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Questions tagged [integral]

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2
votes
1answer
141 views

Integrating nasty gaussian over square root

TLDR: trying to solve, $$\int_1^\infty \exp\left(-\frac{x^2}{2\omega^2}\right) \frac{1}{\sqrt{ax^2+bx-1}}dx$$ After doing some reading and looking at some other questions 1, 2 (and even going through ...
1
vote
0answers
108 views

How to show that this function is continuous (Geometric Measure Theory)

I want to prove that the function $F: \mathbb{R}_+ \to \mathbb{R}$ defined by $$F(t)=\int_{\{d=t\}} g \, d\mathcal{H}^{n-1}$$ is continuous if $g:\Omega \subset \mathbb{R}^n\to \mathbb{R}$ is ...
1
vote
0answers
49 views

Leibniz rule for Hadamard derivative?

I'm having difficulty computing the Hadamard derivative of the following functional $\tau(F)$, where $F$ is a distribution function: $$ \tau(F)=\int_{1-F(0)}^1 F^{-1}(s)ds. $$ The difficulty here is ...
2
votes
0answers
44 views

What restrictions on the form of an integral equation have a unique solution f=0?

We're stuck on the following question in a problem relevant to a physics paper on AdS/CFT that we are working on. Given a Fredholm equation of second kind with the form $f$+$\int_D K f\,dx = 0$, where ...
2
votes
0answers
109 views

How to solve this integral equation?

$$g(p) = \frac{1}{p^3} + \frac{2}{p^2} \int_{p}^{1}{(u-p)g(u)du}$$ Need to find $g(p)$ for $p > 0$. If there is no explicit solution, how to solve it numerically? Maple13 calculates bad results, ...
0
votes
0answers
34 views

Rewriting an elliptic integral in terms of theta functions

I wish to demonstrate the following from arXiv:hep-th/9808043v2 (equations (3.2) to (3.4)). This is rewriting the following incomplete elliptic integral of the third kind $ \Phi_{\tilde{h}}(h) = \...
1
vote
2answers
207 views

Integral of product of Gaussian pdf and cdf [closed]

$\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ is the pdf of a standard normal distribution. $\Phi(x) = \int_{- \infty}^x \phi(t) dt$ is the cdf of a standard normal distribution. How does one ...
1
vote
1answer
177 views

integrate exponential function of quadratic form over unit norm vectors

Let $A$ and $b$ be an $M\times M$ matrix and an $M\times 1$ vector, respectively. I need to solve $$\int_{\|x\|^2=1} \exp\left(-x^\ast Ax + 2\mathcal{R}\{x^\ast b\}\right) \, dx$$ where $(\cdot )^\...
1
vote
1answer
311 views

I want to disprove an equality involving a double integral

I want to show that the following equality does not hold: \begin{equation}\label{at3} \frac{\lambda^2-1}{2}x^2-\int_{-\infty}^{\infty}\!\!\int_{-\infty}^{\infty}K(y_1,y_2,x)\ln g(y_1,y_2)dy_2dy_1=...
0
votes
1answer
81 views

Change of variables for double integral [closed]

Thank you for your time. My basic question is whether the following change of variables allowed $$\int_0^a \int_0^b f(a-b)g(b-c)h(c)\,dc\,db = \int_0^a \int_0^b f(c)g(b-c)h(a-b)\,dc\,db$$ I fail to ...
3
votes
1answer
229 views

Asymptotic behaviour of function from integral representation

In Appendix A of this paper, it is claimed that the asymptotic behaviour of $$\phi_1(y,\lambda)=\frac{1}{\Gamma(\frac{1-\lambda}{2})}\int_0^\infty dt~e^{-t}\cos(2y\sqrt{t})t^{-\frac{1+\lambda}{2}},$$ ...
0
votes
2answers
124 views

Literature about the integral of Bessel $\int_0^x I_{0,1}(u) e^{-a u}du$?

