# Questions tagged [integration]

Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...

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### Integral average near a point of dispersion

Let $\Omega\subset\subset\mathbb R^{n}$ be a bounded domain and let $E\subset \Omega$ be a Lebesgue measurable set. Let $f\in L^{1}(\Omega)$ and let $x\in \Omega$ be a point of dispersion of $E$, that ...
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### What can this $\int_{0}^{t} (\pi(x)-Li(x)) dx$ tell us about primes distribution?

Many papers I have read which are related to primes distribution only it discussed sign and refinement Bounds of $\pi(x)-Li (x)$ with $\pi(x)$ is a prime counting function and $Li (x)$ is the ...
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### Stationary phase in spherical integral

I'm trying to find asymptotics for an oscillatory integral on $\mathbb{S}^{n-1}$, for which my advisor said I should use stationary phase arguments. The particular, he claims that: If $\lambda\gg 1$...
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### Existence of the inverse Fourier transform, Carr Madan

I have a function $C_T(k)$ that is not $L_1$, because its limit in negative infinity is a constant. So I dampened it by $e^{\alpha k}$. Let's call the transformed function (of the dampened function) ...
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Let $f \in L^p(\mathbb{R}^n,\mathbb{R}^m)$. If $m=1$ then the essential support of $f$ is a mainstream definition; see here for example. However, when $m>1$ is the following definition used? $$\... 0answers 80 views ### Closed form for a double integral over the first quadrant of the L^p disk Is there, by any chance, a closed form for the following integral$$ I_p=\iint_{Q_p}(x+y)\log(x+y)dxdy, $$where Q_p=\{(x,y)\in\mathbb{R}^2, x>0,y>0,x^p+y^p\leq1\}, 0<p\leq\infty ? ... 0answers 44 views ### Weak formulation of PDE with weighted inner product In Boyd's book on spectral methods (available here: https://depts.washington.edu/ph506/Boyd.pdf), I stumbled in section 3.5 "Weak & Strong Forms of Differential Equations: the Use-fulness of ... 0answers 179 views ### Asymptotic of a functional as x\rightarrow \infty Consider the following functional :$$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1}, where  F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])... 0answers 85 views ### A conjecture on integrals of infinite products The problem I would like to discuss in this post is about a conjecture on the following integrals, \begin{align} \int_0^\infty \prod_{n=1}^\infty \cos(x/n)\,dx \stackrel{?}= \pi/4 \tag{1}\\ \int_0^\... 2answers 157 views ### Connection between Volkenborn integral and Haar measure on \mathbb{Q}_p This may be a rather elementary question, but I haven't been able to figure it out on my own, and the literature appears to be eerily silent on the topic. Since \mathbb{Q}_p is a locally compact ... 1answer 102 views ### Inequality involving Gaussian integral [closed] I'm looking to prove the following inequality: \left| \int_0^1 e^{-x^2} \sin(x) \, dx \right| \leq \frac{1}{2} \left(1- \frac{1}{e}\right) $$So far I have no idea on how to prove it. Anybody? 0answers 79 views ### L2 norm of the diagonal entries of a random rotation of a fixed matrix? Let X\in\mathbb{R}^{d\times d} be the diagonal matrix with d/2 entries equal to 1 and d/2 entries equal to -1. Let F_U \triangleq \frac{1}{d}\|\operatorname{diag}(U^{\dagger}XU)\|^2_F ... 2answers 543 views ### Reference request: proofs of integrals presented in Erdélyi's *Table of Integral Transforms* I have asked related questions on MSE and received no answer, so: I am looking for proofs of the integrals presented in Erdélyi's Table of Integral Transforms vol. i-ii, specifically proofs of the ... 0answers 63 views ### Coincidence Topologies for L^p spaces If X and Y are compact metric spaces then it is well-known that the compact-open topology on C(X,Y) coincides with the topology of uniform convergence on compacts. Therefore, the latter is ... 0answers 541 views ### On properties on a certain functional Consider the following function:$$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$Here, \omega(z) is a weight we have to construct and c is a constant. The following three conditions ... 0answers 33 views ### Sufficient and necessary condition for the continuity of an improper integral Let f(\cdot) \in \mathscr{C}\left( \mathcal{D}; \mathbb{R} \right) where \mathcal{D} \subseteq \mathbb{R} is open with 0 \in \mathcal{D} and$$ f(0) = 0, \quad \forall x \in \mathcal{D}\...
