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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291 views

Turning a Line Integral into a Contour one

I'm trying to compute an integral appearing in the article "On Determinants of Laplacians on Riemann Surfaces" of D'Hoker and Phong (page 541). It is as following. Fix $B\in \mathbb{R}_+$ and let ...
3
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0answers
85 views

Show that $\int_{\alpha}\frac{1}{z}\, dz=\int_{\beta}\frac{1}{z}\, dz$.

Let $a$ and $b$ be positive real numbers. Define ways $\alpha,\beta\colon [0,1]\to\mathbb{C}$ via $$ \alpha(t):=a\cos(2\pi t)+ia\sin(2\pi t),~~~~~\beta(t):=a\cos(2\pi t)+ib\sin(2\pi t). $$ Show ...
3
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218 views

Understand integral from Gradshteyn and Ryzhik book “Table of integrals, series, products”

I was checking useful integrals in this book. I have found one (6.298) that is what I need, but I don't understand how every step towards the final result works. ...
3
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474 views

spectral integral

I am learning spectral integration for my summer. I am stuck at a point. Having got hold of a spectral measure, we define the spectral integral of a simple function as usual and then approximate any ...
3
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77 views

What is $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$?

I want to compute the following integral: $\int_{-\infty}^{\infty} \frac{e^{-\alpha t} \cos[t + y]}{1+\beta e^{-2\alpha t} } dt$ with $\alpha, \beta, c$ real constants, and $\alpha>0,\beta=0$. ...
3
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117 views

A photon in expanding Universe (a snail on a tree)

I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of ...
3
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0answers
103 views

Differential equation $y'(t) = 1-y(t) e^{y(t)-1}$

I am interested in finding a clean explicit solution (if possible) to the differential equation $$ y'(t) = 1-y(t) e^{y(t)-1}, $$ where $0 \le t < 1$ and $0 \le y \le 1$. This can obviously be ...
3
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144 views

computing a difficult integral using software

I'd like to compute the following integral. I've tried SAGE but it just runs for 15 minutes then stops.. not sure what that means. If anyone wants to take a crack with mathematica or anything, please ...
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117 views

Integral of the Normal Characteristic Function

The characteristic function of the $N$-variate Normal distribution is $$\forall \mathbf{t} \in \mathbb{R}^N \quad \psi(\mathbf{t}) \equiv \mathbb{E}\left( e^{i\mathbf{t}X}\right) = \exp \left( i{ ...
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258 views

Prove Heisenberg uncertainty principle (measure and integration theory)

Here is a question in measure and integration theory, Let $f$ be a continuously differentiable complex function on $\mathbf{R}$ s.t. the functions $x \mapsto xf(x)$ and $f'$ are in ...
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130 views

Integration by parts of a normalized function - [copied from Physics.SE]

By using integration by parts, I need to show for $$A = \frac{\mathrm d}{\mathrm dx} + \tanh x, \qquad A^{\dagger} = - \frac{\mathrm d}{\mathrm dx} + \tanh x,$$ that ...
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188 views

Simplify the integral with error function

$\newcommand{\erf}{\operatorname{erf}}$ I have the following integral and I need to simplify the solution. I have written first two steps. I don't know what is the value of $$ \erf(\infty) $$ I ...
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341 views

Under which hypotheses is switching between sum and integral signs legit?

Which hypotheses are needed to change the order of sum and integral signs? Concrete example: consider the expression $$ ...
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175 views

Simplifying an integral arising in Physical Chemistry

I am struggling to understand the following transition (encountered in a paper on Physical Chemistry). Let $$D=\frac{\tau_0^{-1}\int_0^\infty G(t)dt}{1-\tau_0^{-1}\int_0^\infty G(t)\int ...
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401 views

Is there a way to Fourier transform Cos[Sin[x]]

In my physics problem, I encountered a solution has the form like Cos[Sin[t]], and I need to do the Fourier transform to this solution. Is there a way to do the Fourier transform analytically to ...
3
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95 views

Integration of sine^2 w.r.t. some norm

Let $||x||$ be any norm over $\mathbb R^n$. Let $B_T$ the open ball with radius $T$ w.r.t. to our norm, i.e. all $x\in\mathbb R^n$ such that $||x||<T$. Let $n\in\mathbb N$. How much ...
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404 views

Riemann integral vs Lebesgue integral

Let $f$ be analytic on a domain $\Omega$ of the complex plane, such that the closed disc $\overline{D(0,R)}$ is contained in $\Omega$. What is the difference between $$ \int_{D(0,R)}|f(w)|dm(w)$$ and ...
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133 views

If $f$ is quasi and left continuous on $[a,b]$ and $\alpha$ is increasing and right continuous then $f$ is Riemann-Stieltjes integrable over $[a,b]$

Question: If $f$ is quasi and left continuous on $[a,b]$ and $\alpha$ is increasing and right continuous then $f$ is Riemann-Stieltjes integrable over $[a,b]$ with integrator $\alpha$. ...
3
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182 views

Hazard Rate Probability HW Question

I am working on a homework problem below from Pitman's Probability book: Suppose the failure rate is $\lambda$($t$) = $at$ $+$ $b$ for $t$ $\geq 0$. The problem asks to find the formula for the ...
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460 views

Integral of two error functions (erf)

In my research I came across the following integral: \begin{equation} ...
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64 views

Can I solve for a unique integral kernel?