Thanks to sound remarks here and here, and looking again at these equations, I noticed that my puzzle boils down to these 2 special functions: $$\int_0^x I_i(u) e^{-a u}du, \quad i=0,1$$ where $I_n(u)$...
6
votes
2answers
243 views

What is $\int_{0}^{z} e^{-a^{2} x^{2}} {\rm erf}(bx)\, dx$?

The integral $$\int_{0}^{z} e^{-a^{2} x^{2}} {\rm erf}(bx)\, dx$$ is related to the convolution of two half-normal distributions. This can be inferred from this question on MSE. The following ...
2
votes
1answer
141 views

Integral Geometry

Consider a smooth curve $\gamma$ of finite length in the unit square $[0,1]\times[0,1]$. Is the following statement correct? There exists a Lebesgue null set $N$ such that for all $x\in [0,1]\...
2
votes
1answer
145 views

Nested trigonometric integral

I am trying to solve the following: $$\int_{0}^{2\pi}\mathrm{d}\phi\,\frac{\sin(a\cos\phi)}{1+b\cos\phi}$$ with $-1 < b < 0$. I started with $\cos\phi = \operatorname{Re}[z]$, but it led to ...
-1
votes
1answer
90 views

how to solve the rational exp integral? [closed]

I want to solve this rational exp function but i don't know how! $$\int_{\nu}^{\infty} \frac{ax}{bx+c} \exp\{-\alpha x-\frac{\beta}{x}\}$$
0
votes
0answers
64 views

Closed form expression of multi-dimensional integral across the multi-dimensional cube

Let $\mathbf{y}$ be a complex-valued $M\times 1$ vector and $\mathbf{B}$ a complex-valued $M\times N$ matrix. Let $\mathbf{b}_j$ denote the $j$:th column of $\mathbf{B}$. I am wondering if there is a ...
1
vote
0answers
66 views

Integral equation with a function limit

I am doing a research paper and I faced with this problem. Does anyone know the answer (or an approximation for the answer) for the following integral equation? $$\int_0^{y(x)} e^{-xt}f(t)dt = g(x)$$ ...
13
votes
1answer
418 views

Average measure of intersection of a convex region with its translate

Let $\lambda$ denote the Lebesgue-measure on $\mathbb{R}^n$, and let $C\subset\mathbb{R}^n$ be a convex region. My question is about $$f(C):=\int_{C} \lambda(C \cap (x + C) ) \mathrm{d} x.$$ How ...
0
votes
0answers
54 views

How can I analytically integrate the exponent of a polynomial?

I am trying to derive the mean transition path time across a specific one-dimensional potential of mean force, $U(x) = V_0 (x^2 - a^2)^2 + (1/2)k(x-b)^2$ And I essentially need to evaluate the ...
4
votes
1answer
286 views

Integral involving Laguerre, Gaussian and modified Bessel function

I am trying to prove that the integral \begin{align} \int_{0}^{\infty } e^{-\frac{r^2}{2B}} r^{l-n} L_n^{l-n}\left(\frac{r^2}{C}\right) I_{l-n}\left(\rho r \right) r dr \end{align} has ...
3
votes
1answer
178 views

the Bochner integral about a semigroup of bounded linear operator on a Banach space

Let $T(t)$ be a semigroup of bounded linear operator on a Banach space $X$. When does the following hold $$ \int_0^t T(s)x ds = (\int_0^t T(s) ds)x, x \in X \, , $$ where $ t \in (0,1)$.
52
votes
4answers
3k views

An interesting integral expression for $\pi^n$?

I came on the following multiple integral while renormalizing elliptic multiple zeta values: $$\int_0^1\cdots \int_0^1\int_1^\infty {{1}\over{t_n(t_{n-1}+t_n)\cdots (t_1+\cdots+t_n)}} dt_n\cdots dt_1.$...
4
votes
0answers
226 views

English language and Mathematics

I have a question maybe more relevant to an English language section of StackExchange, but I doubt that anybody but a Mathematician could properly answer my question. Let $\mathcal M$ be a smooth ...