Assuming that $f$ is a continuous function, we have that $$f(x) = \frac{d}{dx}\int f(t)\,dt.$$ Assuming instead that $f$ has a removable singularity at $x=a$, and is otherwise continuous, we have ...
Consider the following oscillatory integral $$I(n):=\int_{-\pi}^\pi\int_{-\pi}^\pi e^{i n(x+y)}\frac {(1 - \cos(2x)) (1 - \cos(2y))} {2k - (\cos x + \cos y)}\ \mathrm{d}x\,\mathrm{d}y.$$ where $... 0answers 74 views ### The mean along the eccentric anomaly of an ellipse log distance to a point within the ellipse Conjecture. Let $$f(r,\alpha,p, \theta) = \ln\left(\left(r\sin\alpha-\sin\theta\right)^{2}\left(1-p\right)^{2}+\left(r\cos\alpha-\cos\theta\right)^{2}\left(1+p\right)^{2}\right).$$ Then for any ... 0answers 90 views ### Is this integral finite and how does it decay to zero? I would like to know if the following is convergent/finite (it represents a bound from a truncated Legendre series approximation) \begin{equation} \varepsilon_n \leq \int_{-1}^1 \left(\int_{n\gg 1}^\... 0answers 24 views ### Convolve a 4D Gaussian function along a plane? There is a 4D Gaussian function$G(u,s)=G(x|c,\mu,\Sigma )$where$x=\begin{bmatrix}u\\ s\end{bmatrix}$,$u$and$s$is all 2D vector. Now I want to blur (convolve) it along with$u$by another 2D ... 0answers 26 views ### Spectrum of large random asymmetric matrices with correlation Background: In their paper, Sommers Crisanti Sompolinsky and Stein derive the spectral distribution of large random matrices$\mathbf{J}$by studying the following integral: \begin{equation} I=\left[\... 1answer 152 views ### Integral involving associated Laguerre polynomial and Bessel function In a quantum mechanics problem I encountered the following integral \begin{equation*} \int_0^\infty t^{\nu+1}J_\nu(\beta t)L_{\mu-\nu}^{2\nu}(t)e^{-t/2}dt\,, \end{equation*} where$L$denotes the ... 0answers 128 views ### Accuracy of Richardson's error estimate in the presence of rounding errors Richardson extrapolation is a well known technique in scientific computing. It is used to compute error estimates when our approximations can be viewed as a function of a parameter$h$. Common ... 1answer 116 views ### Evaluation of$\int \prod_{j=1}^u \frac{x+j}{j-x}~dx$Let$I_{n,k} = \frac{(n+k)!}{k!(k-1)!(n-k)!}$. This is a sort of generalization of the Apéry's numbers, with$I_{n,n} =$the$n$-th Apéry number. I am studying integrals of the form: $$f_u(x)=\int \... 1answer 264 views ### Theory of integration for functions from \mathbb{Z}_{p} to \mathbb{Z}_{q} for distinct primes p,q Let p and q be prime numbers. When p=q, Mahler's Theorem gives a complete description of C\left(\mathbb{Z}_{p};\mathbb{Z}_{p}\right), the space of continuous functions from \mathbb{Z}_{p} to ... 1answer 121 views ### Orlicz-Sobolev Spaces let A an N-function and suppose that$$\int^{+\infty}_1\frac{A^{-1}(\tau)}{\tau^{1+\frac{1}{n}}}d\tau=+\infty $$we denote by \widehat{A} an N-function equal to A near infinity and \widehat{A} ... 1answer 82 views ### Sign of expectation value Consider a multivariate Gaussian-type measure$$d\lambda(x):=\nu_{\mu,\Sigma} e^{-\langle (x-\mu), \Sigma^{-1}(x-\mu) \rangle - \vert x \vert^2} $$with vector \mu \in \mathbb R^n and \Sigma ... 1answer 131 views ### L_p(I,Y)^\perp=L_q(I,Y^\perp)? Let X be a Banach space and Y be a closed subspace of X. For 1<p<\infty consider the p-th power Bochner Integrable functions which takes values in X and defined on the unit interval ... 1answer 1k views ### Existence and uniqueness of Haar measure on compacta; a cohomological approach I am trying to use a modification of group cohomology to prove the existence and uniqueness of Haar measure on a compact Hausdorff group. I think the best way of introducing the idea I am pursuing is ... 0answers 184 views ### The collection of mean value abscissas in the Mean value theorem The integral mean value theorem for continuous f on [0,b] and finite positive continuous measure \mu we have$$\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)=f(c)(*)$$for at least one c\in [a,b]. We ... 1answer 204 views ### Fubini's theorem on arbitrary foliations In what follows \mathbb{R}^{n+m} = \{(x,y): x \in \mathbb{R}^n, \ y \in \mathbb{R}^m \} \ . Suppose G: U \to V is a C^1-diffeomorphism from an open subset of a manifold to an open subset of ... 0answers 125 views ### Functions that are Khinchin integrable but not Henstock-Kurzweil integrable I posed this question on Mathematics SE recently, though by the total lack of attention it has gotten, I do not anticipate an answer and bring it here. What are some Khinchin integrable f which ... 