}^2:||x-\mu||\leq R\}$. I would like to calculate the ...
1
vote
0answers
19 views

Solving ODE with irregular singular point

I want to solve the following ODE $$x''(z)+ \frac{\frac{d}{dz} \left(\frac{f(z)}{z^2}\right)}{\frac{f(z)}{z^2}}x'(z)+\frac{\omega^2}{(f(z))^2}x(z)=0$$ where $$f(z) = 1- 4 \left(\frac{z}{z_*}\right)^...
2
votes
3answers
108 views

Why does $(\sin x)^2=x^2$ and $\sin x=x$ in these contexts?

Contexts (it must also be noted that as $\delta t$ tends to zero, $\delta \theta$ also tends to zero): First context $$ \lim_{\delta t \to 0} \frac{-2v\sin(\delta\theta/2)^2}{\delta t} = \lim_{\delta ...
0
votes
1answer
54 views

Approximation of series using integral

In notes of statistical physics I found the following approximation $$\sum\limits_{n=0}^{\infty}F\left(n+\frac{1}{2}\right)\approx \int_{0}^{\infty}F(x)dx+\frac{1}{24}F'(0)$$ for $F$ such that the ...
0
votes
2answers
18 views

Finding an approximate function using orthonormal basis

I'm trying to take a function in $C_0[0,1]$ space (let's call this $f(x)$) and trying to find the best approximate of $f(x)$ at $P_2[0,1]$ space (let's call this approximate $p(x)$). Note that $P_2[0,...
3
votes
1answer
51 views

Proof of Stirling's Formula using Trapezoid rule and Wallis Product

I need a proof of stirling's formula which uses the riemann's sum and trapezoid approximation to come up with $ \frac {n!}{(n/e)^n \sqrt n}$ $ \rightarrow C$ where $C$ is derived from Wallis product. ...
7
votes
9answers
273 views

How do I prove that $\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$

How do I prove that $$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$$ without using the calculator?
-1
votes
0answers
32 views

How to remap continued fractions from $\mathbb{R}$ to a discrete set of integers [on hold]

Assuming that I have a continuous fraction \begin{equation} x = a_0 + k_1 \cfrac{x_1}{a_1 + k_2 \cfrac{x_2}{a_2 + k_3 \cfrac{x_3}{a_3 + k_4 \cfrac{x_4}{a_4 \ddots } } } } \end{...
2
votes
1answer
45 views

Maxima of $f(x)/e^x$ where $f(x)$ is an approximation of $e^x$ using Stirling's

Let $$f(x)=1+\sum_{n=1}^\infty\frac{x^n}{\sqrt{2\pi n}(n/e)^n}\tag1$$ and let $$g(x)=\frac{f(x)}{e^x}\tag2$$ If we plot $g(x)$ we get a graph that looks like this: Clearly there is a maximum at ...
0
votes
1answer
43 views

Comparing $f(x)$ and the $f\_approx(x)$ in Matlab, Octave or Mathematica/Wolfram

I have a function $f(x)$ and I wrote the approximation for $f(x)$ as $f\_approx(x)$ which is a simple algebraic formula with sums and products . I now want to study the 2 functions side by side and ...
0
votes
0answers
41 views

Getting the DFT of irregularly spaced points

I am trying to estimate the discrete Fourier transform of a discrete surface, $x:\{1,\dots,N\}\times \{1,\dots,N\} \to\mathbf{R}$, given a sparse set of samples on the grid. If we had all the ...
0
votes
1answer
19 views

Bin packing approximation algorithm

I know that bin packing cannot be solved in $\mathrm P$ unless $\mathrm P=\mathrm{NP}$, because we could solve partition problem. However, I do not see why this theorem is a collorary. There is ...
0
votes
1answer
48 views

Analysis of bisection search

http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-00sc-introduction-to-computer-science-and-programming-spring-2011/unit-1/lecture-3-problem-solving/ In the following video i'm ...
1
vote
3answers
641 views

How come $\pi$ is usually approximated as 3.14 or 22/7?

