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.

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12 views

Limit/convergence of a series of two shifted error functions

I try to find the limit (or at least the proof for convergence) of the infinite sum $$\sum_{\substack{k=-\infty \\ k \neq 0}}^{\infty} e^{-\frac{a}{b}ck} \left| \text{erf}\left( \frac{a + i(b-ck)}{\...
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1answer
33 views

Asymptotic rate of decrease of error function

The complementary error function is defined as $$ \text{erfc}(x) = 1 - \frac{2}{\sqrt{\pi}}\int_0^{x} e^{-t^2} dt $$ and is related to the Gaussian (Normal) distribution. Is there an approximation of ...
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2answers
55 views

Evaluating erf(x) using Taylor's series

I tried to evaluate error function using Taylor series by using its definition $$ erf(z) = \frac{2}{\sqrt{\pi}}\int_0^ze^{-t^2}dt$$ I've used Taylor expansion to evaluate this integration and i got ...
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21 views

finding $h$ that make absolute instability for Euler method

I have an Euler method that has this form: $$\hat{I}(t_{n+1}) = \hat{I}(t_{n})+h\beta \hat{I}(t_{n})[1-\frac{\hat {I}(t_{n})}{N}]$$ which can also be written like $$\hat{I}(t_{n+1})=\phi (\hat{I}(t_{...
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8 views

Minimizing matrix cost function

I have a very basic knowledge of minimization problems mainly limited to curve fitting. I need help to understand how to minimize the following cost function with respect to $X$. $$||Y-XD^T||^2_2+\...
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25 views

Interpreting the Weighted Pinball Loss

I am participating in a Kaggle competition and came across the Weighted Pinball Loss function for the first time and I don't quite understand it. Here is a screen shot of the definition provided by ...
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35 views

How to do an Inverse Fourier Transform of this equation?

I need to get the inverse Fourier transform of the following equation so that I can design a FIR filter from it: $${\rm magnitude} = g * e^{-cf^{2} -\frac{a+b\sqrt{hf}}{mf}}$$ Where: $a$, $b$, $c$, $...
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13 views

Finding upper bound and uniform convergence

For every positive integer $m$, let $x_0^{(m)}$ < $x_1^{(m)}$ < ... < $x_m^{(m)}$ be $(m+1)$ distinct points in $[0, \pi]$ and let $p_m$ $\in$ $P_{m+2}$ be the Hermite interpolant of the ...
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1answer
138 views

Proving $\int_{0}^{\frac{\pi}{2}}\text{erf}(\sqrt{a}\cos(x))\text{erf}(\sqrt{a}\sin(x))\sin(2x)dx=\frac{e^{-a}-1+a}{a}$

Assume $a>0$, how can we show that: $$\int_{0}^{\frac{\pi}{2}}\text{erf}(\sqrt{a}\cos(x))\text{erf}(\sqrt{a}\sin(x))\sin(2x)dx=\frac{e^{-a}-1+a}{a}$$ $$\int_{0}^{\frac{\pi}{2}}\text{erf}\ ^2(\sqrt{...
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21 views

Sigmoid function (Computationally Simple/Easy to Integrate)

I am working on a fitting model and I need to use a sigmoid function in the following integral $$\int_0^R S(x)\cdot x \cdot J_{0}(x) dx $$ where $S(x)$ is the sigmoid function and $J_0(x)$ is the ...
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1answer
20 views

integral on erfc function between x to infiity

I need to calculate the next integral: $\int_{x}^{\infty }\operatorname{erfc}(\frac{\xi }{2 \sqrt{D\cdot t}})d\xi $ My attempt: $u=\frac{\xi }{2\sqrt{D\cdot t}}\rightarrow du=\frac{d\xi }{2\sqrt{D\...
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1answer
91 views

How to choose $t$ such that the probability of arriving between 10:40 and 11:00 is maximal?

