Questions tagged [error-function]
Use this tag for the error and complementary error functions (erf and erfc). These are special functions formed by taking definite integrals of the Gaussian/normal distribution function.
611 questions
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I need help with solving a double integral exp of a quadratic variable
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'...
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Multivariate Gaussian integral over a hypercube
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}_+^...
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Integral of a multivariate Gaussian over a cone
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_{...
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Need help to evaluate $\int_0^1 \sin^{2n} \sqrt{-\ln x} d x$.
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^{...
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Can we evaluate $\int \mathrm{erf}(\sin x)dx$ in terms of antiderivatives of elementary functions?
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 ...
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On the extension of a bound in the theory of error correcting codes for the field $\mathbb{Z}_{p}$
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 ...
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How to express an integral involving an erf,rational function and a gaussian in terms of special functions?
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 ...
- 1,707
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Limit of normal CDF
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 ...
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Absolute value of the complex error function
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 ...
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Closed form for an integral involving special functions
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 ...
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Prove $ e^\pi > \pi^e$ geometrically
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 ...
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How to derive $H^{-1}$ elliptic projection error ？
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 \...
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Why is this quadric error matrix correct?
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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How to relate a sequence following an error fuction and another sequence derived from the first sequence using their standard deviations?
I am an engineering student and want to derive a reasonable relation between a sequence derived from a normal distribution and the other seqence derived from the first sequence.
I have a sequence of ...
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Plotting complimentary error function issues
My problem is that the y-axis for when running the matlab code below starts at $x/S_e = 0$ when it should start at $x/S_e = 0.6$. I've attached the correct plot and my wrong plot titled 'Function F(...
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