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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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Integrate $x\operatorname{erf}^{\,3}(x)\,e^{-x^2}\,dx$

Looking for a way to perform this integral related to the error function. I am thinking an answer in closed form cannot be done, but hoping I missed something. $$ \int x\operatorname{erf}^{\,3}(x)\,e^...
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1answer
31 views

Understanding Taylor series error function and Lagrange error bound

I am in high school and find math interesting so lately I have been trying to learn as much about it as I can. I recently began studying Taylor Series as it pertains to the research areas that I am ...
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56 views

An integral leads to complementary error functions

I am reading a paper Albrecher, Constantinescu and Loisel "2011Explicit ruin formulas for models with dependence among risks" and getting stuck at one integral (Example 2.4): $$\int _{\frac{\lambda }{...
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1answer
75 views

On $\int_0^\infty \frac{\exp(-x^2)}{1+x^2}dx=\frac{\pi e}2\text{erfc}(1)$

I was attempting to answer this question, but then I came across a question of my own involving my attempt. Task: Prove $$\int_0^\infty\frac{\exp(-x^2)}{1+x^2}\mathrm dx=\frac{\pi e}2\text{erfc}(1)$$ ...
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2answers
132 views

Seeking Methods to solve $F\left(\alpha\right) = \int_{0}^{1} x^\alpha \arcsin(x)\:dx$

I'm looking for different methods to solve the following integral. $$ F\left(\alpha\right) = \int_{0}^{1} x^\alpha \arcsin(x)\:dx$$ For $\alpha > 0$ Here the method I took was to employ ...
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55 views

Laplace Transform of Complementary Error Function

I need to apply one Laplace transform formula while I have no idea how to prove it: $$\int_0^\infty e^{-st} e^{a k} e^{a^2 t} \operatorname{erfc} \left( a \sqrt{t} + \frac{k}{2 \sqrt{t}} \right) dt = ...
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1answer
37 views

Solve an equation involving the error function

Let $0<a<1$ be given. The equation: $$a = 1 - \frac{2\sqrt{x/\pi}}{\mathrm e^x \mathrm{erf}(\sqrt x)}$$ has a unique root $x$, because the right-hand side is increasing in $x$, and goes to $0,...
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1answer
95 views

Nice result that I can't prove: $\int_{-2}^{2} \tan^{-1} \bigg( \exp(-x²\text{erf}(x)) \bigg) \;dx=\pi$

I'm always trying to find the integral representation of $\pi$ using some interesting special function, at this time I have got the below representation $$I=\int_{-2}^{2} \tan^{-1} \bigg( \exp(-x^2\...
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21 views

BCH error correction 10-6

I am doing a program to correct BCH(10-6) codes errors. As known in the BCH error correction when we find a single error we look for an error position and error magnitude ( which is the difference ...
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1answer
19 views

Mean Absolute Error (MAE) equal or more than 1.

Can be the estimated Mean Absolute Error (MAE) equal or more than 1? If it is possible (which it happened to me), when it ...
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13 views

Asymptotic form of imaginary error function

I am interested in the asymptotic form of the imaginary error function for large, real arguments. I find [1] the following: $$ \text{erfi}(z) = -i + \frac{e^{z^2}}{\sqrt\pi}\left(z^{-1} + \frac12 z^{-...
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1answer
42 views

The Fourier transform of $\frac{\text{erf}(\omega x)}{x}$

Does anyone know the Fourier transform of $\Large\frac{\text{erf}(\omega x)}{x}$? I think it should be something like $\frac{4\pi}{k^2}\exp{(-k^2/4\omega^2)}$. Is this right? How can one go about ...
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Why this :$I(x)=\int_{-x}^x {0.5(\exp({-t² {\operatorname{erf}(t^2)}})}dt$ is not error function for $|x| >3$?