Peter has an appointment at $11:00$. He wants to arrive between $10:40$ and $11:00$. Suppose that, if he leaves from home $t$ minutes after $10:00$, then the time of his arrival has normal ...
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1answer
81 views

Integration of Exponential with Polynomial in its Exponent

I seek to solve the following integral: $$ \int_0^T dt \frac{1}{\sqrt{4 \pi s^2 t}}\cdot \exp\left({-\frac{(x-vt)^2}{4 s^2t}}\right) $$ My first idea was to substitute $1/\sqrt{t}$ using $u=\frac{1}{...
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1answer
42 views

Solving a complex integration

I want to solve this integration $$\int_{-\infty}^{\infty} \mathrm{d}t \ \mathrm{erf} \Big(\frac{t-ic}{T}\Big) \ \mathrm{e}^{-\frac{(t-ib)^2}{T^2}}$$ one can open it by using integration by parts $$u ...
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1answer
35 views

error function integration $\int_{0}^{\infty} \frac{x \operatorname{erf}(a x ) }{x^2+y^2} dx $ [closed]

I'm interested in the following integral, $$ \int_{0}^{\infty} \frac{x \operatorname{erf}(a x ) }{x^2+y^2} dx $$ where, $\operatorname{erf}$ is error function. Does the analytical solution exist to ...
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37 views

Proper name for “non-absolute” error?

Suppose that we have a CDF $F(x)$ and a model that learned this CDF, namely $F_{\theta}(x)$. I know that the error at any point x is $|F(x) - F_{\theta}(x)|$ (the absolute value), which is always $\...
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1answer
69 views

Calculating $\int_0^\infty\, e^{x^2-x} \operatorname{erfc}(x)\;dx$

I am trying to find $$I=\int_{0}^{\infty }{\,{{e}^{{{x}^{2}}-x}}\operatorname{erfc}\left( x \right)dx}$$ where $\operatorname{erfc}$ is the complementary error function. My Work: $${{e}^{{{x}^{2}}-x}...
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25 views

Integration of exponent over a disk

I am looking for solutions to integrals of the form: $\int_{-a}^{a} \int_{-\sqrt{a^2-y^2}}^{\sqrt{a^2-y^2}}x^ny^m \exp(-ibx-idy-cx^2-cy^2)dxdy$ This is integration over the area of a circle with ...
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23 views

Similarity measure between two tetrahedron

Given that there are two tetrahedrons $\mathbf{T} = \begin{bmatrix} \mathbf{A} & \mathbf{B} & \mathbf{C} & \mathbf{D} \end{bmatrix}^{\top}$ and $\mathbf{T}^{\ast} = \begin{bmatrix} \mathbf{...
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26 views

Indefinite integration of multiplication of two error functions

I am trying to calculate the expectation value of the multiplication of two error functions of the form$\def\erf{\operatorname{erf}}$ $$E[\erf(a-x)\erf(a+x)]$$ I firstly assume that $x$ is uniformly ...
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1answer
29 views

Backpropagation of simple model

I've been attempting to grok how backpropagation works. I've therefore come up with a super simple model that I wanted to attempt to optimize: $f_{p}(x) = p x$ For some parameter $p$. My toy ...
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90 views

The easiest way to evaluate $I=\iint_D e^{-(x^2+y^2)}\,dx\,dy,\ \ \ D=\left\{(x,y)\Bigm|2\leqslant |x|+|y|\leqslant 3\right\}$

Evaluate the following integral: $$ I=\iint_D e^{-(x^2+y^2)}\,dx\,dy,\ \ \ D=\left\{(x,y)\Bigm|2\leqslant |x|+|y|\leqslant 3\right\} $$ Well, to make things a bit easier, I can find an integral $J$ ...
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1answer
32 views

Solving integration of Normal CDF problem to get $\sqrt{\frac{2}{\pi}}$

I encountered the following example: (Folded Normal). Let $Y = \vert Z \vert$ with $Z ∼ N(0, 1)$. The distribution of $Y$ is called a Folded Normal with parameters $\mu = 0$ and $\sigma^2 = 1$. At ...
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35 views

Integral of error function times Gaussian

By manipulating equation 4.3.13 from A table of integrals of the error functions, it is possible to derive the following result: $$ \int_{-\infty}^{+\infty} e^{-(ax+b)^2}\text{erf}(cx+d)dx = \frac{\...
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46 views

Is the following matrix whose entry is defined as an integral positive definite?