0answers 85 views ### Conceptual meaning of a non-linear relation connecting 6 Mordell integrals? Define Mordell integral by$$ \phi_\alpha(\theta)=\int\limits_0^\infty\frac{\cos\pi \theta x}{\cosh \pi x}\,e^{-\pi \alpha x^2}dx.\tag{1} $$There are a lot of linear relations connecting integrals ... 0answers 48 views ### Expression of divergent integrals via simplier divergent integrals (or series) Yesterday I just came to a formula that may allow to express divergent improper integrals bounded (on any finite interval) functions in terms of simple improper integrals or series.$$\int_0^\infty f(... 0answers 120 views ### Why is the divergence theorem used in the Eells-Sampson paper slightly different from that in a textbook? I am reading Harmonic Mappings of Riemannian Manifolds by Eells and Sampson. In chapter 2, the author(s) used the divergence theorem, which does not look like the usual divergence theorem for ... 0answers 140 views ### The Poincaré Lemma Let me consider an$L^1(\mathbb R^N)$function$f$such that $$\int_{\mathbb R^N} f(x) dx =0.$$ Then I claim that the$N$-form$f(x) dx_1\wedge\dots\wedge dx_N$is closed, i.e. there exists a vector ... 1answer 211 views ### How can we do a Gaussian integral over matrix elements? I am integrating the following Gaussian over all possible matrix elements$J_{ij}$: $$I=\int \exp{\left\{-a\sum_{ij}J_{ij}^2+b\sum_{ij}J_{ij}+c\sum_{ij}J_{ij}J_{ji} \right\}} \left (\prod_{ij}\mathrm{... 1answer 87 views ### Integration of a particular rational expression [closed] I am trying to solve the following integration, where a,b,c,d,e and f are constants:$$I=\int\frac{x^4+ax^3+bx^2+cx+d}{x^3(x^3+ex+f)}dx$$I tried to solve the integral using the following two ... 3answers 207 views ### Is there a good approximation for this Gaussian-like integration? Is there an analytic solution or approximation for the following Gaussian-like integration? \frac{1}{\eta^{2n}} \frac{1}{\sqrt{2 \pi}} \int_{-\eta}^{+\eta} e^{-x^2/2} x^{2n} dx? The numerical plot ... 0answers 114 views ### Hadamard lemma without integration Let I be the ideal of smooth germs vanishing at zero. Let I^{k+1} be the ideal generated by (k+1)-fold product of such germs. Write F_k for the ideal of k-flat germs at zero. By the product ... 0answers 169 views ### Adaptive Simpsons Quadrature Algorithm for Double Integrals? [closed] I'm currently using Numerical Analysis 10th edition by Richard L Burden as a reference for approximate Integration techniques. In there it describes the Adaptive Simpsons Quadrature rule that inputs ... 1answer 200 views ### Integral g(a)= \int_{0}^{\frac{\pi}{2}} \frac{\arctan(a \tan x)}{\tan x}dx [closed] I am having trouble calculating this integral:$$ g(a) = \int_{0}^{\frac{\pi}{2}} \frac{\arctan(a \tan x)}{\tan x}dx $$I tried calculating g'(a) but then I get stuck. 0answers 71 views ### Can we solve this integral equation? Let (E,\mathcal E,\lambda) be a measure space, p,q_i be positive probability densities on (E,\mathcal E,\lambda) for i=1,2, \mu:=p\lambda, \sigma_{ij}:E^2\to[0,\infty) be \mathcal E^{\... 0answers 79 views ### Is harmonic mean of linear functions a Bernstein function? According to some experiments I've been running, for any n and non-negative a_1, a_2, \ldots a_n, the following function: f(t) = \frac{n}{\sum_{i=1}^n 1/(a_i+t)} is a Bernstein function, ... 0answers 77 views ### Expressing 1-e^{-z} as a Fourier integral According to the theory of screw functions and screw lines by John Von Neumann and Issai Schoenberg (see here), any function F:\mathbb{R} \rightarrow \mathbb{R} such that F(|x_i - x_j|) = \|f(x_i)-... 0answers 38 views ### moment generation function for matrix Gaussian distribution What is the moment generation function for the following function$$ E(e^{\mathbf{X}^T\mathbf{y}\mathbf{W}})=\int e^{\mathbf{X}^T\mathbf{y}\mathbf{W}}\frac{\exp\big(-\frac{1}{2}\mathrm{tr}\big[\mathbf{... 1answer 132 views ### Integration of a particular quartic form I would like to solve the following integral: \begin{equation} \int \prod_i d x_i e^{a x_i^2 + b x_i^4 + c x_i^2 x^2_{i+1}} \end{equation} This integral can be for sure lead back to a common ... 1answer 130 views ### Bounds on expectation of$X/(X^2 + c)$with$X$~ Gaussian and$c > 0$I'm trying to compute expectation of$X / (X^2 + c)$when$X$is normally distributed with mean$\mu$and variance$\sigma^2$, and$c$is some positive constant. I think this cannot be solved ... 1answer 646 views ### On the closed-form of$\int_0^1\int_0^1\int_0^1\frac{dxdydz}{1-\frac{z}{3}(x+\sqrt{xy}+y)}\$
I would like to know if it is possible to calculate in closed-form, or well what work can be done about it, the definite integral $$\int_0^1\int_0^1\int_0^1\frac{3dxdydz}{3-z(x+\sqrt{xy}+y)},\tag{1}$$ ...