Questions tagged [error-function]

Question 1

Since I have to teach a course involving some statistical hypothesis tests, while not being a statistician myself, I am looking for some simple, but possibly accurate, bounds for cumulative ...

Question 2

Is there a closed form for this integral? $$\int_{0}^\infty e^{-ax^2}\operatorname{erf}(px)\operatorname{erf}(qx)\operatorname{erf}(rx)\operatorname{erf}(sx)\,dx$$ Recall the definition of the error ...

Question 3

I have a double integral, which I want to solve: $\mathcal{I}_{au}=\int_0^{t'}e^{(a-c)t''}\int_0^{t''}e^{(-b+c)t'''}\text{exp}\big(-\frac{\lambda^2}{2}\left( S_2(t''')^2+S_1(t')^2+S_1(t'')^2+2S_{12}t'...

Question 4

This is a generalization of another question from here.. Let $n \ge 2$ be an integer. Let $\vec{a}:=(a_i)_{i=1}^n \in {\mathbb R}_+^n$ and ${\bar a}:=(a_{i,j})_{1\le i < j \le n} \in {\mathbb R}_+^...

Question 5

Consider the following integral over $\mathbb{R}^{q+1}$ $$I_q(s) := \int_{\substack{y_1, \ldots, y_q > s \\ y_{q+1} <s}} \exp\left(-Y\Sigma Y^T\right) d\lambda(Y),$$ where $Y = (y_1, \ldots, y_{...

Question 6

Being interested in the result of $$\int_0^1 \sin \sqrt{-\ln x} \ \mathrm dx= -\frac{\sqrt{\pi}}{2 \sqrt[4]{e}} ,$$ I tried the substitution $y=\sqrt{-\ln x}$, then $x=e^{-y^2}$ and $\mathrm dx=-2ye^{...

Question 7

I'm trying to evaluate $$\int \mathrm{erf}(\sin x)dx$$ where $\mathrm{erf}(x) := \frac{2}{\sqrt{\pi}}\int_0^t e^{-t^2}dt$ is the error function. I want to write the answer as in terms of an ...

Question 8

Recently, I have been studying estimates on error correcting codes in relation to $\mathbb{Z}_{2}$ actions on manifolds positive sectional curvature. In this paper, in particular, I came across the ...

Question 9

I am trying to find a version of the following integral in terms of special functions: $$ \int _{\alpha }^{\beta }\frac{e^{-r^2 \sigma ^2} \text{erf}(\gamma \sigma )}{\sigma }d\sigma \quad (1) $$ for ...

Question 10

I'm trying to prove that $\Phi\left((1-\epsilon)\sqrt{2\log(n)}\right)^n \to 0$ as $n \to \infty$ (where $\Phi$ is the standard normal CDF) for any $\epsilon > 0$. I'm having problems. I've tried ...

Question 11

I have the error function: $$\text{erf}(x + iy) = \frac{2}{\sqrt{\pi}} \int_0^{x + iy} e^{-t^2} dt$$ I'm looking for an upper bound on $|\text{erf}(x + iy)|$ in terms of $y$. I know that for $y=0$ a ...

Question 12

I have been studying heat kernels for elliptic operators and working with the eigenfunction expansion method. During my asymptotic analysis of the Green’s function, the following integral emerged from ...

Question 13

A classic exercise is to demonstrate that $ e^\pi > \pi^e$ without direct computation, and there are alot of ways to do that. I'm curious to see if there is a geometric way to prove it. My idea is ...

Question 14

Suppose that $\Omega$ is a convex polygonal domain in $\mathbb{R}^d (d=2,3)$ with the Lipschitz continuous boundary. Define elliptic projection $$(\nabla u,\nabla v)=(\nabla Pu,\nabla v),v \in W^h \...

Question 15

I am used tot he following formulation of a quadric error matrix. Given a plane normal $n$ and a distance $d$ from the origin along the normal, we can define an implicit equation for the plane as $p \...

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Questions tagged [error-function]

Good known bounds for cumulative distribution functions

Closed form for $\int_{0}^\infty e^{-ax^2}\operatorname{erf}(px)\operatorname{erf}(qx)\operatorname{erf}(rx)\operatorname{erf}(sx)\,dx$?

I need help with solving a double integral exp of a quadratic variable

Multivariate Gaussian integral over a hypercube

Integral of a multivariate Gaussian over a cone

Need help to evaluate $\int_0^1 \sin^{2n} \sqrt{-\ln x} d x$.

Can we evaluate $\int \mathrm{erf}(\sin x)dx$ in terms of antiderivatives of elementary functions?

On the extension of a bound in the theory of error correcting codes for the field $\mathbb{Z}_{p}$

How to express an integral involving an erf,rational function and a gaussian in terms of special functions?

Limit of normal CDF

Absolute value of the complex error function

Closed form for an integral involving special functions

Prove $ e^\pi > \pi^e$ geometrically

How to derive $H^{-1}$ elliptic projection error ？

Why is this quadric error matrix correct?

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