For any given $\xi=(\xi_1,\xi_2,\cdots,\xi_n)\in\mathbb{R}^n$, define a matrix $M_{n\times n}$ by $$M_{i,j}:=\int_\mathbb{R}e^{-(x-\xi_i)^2}e^{-(x-\xi_j)^2}\mathcal{X}_{[-\pi,\pi]}(x)dx$$ where $\...
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1answer
75 views

What is the indefinite integral of the error function times a gaussian?

Wikipedia lists $$\int\limits_{-\infty}^{\infty} \operatorname{erf}(ax+b) \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left( \frac{-(x-\mu)^2}{2 \sigma^2}\right) \, dx \\ = \operatorname{erf}\left(\frac{a\mu+b}...
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100 views

Unifying the sum of two error functions.

Let's propose you have the sum of two error functions: $f(x) = \text{erf}(ax)+\text{erf(bx)}$ If you wanted to solve for x, you'd first unify them into a third function $f(x) = \text{erf}(cx)$ Or ...
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1answer
18 views

Solving an inequality wirh Erfc

I am trying to prove the following statement. $$ 2\sqrt{2}\sigma - k \exp{\frac{k^2}{2\sigma^2}}\sqrt{\pi} \textit{Erfc}\Big[ \frac{k}{\sqrt{2}\sigma} \Big] >0 $$ My approach is as follows: It ...
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59 views

How to integrate the following expression?

I want to know what is the best way to integrate the following expression and get the simplest answer $$ \int_{u_-}^{u^+} \frac{1}{r_1r_2} \exp\left(\left[r_1^2+r_2^2-2u\right]^{1/2}+4su\right)...
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1answer
120 views

Integrals involving Gaussian Q function

I am trying to find the following definite integral: \begin{equation} I = \int_{0}^{b} Q\left((b-x)\, a \right)\,\frac{x}{\sigma^2}\,\exp\left(-\frac{x^2}{2\sigma^2}\right)\,dx, \end{equation} where $...
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72 views

Asymptotic of $I(x)=\int_{-\infty}^{x} \exp(-t^2 \operatorname {erfi}(t \sqrt{2\pi}) \operatorname {erf}(t \sqrt{2\pi})) dt$ for $x\to \infty$?

Really I have tried to get any closed form of asymptotic series ( Taylor series ) of the below integrand function: $$I(x)=\int_{-\infty}^{x} \exp(-t^2 \operatorname {erfi}(t \sqrt{2\pi}) \operatorname ...
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1answer
37 views

Maclaurin Series for $\operatorname{erf}(z)$

I am attempting to compose the Maclaurin series for $\operatorname{erf}(z)$. My disclaimer is that I am not an expert in the field of complex analysis. Below is my attempt. I am worried about ...
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1answer
82 views

Analytical Solution for 1D Heat Diffusion on Cylindrical/Spherical Coordinates

I would like some insight in how to solve the following equations analytically: $$ \frac{\partial T}{\partial t} = \frac{\alpha}{r} \frac{\partial^2 (r T)}{\partial r^2} \\ \frac{\partial T}{\partial ...
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44 views

Erf/gaussian integral

I'm working on algorithm that identifies clusters of objects together in space. One of the calculations in the algorithm is of the form $$ P = \int_0^\infty e^{-ax^2}{\rm erf}\left(\frac{x}{c} + b\...
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23 views

How to calculate the expected error given the error function and posteriori distribution?

Given the error function: $$ e(y,\hat{y}) = \begin{cases} 2,& \text{if } y<\hat{y}-1\\ 0,& \text{if } |y-y|\leq 1\\ 1,& \text{if } y>\hat{y}+1\\ \end{cases} $$ I need ...
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1answer
36 views

Relationship between incomplete gamma function and error function

WolframAlpha verifies the following result: For $a > 0$, $$\int_a^\infty x^{1/2} e^{-x} \, dx = \int_{\sqrt{a}}^\infty e^{-x^2} \, dx + \sqrt{a} e^{-a}.$$ Is there a simple proof of this fact? ...
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13 views

Kinda confused about Laplace Transform of exponentials

Well, I have searched a bit about the Laplace Transform (LT) of $e^(x^2) $ and I'm very confused about the answer. Some say it does not exist, due to the greater exponential order; on the other hand, ...
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12 views

why iterate through different weight and bias for linear function when using LMS single value can be derived

