Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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48 views

Subdifferential of integral

I am currently trying to extend my knowledge about subdifferentials. Now I am stuck at a particular property of the subdifferential. In this "paper" ...
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58 views

A integral inequality

Let $g\in C_0^\infty((-1,1))$.Prove $\forall t\in (-1,1)$,$${g^4}\left( t \right) \le 16\int_{ - 1}^1 {\left( {{{\left| {g'\left( s \right)} \right|}^2} - \frac{{{g^2}\left( s \right)}}{{4{{\left( ...
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42 views

Swapping the integrand and variables being integrated

I am reading a paper in which the following integral is solved, but I am not sure how to derive the answer myself. The integral is: $$\int_{F_{x-1}}^{F_x} du {\int_{G_{y-1}}^{G_y}{dv}\ C^*(u,v)} \ ...
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25 views

Derivatives of (vanishing) infinite integrals

Probably this is a very stupid question, but if an infinite integral of some exponented expression vanishes, e.g.: $$\int_{\mathbb{R}^D} d^D \mathbf{x} \, P^r(\mathbf{x}) \cdots = 0,$$ does this imply ...
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99 views
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82 views

Calculating in closed form $\int_0^{\infty} \frac{\text{PolyLog}^{(1,0)}(1,-x)}{1+x^2} \, dx$

Can you confirm the following result? Mathematica and other computational stuff I used seem unable to do anything about this result. Maybe to confirm it numerically? $$\int_0^{\infty} ...
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44 views

Integral of an expression involving sine and cosine powers

For integers $a,n\in \mathbb N$, consider the following integral $$ I_n(a) = \frac{(-i)^x}{\pi}\int_0^\pi e^{i\theta(n-2a)} \sin^x \theta \cos^{n-x} \theta\; \mathrm d\theta\;. $$ How would one go ...
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112 views

A difficult integral $\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$

Here is an integral that I want to see a different approach: $$\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$$ Well, for someone who is deeply aware of the exponential integral function and the ...
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57 views

Transform an improper integral with $\sin,\cos,\ln$ into an equal integral over $[0,2\pi]$

As there are many properties of integrals and methods of integration often some seem to escape being "readily seen how to...". This is one of the many. The integral in question is \begin{align} ...
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33 views

Can an integral be certifiably non-elementary?

If you want to convince somebody that a particular natural number is prime, you can hand them a primality certificate--a small bundle of data which can be used to efficiently generate a proof of ...
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33 views

Estimating the sum of a series within to arbitrary certainty.

Find the sum of the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^5} = a_n$ within three decimal places. The sum is estimated by $\displaystyle a_n \approx \sum_{k=1}^{n}\frac{1}{k^5}+R(n)$ ...
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82 views

When can we use Differentiation under the Integral sign?

Let me elaborate, 'Feynman's' trick ranks up in the top ten on most people's list, right behind contour integration, for best ways to evaluate definite integrals. However, unlike contour integration ...
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61 views

how to solve this inverse fourier $ f(x) =\int^{\infty}_{-\infty} 1/\sqrt{2\pi}\ e^{-2\pi^2/s^2} e^{ i \ s\ x}ds$

I have two functions f(x) and f(s). f(s) is the fourier transform of f(x) and tends to $$e^{-2\pi^2/s^2}$$ I need to take inverse transform of this f(s) to get to f(x). (i need to prove f(x) tends to ...
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165 views

Using a sequence of measures to create simple functions which approximate the Radon-Nikodym derivative of the limiting measure

I have a bunch of discrete probability measures with finite support: $\mu_1,\mu_2,\dots$, which strongly converge to an absolutely continuous probability measure $\mu$ in $\mathbf{R}^2$. That is, for ...
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51 views

Do we have a inverse Laplace transform of $\frac{1}{\arctan s}$

Do we have a closed form of this seemingly very simple inverse transformation? If no closed form, what about its asymptotic form? Does this satisfies the criterion to have its inverse ...
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31 views

First mean value theorem for integration and Lebesgue measureability

According to first mean value theorem for integration, if $G \ : \ [a,b] \to \mathbb{R}$ is a continuous function, there exists $x \in (a,b)$ such that $$\int_a^b G(t) dt = G(x)(b-a)$$ Assume $G$ is ...
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44 views

How to Solve this Improper Integral with six poles?

