# Questions tagged [integral]

The tag has no usage guidance.

24 questions
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 ...
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 ...
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 ...
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 ...
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, ...
34 views

311 views

### I want to disprove an equality involving a double integral

I want to show that the following equality does not hold: \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=...
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 ...
229 views

### Asymptotic behaviour of function from integral representation

In Appendix A of this paper, it is claimed that the asymptotic behaviour of $$\phi_1(y,\lambda)=\frac{1}{\Gamma(\frac{1-\lambda}{2})}\int_0^\infty dt~e^{-t}\cos(2y\sqrt{t})t^{-\frac{1+\lambda}{2}},$$ ...
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 $I_n(u)$...
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 ...
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]\... 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 ... 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! $$\int_{\nu}^{\infty} \frac{ax}{bx+c} \exp\{-\alpha x-\frac{\beta}{x}\}$$ 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 ... 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? $$\int_0^{y(x)} e^{-xt}f(t)dt = g(x)$$ ... 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 ... 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,$U(x) = V_0 (x^2 - a^2)^2 + (1/2)k(x-b)^2And I essentially need to evaluate the ... 1answer 286 views ### Integral involving Laguerre, Gaussian and modified Bessel function I am trying to prove that the integral \begin{align} \int_{0}^{\infty } e^{-\frac{r^2}{2B}} r^{l-n} L_n^{l-n}\left(\frac{r^2}{C}\right) I_{l-n}\left(\rho r \right) r dr \end{align} has ... 1answer 178 views ### the Bochner integral about a semigroup of bounded linear operator on a Banach space LetT(t)$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)$. 4answers 3k views ### An interesting integral expression for$\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.$...
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 ...