Consider, for $\mathbf{v},\mathbf{w} \in \mathbb{R^3}$, $$ f(\mathbf{w}) := \int K(\mathbf{w,\mathbf{v}}) g(\mathbf{v}) \, d\mathbf{v} \, .$$ Is it possible to solve for the integral kernel, ...
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884 views

Integration of nontrivial trigonometric functions

First an example which I know how to solve. If we have the following integral $$\int_{-\pi}^{\pi}\frac{1}{1+3~\cos^2(t)}dt$$ there is a very practical way to evaluate it by interpreting it as some ...
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130 views

Equivalence of integrals

Let $x_1, \ldots, x_n$ be vectors in the normed space $(X, \|\cdot\|)$. Let $\mu$ be the Lebesgue measure on the cube $[-1,1]^n$. Denote vectors in $[-1,1]^n$ by $y=(y_1, \ldots, y_n)$. Are the ...
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107 views

integral equation solution for two functions $ f(x) $ and $ g(x) $ and see if they are related

given two functios $ f(x) $ and $ g(x) $ related by $$\frac{ \Gamma(s-1/2)}{\Gamma(s) \sqrt{ \pi}}\int_{0}^{\infty}dx \frac{g(x)dx}{(x+y)^{s-1/2}}=\int_{0}^{\infty}dx \frac{f(x)dx}{(x+y)^{s}}$$ what ...
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202 views

Integral of a product

Given the following integral: $$I(N)=\int_{0}^{t}\prod_{k=1}^{N}\sin(k\omega \tau)d\tau$$ does someone know if is it possible to find the solution of $I(N)$ in a closed form? I'm able to find the ...
3
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64 views

How to approach an integral over $g(\cos(t))$ from $0$ to $2\pi$, where $g(x)$ is nasty?

For notational convenience, let $f(t) = a^2 + 2 a b \cdot \cos(t) + b^2$, where $a,b$ are both positive real constants and $t$ will be the integrand of the integral, which is supposed to be carried ...
3
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200 views

Pontryagin's maximum principle problem, minimising an integral

I am really stuck on the following: Use Pontryagin's maximum principle to show that $\pi$ is the minimum value of the integral $\frac{1}{2} \int _{0} ^{1} u^{2} + v^{2} \,dt$ subject to constraints ...
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334 views

Taking derivative below an integral

I am trying to solve the following question: If $t>0$, then \begin{align*} \int_{0}^{+\infty} e^{-tx} \; dx = \frac{1}{t} \end{align*} Moreover, if $t \geq a > 0$, then $e^{-tx} \leq ...
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183 views

What is the Dunford Integral and why is it useful?

Wikipedia defines the Pettis Integral for Banach space valued functions on a measure space by duality. Apparently there is a Dunford integral which specializes to the Pettis integral. What is its ...
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427 views

Lebesgue Line Integrals - Parametric Change of Variables

Consider the following Lebesgue integral in $\mathbb{R}^n$ $$ \int_C f(x) dx $$ Where $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is measurable and $C$ is a measurable subset of $\mathbb{R}^n$ that ...
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94 views

Asymptotics of Riemann-Lebesgue type integral

How to show that for $u \in L_{\mathbb{C}}^2$ and $a>0$, $$\int_0^a u(t) \sin{\sqrt{\lambda}t} \,dt = o(e^{|Im\sqrt{\lambda}|a}),\text{ as } |\lambda| \rightarrow \infty$$ Note that $\lambda$ ...
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54 views

understanding of the rank-2 tensor as an integral kernel

Let $G\subset{\mathbb R}^m$ be a nonempty compact and Jordan measurable set that coincides with the closure of its interior. Let $K:G\times G\to{\mathbb C} $ be a continuous function. Then we can ...
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56 views

Integrability of $\frac{k-1}{2(1+|x|^k)}$

Is the following function known to be integrable? It is supposed to be a probability density function, i.e., integrates to one. However, it leaves the online Mathematica integrator stumped: ...
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224 views

Change of variables in line integral with abs. value

Let $\gamma : I \rightarrow \mathbb C$ be a path. Let $g: \mathbb C \rightarrow \mathbb C$ be a biholomorphic map. Let $f$ be a holomorphic function. Consider the integral $$ \int_{g\circ \gamma} ...
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378 views