I'm trying to solve the following integral, where $a>0$, $b>0$, $y\in\mathbb{R}$ and $z\in\mathbb{R}$ are given constants: $$ \int_{-\infty}^{0} \left[ ...
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51 views

Double integral of symmetric polylogarithmic function over rectangular region

This question was inspired by M.N.C.E.'s wonderful response here. While exploring the possibility of generalizing his result, I found that a significant part of the problem reduced to evaluating the ...
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62 views

Solution of $\int_0^{\pi} \frac{ y \cos y}{s^2+y^2} dy$

Is there a solution for the following integral (even in terms of Bessel or Struve functions)? $$ \int_0^{\pi} \frac{ y \cos(y)}{s^2+y^2} \,dy $$
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60 views

Integral involving Whittaker function

Consider the following integral: $$ \int_1^{\infty} \frac{e^{u/2}}{u}[-\mathrm{Ei}(-u)]\,W_{1,\imath p}(u)\,du, $$ where $\imath=\sqrt{-1}$ and $p>0$ selected so that $W_{1,\imath p}(1)=0$; here ...
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55 views

Strange triple integral of an inverse function

Let $$ \Omega(a, b, c) = \min\left\{\theta\ge0\ \text{s.t.}\ \tan(a\theta) + \tan(b\theta) + \tan(c\theta) = 1\right\} $$ What is the value of the following integral $$ I = ...
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78 views

Fast Hankel Transform

Can someone please explain what would be the expression for weights(Ho) in a Fast Hankel Transform.I found this in a paper and could not find any satisfactory answers .
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910 views

Solving an integral for a characteristic function

For $L>0, H>0,\alpha>0,\sigma>0,$ $$f(t)=\int_L^H \frac{ e^{i t x} \alpha H \left(\frac{\sigma -H \log \left(\frac{H-x}{H-L}\right)}{\sigma }\right)^{-\alpha -1}}{\sigma (H-x)} \, ...
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100 views

How to evaluate $\int_1^\infty \frac{1}{z} e^{-\left(\frac{z-1}{b}\right)^{\frac{1}{a}}} dz$?

I am really stuck on this integral: \begin{equation} \int_1^\infty \frac{1}{z} e^{-\left(\frac{z-1}{b}\right)^{\frac{1}{a}}} dz \end{equation} where $a,b$ are real constants. Is it a special ...
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23 views

Stieltjes Integral - If $f, f^2, g, g^2\in R(\alpha)$ for an arbitrary integrator $\alpha$, then is $fg\in R(\alpha)$

My question is if $f, f^2, g, g^2\in R(\alpha)$ on $[a,b]$ for an arbitrary integrator $\alpha$, then is $fg\in R(\alpha)$ as well? This question stemmed from a problem in Apostol's Analysis, in ...
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60 views

Is there a spherical coordinates system for vectors of complex numbers?

Suppose I have a scalar field $f(\vec{x})$, where $\vec{x}\in\mathbb{R}_3$, and I wish to average $f$ over a sphere $|\vec{x}|=R$: $\displaystyle\langle f\rangle_{R} = \frac{\int_{S} f(\vec{x})\, ...
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53 views

Implementing the Risch algorithm to integrate $\dfrac{\log(x)+2}{x^{2}\log^{3}(x)}$

Following the work of Andreas Wurfl i am trying to implement the Risch algorithm on $\int{\dfrac{\log(x)+2}{x^{2}\log^{3}(x)}dx}$ following his method for extensions that are purely logarithmic, we ...
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76 views

If $y'=\frac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) $

If $y'=\dfrac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) = ?$ By integrating I am getting $$y = \ln (x+1)+C$$ I am stuck somewhat as it looks tricky from here. Any help ? Thanks!
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Change of variables in $\iiiint$ or $\idotsint$?

I apologize if this is a question that has been asked before, but I have seen that it is possible to change variables in single, double and triple integrals. Now what about quadruple integrals, or ...
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124 views

Visualising surface integrals

For a current problem I am working on, I have run into angular surface integrals, i.e. the differential solid angle $\text{d}\Omega$. Specifically the surface integrals are defined by ...
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55 views

closed form of improper integral

I would like to evaluate an integrale which depend on 2 parameters. The goal is to obtain an expression of the integrale depending on theses 2 parameters ($x_d$ and $z_d$) such as $f(x_d,z_d)= ...
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101 views

Graphs of interesting integrals of the form: $\int \sin^a(x^a)\cos^a(x^a)$

Here are a few graphs of the form:- $$\int \sin^a(x^a)\cos^a(x^a)dx$$ Where $a$ is an even, positive integer. $a = 2$ $a = 4$ $a = 6$ Now, a few graphs of the form:- $$\int ...
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31 views

Riemannian Volume vs. Euclidean Volume

If $M$ is a Riemannian manifold and $\phi$ is a Borel measurable function into $M$, then what is the relationship between $\int_M \phi \,d\mu$ and $\int_{\mathbb{R}^d} \psi\circ \phi\, d\lambda$; ...
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30 views

How to recover complex function on $\mathbb C$ from integral equation?