This integral : $$I(x)=\int_{-x}^x {0.5(\exp({-t² {\operatorname{erf}(t^2)}})}dt$$ close to $x$ for $|x|<3$ and converge to $1$ for $|x|>3$ from $-\infty \to +\infty$ as shown here such that ...
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1answer
16 views

Is the parity of error function enough to show :$\int_{-l}^{l} \exp ({\operatorname{-x^2erf(x)})dx=\int_{-l}^{l} \exp({\operatorname{x^2erf}}(x)})dx$?

I have tried to show the below identity using the parity of both error function and exp function but I didn't succeed, then my question here is there any analytical way to show this identity or Is ...
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calculate $(a+b)^2 $ with error for a and b under square root

I have a data set with a lot of values each has a different error value. I need to combine the values as well as the errors and find the mean of it. The formula for the combined value is $$ z =\sqrt{...
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1answer
68 views

The integral $\int_0^∞ e^{-f(x^2)} dx$

We know that : $$\int_0^∞ e^{-x^2} dx = \frac {\sqrt{π}}{2}$$ $$\int_0^∞ e^{-x^2-\frac {a^2}{x^2}} dx = \frac {\sqrt{π}}{2}e^{-2a}$$ Both the above results can be easily proved by integration under ...
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1answer
28 views

Handling overflow in Matlab with exponential and imaginary error functions

With Matlab, I want to verify numerically whether the following inequality holds $$\exp(-27^2)\text{erfi}(27)<\frac{21}{1000}.$$ However, I obtain "Inf" with Matlab. The reason is due to ...
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16 views

Coming up with a custom error function

Say I have two variables $e_1, e_2$ which denote some errors. I want to come up with a function $f(e_1,e_2)$ which will increase if either $e_1$ or $e_2$ increases, irrespective of what's happening ...
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23 views

Integral of $\exp[\text{erfc}[C x]]$

I have the complementary CDF of a continuous random variable $X$ that looks like... $$1-F_{X}(x)=\frac{\exp\left(\frac{\pi}{2\sqrt{e}}\text{ erfc}\left(\sqrt{2}x\right)\right)-1}{\exp\left(\frac{\pi}{...
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58 views

Can it be proved that the integral $I_1 = \frac{1}{2}$ iff $A=0$?

I have the following integral: $$ I_1= \int_{-\infty}^{\infty} \frac{d\tau}{2\pi i} \int_{-\infty}^{\infty} \frac{d\tau'}{2\pi i} \frac{1}{(\tau - i \epsilon)(\tau' - i\epsilon')}. M(\tau, \tau')$$ ...
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1answer
22 views

Calculating the error function

If I want to calculate the following integral in terms of the Error function, is this correct? $$\frac{1}{\sqrt{2\pi}}\int_{f(x)}^{-\infty}e^{-p^2}\mathrm{d}p = \mathrm{Erf}(-\infty) - \mathrm{Erf}(f(...
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2answers
66 views

Analytic solution exists for integral involving product of two Erf functions, but Mathematica can't find it. Why?

Consider the following integral and its analytic solution: $$ \int_{-\infty}^\infty \frac{dx}{\sqrt{2\pi}} e^{-x^2/2} \text{erf}(ax)\text{erf}(bx)=\frac{2}{\pi} \sin^{-1}\left(\frac{2ab}{\sqrt{(1+2a^2)...
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1answer
28 views

A quick question concerning error function

Why $\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{\infty}e^{tx}e^{-x^{2}/2}dx$ equals to $e^{t^{2}/2}$ ? I know it is error function. but I just do not have any basic knowledge about error function and ...
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9 views

Compute final error covariance matrix using measurements from two independent set

I have two independent sets of two-dimensional measurements ($X_m^1$ and $X_m^2$) and i know their corresponding ground truth data ($X_g^1$ and $X_g^2$). So, i can calculate the error statistics for ...
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4answers
50 views

Proving approximation of $\text{erf}$ with Taylor expansion

I am asked to show that $$\text{erf}(x) \approx 1 - \frac{1}{\sqrt{\pi}}\frac{1}{x}e^{-x^2}$$ in a computational project. Numerically it is really easy to show that this approximation makes sense. ...
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2answers
43 views

How can I express this solution in terms of the error function?

