Tagged Questions

0
votes
1answer
24 views

Function with $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$

I am looking for a function $f:\mathbb{R}\to \mathbb{R}$ and $\epsilon>0$ such that there is no $\delta>0$, for him any $x\in\mathbb{R}$: $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$ ...
2
votes
4answers
91 views

Analytical approximation of an integral

I think there is no analytical solution for $$ \int_{K}^{\infty} \frac{exp(-x)}{x} dx $$ where $K > 0$. Instead, is there an analytical approximation?
2
votes
2answers
73 views

Asymptotic expansion for $\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du$?

Is there an asymptotic expansion for the function: \begin{equation} g(x)=\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du, \end{equation} over the domain $x\in [0,\infty)$ in terms of ...
1
vote
0answers
72 views

Approximations of the incomplete elliptic integral of the second kind

For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I know that this is a ...
5
votes
0answers
275 views

Show that the function is constant

Let $S^n$ be an $n$-dimentional unit sphere. Consider $f: S^n \longrightarrow R_+$ even continuous function. Denote $$ ...
0
votes
1answer
71 views

Integral approximation.

Can you help me to show that $$\int^{\pi/2}_{0}{d\theta\over (1-m^2\cos^2\theta)^2} \approx {(2-m^2)\pi\over4(1-m^2)^{3/2}}$$ to first order, such that $0 \lt m \lt 1$
2
votes
1answer
73 views

approximate error between integral an sum

I am new here. My problem: There is an integral $I:=\int_0^1 f(x)\,dx$ for $f\colon [0,1]\to\mathbb{R}$ and I want to compute it by ...
2
votes
0answers
170 views

Approximation of integral using series expansion of the integrand.

I have a smooth function $x \rightarrow f_\epsilon (x)$ on $x\in[-1\ldots 1]$ (dependent on the continuous parameter $\epsilon$) and I want to approximate the integral $$ I=\int_{-1}^1 f_\epsilon ...
6
votes
2answers
364 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 ...
1
vote
0answers
80 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 ...
3
votes
1answer
142 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 ...
1
vote
2answers
104 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
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 ...
1
vote
1answer
447 views

How to approximate an integral using the Composite Trapezoid Rule

I'm trying to estimate the value of the following integral on the interval $[0,1]$ $$ I = \int_0^1 \frac{1}{1+x} dx $$ So, using the composite trapezoid rule (and with $n=4$, ie I'm only using the ...
5
votes
3answers
196 views

Approximate $\int_{0}^{\infty} \frac{\text{d} x}{1 + x^4}$

Now, I have been given this integral. And need to approximate it. My first idea was to use a Taylor series, but this series explodes, as x reaches infinity. Does anyone know how to approximate ...
1
vote
0answers
73 views

Riemannian sums approximation: bounds on the argument

I'd like to approximate a sum of the form $ S(n)=\sum\limits_{k=1}^{n}\phi\left(\frac{k}{n}\right)$ with an integral using Riemannian sums: $$S(n) \approx n \int_{0}^{1}\phi(x)dx +o(n).$$ My ...
2
votes
0answers
100 views

Approximating sums like $\sum_{j=1}^n\sum_{k=1}^{\lfloor\frac{n}{j}\rfloor}\int_1^{\frac{n}{jk}}dx$

Can anyone tell me how to approximate the following functions? $f_3(n) = \displaystyle\sum_{j=1}^n\sum_{k=1}^{\lfloor\frac{n}{j}\rfloor}\int_1^{\frac{n}{jk}}dx$ $f_4(n) = ...
4
votes
1answer
92 views

How can we approximate $\sum_{j=0}^n{\sum_{k=0}^j{c^j k^{1/2}}}$ by integrals?

"Difference Equations" by Walter G. Kelley and Allan C. Peterson, 2nd Edition, gives an example on how to approximate $\sum_{k=1}^n{k^{1/2}}$ using integrals and Bernoulli numbers. I'm interested in ...
0
votes
1answer
184 views

Solve $\displaystyle\int_{0}^a \left(3^{\frac{1}{3} \left( x^3 - 3x \right) }-1\right)\, dx = 0$ using elementary methods

A friend of mine came upon the following problem. Solve for $a$ the equation $\displaystyle\int_{0}^a \left(3^{\frac{1}{3} \left( x^3 - 3x \right)}- 1\right)\, dx = 0$. By typing the problem ...
1
vote
1answer
370 views

Approximate the integral $\int_0^1 \sin(x^2) dx$

I'd like to ask if someone can please give me a little push with this assignment: Approximate the value of the integral $\int_0^1 \sin(x^2) dx$ using only $\mathbb{N}$ numbers and basic operations ...
1
vote
1answer
128 views

Midpoint Rule, Trapezoidal Rule, etc.: When the number of intervals increases by a factor of $q$, the approximation error decreases by $r(q) =\;$?

I'd like to look at this problem in terms of the definite integral $I = \int_0^5 e^{\sin\sqrt x}dx$, and in terms of the Midpoint Rule. (Then, hopefully, I'll be able to figure out the left-point ...
4
votes
1answer
190 views

Stuck on complex integral, approximate?

I've been stuck on a particular integral I encountered. I don't need an exact solution, I doubt it even exists. $$f(x)=\frac{e^{-i (r+R-k) x} \left(i-2 e^{i (r+R) x} r x-R x+e^{2 i r x} (R ...
0
votes
2answers
654 views

efficient and accurate approximation of error function

I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2} \,\mathrm dt$$
2
votes
0answers
182 views

When is it valid to convert a function inside a probability integral to the indicator function?

I am faced with an approximation that replaces a probability density function with the indicator function and I am at a loss as to why this is valid. We want to model the lifetime $T$ of a website ...
2
votes
4answers
626 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 ...
6
votes
4answers
403 views

An approximation of an integral

Is there any good way to approximate following integral? $$\int_0^{0.5}\frac{x^2}{\sqrt{2\pi}\sigma}\cdot \exp\left(-\frac{(x^2-\mu)^2}{2\sigma^2}\right)\mathrm dx$$ $\mu$ is between $0$ and $0.25$, ...
2
votes
1answer
632 views

Numerical integration - Gauss quadrature rule

How do we numerically integrate a rapidly decaying exponential function? A simple Gauss quadrature which is based on approximating the function by polynomial, I think will not work, since rapidly ...
2
votes
2answers
366 views

Is this integration approximation method known/used?

I'm approximating an integral with only exponentials. i.e., it is equal to $\displaystyle \int_{-\pi}^\pi{\frac{\displaystyle\sum_{j=a}^b{c_j e^{i\cdot d_j \cdot t}}}{\displaystyle\sum_{k=a}^b{r_k ...
1
vote
0answers
275 views

How effective is this alternative to integration?

I have a function that is difficult to integrate. So I elect to work with power series representations. Suppose the power series representation for this function is the following: $f(x) = ...
4
votes
1answer
519 views

Approximating $\pi$ using Monte Carlo integration

I'm trying to approximate $\pi$ using Monte Carlo integration; I am approximating the integral $$\int\limits_0^1\!\frac{4}{1+x^2}\;\mathrm{d}x=\pi$$ This is working fine, and so is estimating the ...
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 ...
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 ...