Let $f:\mathbb C \to \mathbb C$ be a continuous function with the form $f(z)= z\tilde{f}(|z|)$ for all $z\in \mathbb C,$ where $\tilde{f}$ is a real function defined on $(0, \infty).$ We define ...
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66 views

Help me to figure out this integral

$$\int_{0}^{\infty} \frac{x^3\sin\left(\frac{1}{2}\pi x\right)}{e^{2\pi\sqrt{x}} - 1}~dx$$ I've been thinking a long time ,but I have no idea how to do it.
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Understanding why $\int_{-a}^a \sin^{100}x\,\mathrm dx = \frac{99}{100} \int_{-a}^a \sin^{98}x\,\mathrm dx$

In an answer here on Math.SE it is claimed that $$\int_{-a}^a \sin^{100}x\,\mathrm dx = \frac{99}{100} \int_{-a}^a \sin^{98}x\,\mathrm dx$$ but I don't understand how it could be. Since $$ ...
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46 views

A Step in the Proof of Green's Theorem

There is a step in the proof of Green's Theorem where we must combine the line integrals of each curve in the same direction. $$\oint\limits_C P(x,y)\,dx = \int_a^b P(x_1,y_1)\,dx + \int_b^a ...
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84 views

Equivalent Vitali Covering Properties for Differentiation Bases

N.B. In what follows below, we work exclusively with the Lebesgue measure on $\mathbb{R}^{n}$. Def 1 For each $x\in\mathbb{R}^{n}$, let $\mathcal{B}(x)$ be a collection of bounded open sets $R$ ...
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46 views

Center of Mass of objects with infinite length?

Suppose you have $f(x) = \frac{\sin(x)}{x}$ And you have that shape, find the center of mass of $f(x)$ in $x \in (-\infty, \infty)$ Is it possible considering $f(x)$ is an even function?
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How good is my approximation of this complicated sinc function? (plot included)

Part 1: The following function (for $N=256$) has the plot shown nexr $$ G(x) = \frac{1}{N}\text{exp} \bigg( j \frac{\pi}{2} \, x(N-1)\bigg) \frac{\sin (\frac{\pi N}{2} x)}{\sin (\frac{\pi}{2} x ...
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104 views

Any smart tricks to simplify my nasty integration?

I am trying to solve for the following unpleasant integral $$\int_{\gamma}^{\infty} \bigg[t- \int_{-2}^{2}\frac{ t \ f_X(x)}{1+N \ \big|G(x)\big|^2 \ t^{-3}}\ dx\bigg] \ dt$$ where $N$ is a ...
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49 views

Understanding averaging of symplectic matrices via Haar measure

In McDuff and Salamon's Intro. to Symplectic Topology (2nd edition), there's a proof that $U(n)$ is a maximal compact subgroup of $Sp(2n)$ which I'm trying to understand. The proof uses the Haar ...
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31 views

Issues proving a basis via wedge product

On a quiz I was given the problem" a series that is a basis for $[-1,1]$ is $ \sum_0^{\infty} c_n P_n $, where $ P_n $ is a polynomial and each polynomial $P_n$ is orthonormal to the others. Using the ...
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55 views

$\lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$

I am just having fun with this question: Is this true that $\displaystyle \lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$? I thought to change this to an integral, namely ...
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76 views

how to calculate this line integral $\int_{0}^{2\pi} (16\sin^2 3t +16\cos^2 4t)\sqrt{(144\cos^2 3t +256\sin^2 4t)}dt$

I am working on a line integral to calculate the amount of chocolate to cover a pretzel. the density of the pretzel is given by this formula $\lambda=3(x^2+y^2)$ and the parameter equation of a ...
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50 views

Prove the function is integrable

For a point $x \in [1,2]$, define $f(x) = 0$ if $x$ is irrational and define $f(x)= \frac 1n$ if $x$ is rational and is expressed as $x = \frac mn$ for natural numbers $m$ & $n$ having no common ...
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73 views

Fatou's Lemma and Counting Measure

I have a vague problem in a Measure and Integration book here. They ask me to consider $\mu$ the counting measure in $\mathbb{N}$ and interpret Fatou's lemma, monotone and dominated convergence ...
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81 views

Klein-Gordon field commutator integral?

Consider a Klein-Gordon field $\phi$, which satisfies $$(\Box+ \omega_0^2)\phi=0$$ on points $x \equiv \{x_0,\vec{x}\},y\equiv \{y_0,\vec{y} \}$ of 4D Minkowski-spacetime. The field commutator is $$ ...
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63 views

Differentation uder the integral sign

Let $F(x)=\int_{\sin x}^{\cos x} e^{x\sqrt{1-y^2}} \, dy $. My task is to calculate $F'(x)$. My idea is to use http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign and I get: ...
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107 views

Integration involving square root function

First , i multiply numerator and denominator by (2-x-x^2)^(1/2), then I split the integral into 2 parts , using trig substitution , part 2 is easy to be integrated .. but when I tried to ...