I've heard that $\pi$ is usually approximated as 3.14, but it can also be approximated as 22/7, which is equal to 3.142857142857142857.... Guess what? $\pi$ can also be approximated as 355/113, ...
10
votes
6answers
15k 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: ...
0
votes
0answers
22 views

Aproximation for the variance (sum)

Given that we know The mean of a population $\tilde W(t) = \sum_{i=1}^{n}f_{i}(t)*W_{i}$ The variance of the population in the previous step $Var(0) = \sum_{i=1}^{n}f_{i}(0)*(W_{i}-\tilde W(0) )$ ...
1
vote
2answers
114 views

Compute $\lim_{n\to\infty} n \bigg( \sum_{k=0}^n f(\frac{k}{n}) - n\int_0^1 f(x) \, dx \bigg)$

The task is to show the following limit exists, and then compute it. Here, $\,\mathrm{f}:\left[0,1\right] \to \mathbb{R}$ is a continuously differentiable function. $$ \lim_{n \to \infty}\left\{n\...
1
vote
1answer
69 views

Sequence of polynomials converging to zero, but not uniformly on unit disc

I have been trying to solve the following without success so far: Show that there exists a sequence of polynomials satisfying $P_n(z)\rightarrow 0$ for every $z\in \mathbb{C}$, but the convergence is ...
14
votes
1answer
647 views

Approximation of Semicontinuous Functions

Assume that $k \in \mathbb{N}$ and $f : \mathbb{R}^d \rightarrow [0,\infty)$ is lower semicontinuous, i.e. $f(x) \leq \liminf_{y \rightarrow x} f(y)$ for all $x \in \mathbb{R}^d$. Does there exist a ...
2
votes
1answer
29 views

Local quadratic approximation

I wanted to implement some penalized regression parameter estimation algorithm by Fan&Li (http://sites.stat.psu.edu/~rli/research/penlike.pdf, section 3.3, [1]), but cannot catch the idea of some ...
0
votes
1answer
26 views

Calculating approximate growth from three numbers [closed]

I have a set of three numbers 3600, 5200,12000; how do I calculate an approximate 4th number ...
2
votes
1answer
38 views

A square-root approximation method that would halt on $\sqrt{378}$

Back in the early $'90$s, I used to program in a (now obsolete) scripting language called LOGO. Now, one peculiar glitch that I encountered at the time, was the interpreter halting on $\sqrt{378}$. ...
1
vote
1answer
1k views

Smooth approximation of maximum using softmax?

Look at the Wiki page for Softmax function (section "Smooth approximation of maximum"): https://en.wikipedia.org/wiki/Softmax_function It is saying that the following is a smooth approximation to the ...
0
votes
1answer
20 views

Approximate perfect matching through MST

If I compute a minimum spanning tree T in a graph with an even number of vertices and T contains a perfect matching M (which is unique in this case), can I get some approximation guarantee on the ...
4
votes
1answer
57 views

Asymptotic vlaue of $ f(n)=\sum_{i=0}^n\lfloor \sqrt{i}\rfloor\binom{n}{i} $

Inspired by this question I tried to find an asymptotic formula for $$ f(n)=\sum_{i=0}^n\lfloor \sqrt{i}\rfloor\binom{n}{i} $$ With the observation: $$ f(n)=\sum_{i=0}^n\frac{\lfloor \sqrt{i}\rfloor+\...
0
votes
0answers
33 views

On the inverse of the regularized upper incomplete gamma function

I'm interested to bound/approximate the the inverse of the regularized upper incomplete gamma function $Q^{-1}(a,z)$, where $Q(a,z) = \frac{\int_z^\infty t^{a-1} e^{-t} \mathrm{d} t}{\Gamma(a)} $. I ...
21
votes
7answers
4k views

How to show this formula to get a square root of a number in “just few seconds” is true?

I don't remember in which topic I found it but I know it was there. And I still have not find a proof of this nice approximation. Let $x$ be a non perfect square number. If $y$ is the closer ...
2
votes
0answers
16 views

Central Limit Theorem Heuristics

Surrounding the central limit theorem there exist several heuristics which say when a normal distribution is a reasonable approximation to the mean $\frac{X_1 + \cdots + X_N}{N}$ of $N$ independent (...
2
votes
1answer
151 views

How does one show that $\cos {\left (\ln 2 \right )}\approx \frac{10}{13}$?

