1
vote
0answers
44 views

Approximate an integral

In a physics textbook, I came across the integral $$I(r_1,r_0)=\int_{r_0}^{r_1}\frac{1}{1-2m/r}\left[1-\frac{r_0^2(1-2m/r)}{r^2(1-2m/r_0)}\right]^{-1/2}dr$$ The author said that the integrand can be ...
0
votes
0answers
10 views

Correlation 4-point

I need to calculate $\langle x_{i}x_{j}x_{k}x_{l}\rangle $, where $$ \langle f(x) \rangle = \int e^{-\frac{1}{2}A_{ij}x^{i}x^{j} - \frac{\lambda }{4!}\sum_{i}x_{i}^{4}} f(x)d^{n}\mathbf x , $$ for ...
1
vote
0answers
19 views

Copying the Curvature of One Function onto Another: Approximation

I have a polar function $$ r(\theta)=\left(r+\epsilon\right)\cos(\theta)-\sqrt{r^{2}-\left(r+\epsilon\right)^{2}\sin^{2}(\theta)} $$ Is it possible to methodically conjure another polar function ...
2
votes
0answers
30 views

Using Polars to Approximate a Cartesian line: Approximating an Integral

I have the equation of the lower semicircle of radius $r$ centred at a distance $a+r$ above the x-axis $$ f(x)=r+a-\sqrt{r^{2}-x^{2}} $$ which I can approximate (for small $x$) as $$ f(x)\approx ...
3
votes
1answer
57 views

Why is Simpson's rule exact for cubics?

I can't understand: Why is Simpson's rule exact for cubic polynomials?
0
votes
1answer
32 views

Show $\int_{M}^{\infty}\left|\frac{e^{-x^a}\cos(Cx)}{x^{b+1}}\right|dx<\frac{1}{aM^{a+b}e^{M^a}}$?

is there a way to show the inequality \begin{equation}\int_{M}^{\infty}\left|\frac{e^{-x^a}\cos(Cx)}{x^{b+1}}\right|dx<\frac{1}{aM^{a+b}e^{M^a}}\end{equation} for positive constants $M$ and $C$ ...
3
votes
0answers
273 views

Taylor Series of Integral

I'm trying to come up with the Taylor expansion of an integral expression. For simplicity, consider the toy integral $$ ...
0
votes
0answers
38 views

Efficient approximation of derivatives of an integral

Suppose $ \phi(z) $ is the probit function (http://en.wikipedia.org/wiki/Probit). And $$ Z = \int \phi(\mathbf{w}^\top \mathbf{x}) \mathcal{N}(\mathbf{w}; \mathbf{\mu}, \mathbf{\Sigma}) d\mathbf{w} ...
1
vote
0answers
39 views

Numerically integrating in to Chebyshev polynomial

I'm trying to find the Chebyshev interpolate for an ODE in a given interval. That is, given an ODE that looks something like: $$y'' = g(y) \ y'$$ I want to numerically integrate it inside the ...
1
vote
1answer
223 views

Midpoint approximation over/under estimation

So left handed approximation underestimates the area under a increasing curve and over estimates for decreasing curves. And right handed approximation overestimates for increasing curves and ...
2
votes
1answer
55 views

Approximating a Gaussian Integral: Can you do better?

I have attempted to approximate this Gaussian: $$ I =\int_{0}^{\lambda}dx\left(r+x\right)\exp\left(-\rho\left(ax^{1/2}+bx^{3/2}\right)\right)\ $$ using $$ I ...
0
votes
2answers
39 views

Help evaluating or approximating this integral

For a thermodynamics project I'm working on, I need to evaluate this integral: $\int \frac{(a-bx)(x-c)^d}{x^3}dx$, where $a,b,c,$ and $d$ are all positive constants. I tried evaluating it on ...
0
votes
0answers
38 views

Bounding a convolution with a maximal function

Consider a family of kernels $\{K_{\epsilon}\}_{\epsilon>0}$ such that: $\int_{\mathbb{R}^d}K_{\epsilon}\ dx=1$ $|K_{\epsilon}(x)|\leq A\delta^{-d}$ for all $\delta>0$ $|K_{\epsilon}(x)|\leq ...
4
votes
2answers
80 views

How to approximate $n \int_{0}^{1} [1 - x^m ]^n x^m dx $ near infinity?