If I have this expression: $$u(x,t) = \frac {U_o}{\pi} \int_{-\infty}^{\infty} \!\frac{\sin(\alpha) \cos(\alpha x) e^{-k\alpha^2 t}}{\alpha} \,d\alpha, $$ how can I rewrite it in terms of the error ...
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65 views

why do we define $\int{e^{-t^2}dt}=\frac{\sqrt{\pi}}{2}erf(t)+c$?

I am relearning differential equation on my own, and came across a problem that gives the integral as erf function. Why is it defined in this way $\int{e^{-t^2}dt}=\frac{\sqrt{\pi}}{2}erf(t)+c$? I ...
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1answer
41 views

First order non-linear ODE with error function

I have to solve $ y'(x)=-2xy(x)+ey^2(x) $. Using $ z=y^{-1}$ and $-z^{'}=\frac{y^{'}}{y^{2}}$ i arrive to prove that $ z^{'}=-2xz+e $, but when i apply the variation of constants method i obtain $ ...
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72 views

Evaluating $\int_{0}^{\infty} \mathrm{erfc}(ax)\exp(bx^2+cx)dx$

I tried to evaluate the integral below using differentiation under the integral sign and error function tables [1,2,3]: $$I = \int_{0}^{\infty} \mathrm{erfc}(ax)\exp(bx^2+cx)dx.$$ Also, the approach ...
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12 views

Standard notation for Dawson's integral?

Is there a standard notation for Dawson's integral? This is the function defined by: $$F(x) = e^{-x^2}\int_0^x e^{y^2} \mathrm{d}y$$ I have seen the symbols $F(x)$ (Mathworld) or $D_+(x)$ (Wikipedia)...
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38 views

Integral of combination of error function with exponential

I am trying to find analytical solution for the following integral: $$\int_{0}^\infty x^{p-1}\exp(-x^p)*\text{erf}(ax+b)dx $$ I found that for the special cases of $p=1$ and $p=2$, solution exist. ...
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1answer
121 views

Integral involving erf and exponential

Problem I would like to compute the integral: \begin{align} \int_{0}^{+\infty} \text{erf}(ax+b) \exp(-(cx+d)^2) dx \tag{1} \end{align} I have been looking at this popular table of integral of the ...
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1answer
65 views

Doubt on differential equation involving the complementary error function

How to find the general solution of $y''+2xy'-2ny=0$? I was solving a problem in this thread. The posted solution involves the error function which I am not aware of. I found this link. How to find ...
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2answers
75 views

Double Gaussian definite integral with one variable limit

I am interested in solving a definite double integral of the following form: \begin{align} f(a,b) &= \int_0^\infty \exp\Big(\frac{-x^2}{2a}\Big)\int_{x}^{\infty} \exp\Big(\frac{-y^2}{2b}\Big) ...
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1answer
82 views

Expressing a Gaussian-like integral in terms of error function, faliure of Glasser's master theorem?

So I have this integral $$ I=\int_0^c\exp\left(-a^2x^2-\frac{b^2}{x^2}\right)\,dx, \quad(a,b>0) $$ This is what I tried to write it in terms of error function. $$ I=e^{-2ab}\int_0^c\exp\left(-a^2\...
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1answer
112 views

Integral involving numerous erf functions

As a part of a bigger problem, I am puzzled with computing: $$\int_0^\infty e^{-x^2}\cdot \operatorname{erf}(s_1 x)\cdot \operatorname{erf}(s_2 x)\cdot \operatorname{erf}(s_3 x)\cdot \ldots \cdot \...
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Solution for the Fourier transform of multiplication of two error functions.