How does one approximate the value of something like this? Apparently Euler found the value of $\large \frac{2^i+2^{-i}}{2}\large $ [which equals $\cos {\left (\ln 2 \right )}$] to be close to $\...
0
votes
1answer
468 views

Need some help understanding notation for composite gauss quadrature formula

Reading through some notes on 2-point gauss quadrature, I came across the following general formula. I'm currently doing an assignment with 3-point quadrature, and have gotten to a similar formula, ...
1
vote
1answer
31 views

How to use Taylor's Theorem to obtain an upper bound for an error approximation

$e \approx 1 + 1 + \frac{1^2}{2!} + \frac{1^3}{3!} + \frac{1^4}{4!} + \frac{1^5}{5!}$ must find upper bound for this but I don't see what I should be doing. The remainder/error is given by $\frac{f^{n+...
0
votes
1answer
58 views

(Terminology_Taylor Series) “expand at $x_0$, evaluate at x, affine approximation”

I am reading one-variable calculus book where it explains Taylor series and little confused with the following terms: (1) Expand $f(x)$ at $x_0$ (2) Evaluate $f(x)$ at x (3) Best Affine, ...
-1
votes
1answer
26 views

Iteration: Approximation and Errors, finding all possible iterative arrangements

I am looking at a relatively simple problem to reiterate: $x^4=e^x$ I've found 5 different possible forms 1: $x_{r+1}=\frac{e^x}{x^3}$ 2: $x_{r+1}=(\frac{e^x}{x^2})^{0.5}$ 3: $x_{r+1}=(\frac{e^x}...
2
votes
0answers
24 views

(Conceptual_Calculus) Differential Conditions v. Derivative Conditions

I have few questions regarding the reason we learn about the differential conditions in higher dimensions and dealing with multivariable calculus. In the context of optimization (e.g. finding ...
0
votes
0answers
21 views

How can we use the Lindley's method to approximate the following expression?

The Lindley's(1980) approximation is one of the most popular methods that is used to obtain Bayes estimates. In this method we need to maximum likelihood estimators(MLEs) of the unknown parameters. ...
0
votes
0answers
15 views

Approximating $\chi_0$ pointwise with holomorphic functions

Define $\chi_0:\mathbb{C}\to \mathbb{C}$ as $$ \chi_0(z)=\left\{ \begin{gathered} 1 \quad z=0\hfill \\ 0 \quad z\ne 0 \hfill \\ \end{gathered} \right.$$ Does there exist a sequence of ...
2
votes
1answer
63 views

Taylor Series for a Function of $3$ Variables

The Taylor expansion of the function $f(x,y)$ is: \begin{equation} f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)...
2
votes
0answers
29 views

Chebyshev's inequality and quadratic function

I am trying to use Chebyshev's inequality in order to find sample sizes $n$ such that some condition is met with probability $p$ or larger. That is, find $n\in \mathbb{N}$ such that $\mathbb{P}(X_n &...
1
vote
2answers
44 views

Partial Derivatives Approximation

By definition we know the following: \begin{equation} \frac{\partial f(x,y)}{\partial x} \approx \frac {f(x+ \delta x,y)-f(x,y)}{\delta x} \end{equation} \begin{equation} \frac{\partial f(x,y)}{\...
0
votes
1answer
23 views

finding a close enough point with implicit function theorem

Let $a_1, ...,a_n ,B\in \mathbb{R}^n$ , not all on the same plane. Prove that for a small enough neighborhood of zero $U$ and $\forall u_1,...,u_n \in U $ there is a point $C \in \mathbb{R}^n$ that $...
0
votes
0answers
25 views

Formal approximation for second-order ODE with varying coefficients

I have a differential equation of the form $$0=a+by(x)+cf(x)+z(x)f''(x)$$where the functions $y$ and $z$ are known and we want to find $f$. If $z$ is constant, i.e. $z(x)=Z$, it is straightforward to ...
2
votes
3answers
73 views

Finding an approximation of a function's root

I have the polynomial function $f (x) = x^5+2x^2+1$. I am trying to find an approximation to its root in $[-2,-1]$, with the precision of $0.1$, and with a minimal number of steps. The answer I was ...
0
votes
1answer
32 views

Behaviour of the Spectral Weight Function $\frac{\sin^2{(\pi f t)}}{(\pi f)^2}$

I'm looking into the properties of the so called spectral weight function $W_0 = \frac{\sin^2{(\pi f t)}}{(\pi f)^2}$. While not important for the question, this function is is encountered in the ...
0
votes
0answers
35 views

Maximum of Contour Line

Consider the potential $U(x,y)=ay^{2}+b(e^{x-y}-1)^{2}+c(e^{x+y}-1)^{2}$ where $a$, $b$ and $c$ are known constants. I want to move through a contour line of this potential $U(x,y)=k$, say $y=g(x)$. ...
1
vote
0answers
11 views

Estimate the drift and diffusion function numerically

I have a 1D problem as following $$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x} \Big[ \frac{1}{2} \frac{\partial (g(x) f)}{\partial x} -\mu(x)f \Big]$$ I have a time-series of ...