I have a hypothesis that if: $$ I_{n,m} := n \int_{0}^{1} [1 - x^m ]^n x^m dx $$ where $m,n \in \mathbb{N}$ then $$ \lim_{n \rightarrow \infty} \frac{I_{n,m}}{n^{-\frac{1}{m} } } = c_m $$ But I ...
2
votes
0answers
56 views

Approximating a function using its integral

Question: Let $f:\Bbb R \to \Bbb R \in C^{1}, \forall \delta>0:$ $$F_\delta = \frac 1{2\delta}\int^{x+\delta}_{x-\delta} f(t) \, d(t)$$ in $[a,b]$ prove that $\forall \varepsilon>0 \exists ...
0
votes
2answers
195 views

How to find upper bound on absolute error with composite trapezoid rule

Obtain an upper bound on the absolute error when we compute $\int_0^6 \sin x^2 \,\mathrm dx$ by means of the composite trapezoid rule using 101 equally spaced points. The formula I'm trying to use ...
4
votes
0answers
109 views

Approximate the integral $\int_0^\pi \sin(x^3)\mathrm{d}x$ with a standard pocket calculator

I came over the following integral $$ \int_0^\pi \sin(x^3) \mathrm{d}x $$ when a friend of mine tried to approximate it. The most obvious way is to use taylors formula, and then turn the integral ...
1
vote
1answer
43 views

Approximate an Integration by a linear formula

I just wonder are there any methods to approximate the following integration by a linear formula ? $$ \int_{x_1}^{x_2} \int_{y_1}^{y_2} f( x,y,w_1,\dots,w_n ) \, dx \, dy \approx \sum\limits_{i = ...
4
votes
1answer
66 views

Taylor series of an integral

I have the following integral $$ 2\int_r^\infty \frac{x g(x)}{\sqrt{x^2-r^2}}\text{d}x $$ where $g$ is a probability distribution (normalized and symmetrical around its only maximum in 0). I'm ...
0
votes
1answer
40 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
176 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?
1
vote
2answers
89 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
185 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
292 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 $$ ...
1
vote
0answers
53 views

alternating series estimation with integral?

We know that there are some approximation like Abel's identity. If $\lambda_n$ is increasing and $$ C(x)=\sum_{\lambda_n\le x}c_n,\qquad(c_n\in\mathbb{C}) $$ Then if $X\ge\lambda_1$ and $\phi(x)$ has ...
2
votes
2answers
272 views

Numerical approximation of the modified Bessel function $I_0$ with radical argument for integration purposes

I have to numerically calculate the following definite integral $$\int_{\alpha}^{\beta}I_0(a\sqrt{1-x^2})dx$$ for different values of $\alpha$ and $\beta$, where $a$ has a value of, say, $30$. I'm ...
0
votes
1answer
102 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
111 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
339 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 ...
3
votes
1answer
183 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 ...
7
votes
2answers
535 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
84 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 ...
2
votes
1answer
126 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 ...
3
votes
1answer
187 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
0answers
147 views

integral with Bessel function

Let $n$ be half an odd integer, say $n=k+1/2, k \in Z$. Let $q\geq 1$. I would like to calculate (or approximate) the following integral $$ \int_0^{\infty}\left(\sqrt{\frac{\pi}{2}}\cdot 1\cdot 3\cdot ...
1
vote
2answers
121 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
108 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 ...
0
votes
1answer
94 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 ...
1
vote
1answer
219 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 ...
1
vote
2answers
283 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 ...
1
vote
1answer
995 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 ...
3
votes
1answer
982 views

Projection of Gaussian in Spherical Coordinates

Consider a point with spherical coordinates $\vec{r}_0=(r_0, \theta_0, 0)$. The spherical gaussian distribution centered at $\vec{r}_0$ is $f(\vec{r})=Ne^{|\vec{r}-\vec{r}_0|^2/A}$, where $N$ is the ...
5
votes
3answers
214 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
74 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
110 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
101 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
205 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
603 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
181 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
197 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 ...