I am learning Linear regression and have following questions and doubts please clarify- If I am given data points and I must find coefficients i.e. weight and bias in ML for the given equation say y=...
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1answer
43 views

finding Laplace transform of $\operatorname{erf}(\sqrt{t})$

I am trying to show that the Laplace transform of erf(√(t)) is equal to 1/(s√(s+1)) I have started with the definition of erf(t) as (2/√𝜋)times the integral from t-0 of e^(-x^2) and substituted √(t) ...
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80 views

Gauss Quadrature Error in 2D

Could someone tell me what is the error when applying the Gauss Quadrature rule in 2 dimensions? I know that for one dimension the error is $$\frac{(n!)^{4}}{(2n+1)[(2n)!]^{3}} \cdot f(\xi)^{2n}(b-a)...
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1answer
116 views

Is the integration of a gaussian function divided by polynomial possible?

I am trying to evaluate the following integral: \begin{equation} I=\int_{-\infty}^{\infty}\exp\left \{-\frac{(u-1)^2}{2\sigma^2}\right\}\frac{1}{(u-x)^2+y^2}\mathrm{d}u \end{equation} where $x,y\in\...
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1answer
40 views

Expanding the Error Function for (very) small values

According to WolframMathWorld (formulae $(9)$ and $(10)$ there) the Error Function may be expanded for $x\ll1$ by $$\operatorname{erf}(x)~=~\frac{e^{-x^2}}{\sqrt\pi}\sum_{n\geqslant1}\frac{2^n}{(...
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1answer
32 views

Inaginary error function with imaginary operand

Wikipedia has the following definition for imaginary error function: $$\textrm{erfi}(x) = i ~\textrm{eft}(ix) $$ I have two questions, Is the following correct? $$\textrm{erfi}(ix) = i ~\textrm{...
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1answer
38 views

How do I solve equations involving the error function?

In my heat transfer classes, we encounter the Gaussian error function when dealing with unsteady heat conduction of a semi-infinite surface. The equation I have to solve is usually of the form: $$\...
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64 views

Evaluation of $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \left( \frac{(\frac{\pi}{2}-\frac{1}{x})dx}{(1+x^2)(\arctan^2(x) \sqrt{\log(\arctan x)}} \right)$

I have tried to evaluate the following integral \begin{eqnarray*} \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \left( \frac{(\frac{\pi}{2}-\frac{1}{x})dx}{(1+x^2)(\arctan^2(x) \sqrt{\log(\arctan x)}} \right) ...
2
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2answers
159 views

Proving that $\int_0^\infty \frac{\operatorname{erf}(1/x)\operatorname{erfc}(1/x)}{x}dx=\frac{2G}{\pi}$

In my research about distribution theory in the topic of probability and statistic, I came across the following integral: $$\int_0^\infty \frac{\operatorname{erf}(1/x)\operatorname{erfc}(1/x)}{x}dx$$ ...
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1answer
48 views

Is really this :$\int_{0}^{t}\operatorname{erf}(x+\sqrt{1-\log (x)} )dx \sim t$ true for every $t$?

I have accrossed this integral when I run some of my computation in Wolfram alpha with many values of $t$ , Really seems to conjecture that : $$\int_{0}^{t}\operatorname{erf}(x+\sqrt{1-\log (x)} )dx ...
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1answer
63 views

What is the closed form of $\int \exp ( \sqrt{x}-x^2)\ dx$?

My attempt to get the closed form of $\int \exp ( \sqrt{x}-x^2)\ dx$ using integration by part to get something related to error function is failed, I believe that function has a closed form because ...
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0answers
30 views

Signification of $\varepsilon(h)$ and $|||A|||$

Consider $f:x\in \mathbb{R}^n \rightarrow f(x):= \frac{1}{2}x^\top Ax +b^\top x,$ with $n\geq 1$, $b\in \mathbb{R}^n$ and $A\in \mathbb{R}^{n\times n}$ Check that the gradient does fulfill the ...
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166 views

Integral of product of error function and Gaussian

I found the following relation from a 1968 paper: $ \DeclareMathOperator\erf{erf} \int_{-\infty}^{\infty}\erf(x)\exp^{-(ax+b)^2}dx=-\frac{\sqrt\pi}{a}\erf\big(\frac{b}{\sqrt{a^2+1}}\big), Re(a^2)>...

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