Seeking rationale for Hadamard's finite part of a divergent integral

I have a problem justifying the throwing away the divergent term in order to obtain Hadamard's finite part. I find this step to be highly unusual and it is not obvious to me how the resulting ...
3
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235 views

Series of nested double integrals

This is kind of a follow-up of my previous question. I'm investigating the following infinite series of nested two-dimensional integrals $$\sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 ...
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368 views

What is the expected value of modified Dirichlet distribution? (integration problem)

It is easy to produce a random variable with Dirichlet distribution using Gamma variables with the same scale parameter. If: $ X_i \sim \text{Gamma}(\alpha_i, \beta) $ Then: $ ...
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26 views

Imaginary number and absolute value integral - Fourier transform

I came across this integral problem: $$\hat f(\xi)=\int_{-\infty}^{+\infty} e^{-|x|+xi\xi}dx$$ Now I know how to integrate simple absolute value functions like: $\int_{-2}^{4}|x-2| dx$, we just ...
2
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0answers
36 views

Asymptotic behaviour of an integral depending on a parameter

I am trying to compute the asymptotics on $t$ of the following integral: \begin{equation} I(t)=\int_{\mathbb{R}^{n}}e^{-|\lambda|^{2}/2t}\prod_{i<j}\left( e^{\lambda_{j}/t}-e^{\lambda_{i}/t} ...
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17 views

Changing integration bounds

I came accross this line: $$x\in [0,1],y\in [0,1]$$ $$E(Y|x)=\int_y y*g(y|x)dx=\int _0^1y*g(y|x)dy$$ Can someone please explain how the second equality holds! Thanks.
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42 views

How to prove that the following function has a unique mode?

I am trying to prove that the function $$f(\alpha)=n\ln \alpha-n\ln\Big(\sum_{i=1}^{n}t_i^\alpha+\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx\Big)+(\alpha-1)\sum_{i=1}^{n}\ln t_i,$$ where ...
2
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0answers
97 views

How do I evaluate this special type of integral

Does the following function can be simplified or solved? $$R(i) = \int_{y\in S} {\frac{{w(y) g(y,i)_{}^\sigma }}{{\int_{x\in S} {h(x)g(x,y)_{}^\sigma f(x,y)_{}^\sigma dx} }}dy} $$ where S is a ...
2
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25 views

Volumes by Cylindrical Shells - What am I doing wrong?

I am trying to solve this exercise from a textbook: $y = x^4, y = 0, x = 1;$ rotated about $x=2$ This is my attempt at solving the problem: Shell radius: $2 - x$ Shell height: $x^4$ $a = 1$ $b = 2$ ...
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30 views

Does the quadratic covariation process define a measure?

In the context of stochastic integration (when we define the space $L^2(M)$), we define the (possibly infinite) measure $$P_M = P \otimes [M]$$ by $$E_M[Y] = E\left[\int_0^\infty Y_s(\omega) ...
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64 views

Conjecture: $\int_0^{\infty}dx\frac{e^{i\alpha\sqrt{x^2+1}}}{\sqrt{x^2+1}}J_1(Qx)=\left(e^{i\alpha}-e^{i\sqrt{{\alpha}^2-Q^2}}\right)/Q$

Here $\alpha>0$, $Q>0$, and $J_1$ is a Bessel function. I'm fairly certain the closed form in the title is accurate for a couple of reasons. First, I've evaluated the integral numerically in ...
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48 views

Sine improper integral

Suppose the following integral $$ \int\limits_{-\infty}^{\infty}\sin{x}dx $$ In mathematical rigor, the following is the definition $$ \int\limits_{-\infty}^{\infty}\sin{x}dx = ...
2
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60 views

Inner Product of Chebyshev polynomials of the second kind with $x$ as weighting

I have tried to solve the integral $${\int_0^1 U_n (x) U_m (x)x dx },$$ where ${U_n (x) }$ denotes Chebyshev polynomial of the second kind. Solving the integration and checking the result, I ...
2
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38 views

How to evaluate this integral with a bessel function

I have to evaluate the following two integrals that I would like to solve, Sin[θ] 2 π BesselJ[0, k Sin[θ] (μ[j] - μ[i])] Exp[2 k^2 σ^2 Cos[θ]^2], and ...
2
votes
0answers
18 views

integral derivation?

i have a problem in the partial derivatives of this function : so consider this integral : $u(x,t)=\int_0^t(\int_{\mathbb{R}^n}\phi_n(y,s)f(x-y,t-s)dy)ds$ can somone tell me why we have this ...
2
votes
0answers
19 views

proofing pyramid volume formula using integration.

I'm studying the proof of pyramid volume formula using integration. In the video minute 12:10 when the professor already got the $S$ variable which is the side of the pyramid base he continues to do ...