I'm hoping someone can help me with what a Fourier transform problem. I seek the Fourier transform of $f(t)$, where: $$f(t) = a\left(1+\mathrm{erf}\left(\frac{\ln(t)-u_1}{\sigma_1\sqrt{2}}\right)\...
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40 views

Picking between SVD and Gauss Newton for least sqaures error minimization

I have seen examples where least squares error minimization has been done using SVD. One such example is camera calibration. The system to solve is $M \times p = \omega$ Ideally $\omega$ has to be ...
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18 views

Optimization and splitting the problem by dependent/independent variables

I have the following nonlinear function: $$f_{(a,b,c,d)}$$ and measurements : $$f_{measured}^{i}$$ for $i = 1, 2, 3, 4 ...$ The problem is defined as minimization of : $$\min_{a,b,c,d}\bigg(\sum_{...
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1answer
104 views

Asymptotic expansion of Erfi(x)

I know that the imaginary error function, $\mathrm{Erfi}(x)=(2/\sqrt{\pi})\int_0^x \exp{t^2} \mathrm{d}t$, has the asymptotic expansion given in the answer to this question: Asymptotic order of $\frac{...
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1answer
79 views

writing the “error function” $\text{erf}(x) = \frac{1}{\sqrt{2\pi}} \int_0^x e^\frac{-t^2}{2} dt$ as power series

I'm working on a problem which asks to write the error function $$ \text{erf}(x) = \frac{1}{\sqrt{2\pi}} \int_0^x e^{-t^2/2} dt $$ as a power series -- i.e., a series in the form $$ \text{erf}(x) = \...
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1answer
30 views

Integrating the errorfunction using the method of undetermined coefficients

$\DeclareMathOperator\erf{Erf}$ I am trying to solve using the method of undetermined coefficients: $$\int\erf(x)dx$$ With the method of undetermined coefficients one would start by "simply" ...
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32 views

How do they find out the definition of cross entropy error function?

The definition of cross entropy error function is $$J = \sum_i y_i ln (\hat{y}_i) $$ I wonder how can they come up with this definition?
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1answer
50 views

(possibly indefinite) integral of product of error functions

As given in this question, the convolution of a uniform and Gaussian distribution gives $$ (*) \quad \quad p(t) = \frac{1}{(b-a)\sqrt{2 \pi \sigma^2}} \int_{a+\mu}^{b+\mu} \exp \left\{-\frac{(\tau -t)...
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1answer
30 views

$\dfrac{ \partial{E} }{ \partial{s} } = \dfrac{ \partial{E} }{ \partial{z} } \dfrac{ \partial{z} }{ \partial{s} } = z - t$?

The following is a lecture slide from a machine learning class: Cross Entropy For classification tasks, target $t$ is either $0$ or $1$, so better to use $$E=-t\log(z)-(1-t)\log(1-z)$$ ...
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56 views

Correctly Differentiating (Cross Entropy and Other) Error Functions

The following is a lecture slide from a machine learning class: Cross Entropy For classification tasks, target $t$ is either $0$ or $1$, so better to use $$E=-t\log(z)-(1-t)\log(1-z)$$ ...
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0answers
82 views

Integral of product of error function difference

In the course of my research I came across the following integral: $$\int_{-\infty}^{\infty}\,\left(\operatorname{erf}\left(ax-b\right) -\operatorname{erf}\left(\frac{a}{\gamma} x-b- \dfrac{ar}{\...
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1answer
69 views

Integrating the error function of $\frac{a}{\sqrt{x^2+b^2}}$

I'm faced with trying to solve this integral: $ \int_{0}^{\infty} \operatorname{erf} \left( \frac {a} {\sqrt{x^2+b^2}} \right)\;dx$ My integration muscles are rather atrophied, though so I'm faced ...
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0answers
30 views

Carrying forward the summation symbol when differentiating $E = \dfrac{1}{2} \sum (z - t)^2 $?

My lecture notes state the following: $z = \dfrac{1}{1 + e^{-s}}$ $E = \dfrac{1}{2} \sum (z - t)^2$ $\therefore \dfrac{ \partial{E} }{ \partial{s} } = (z - t)z(1 - z)$ In this case, $E$ ...
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1answer
57 views

why is it equal to $\int^{\infty}_{1} \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{x^2}{2\sigma^2}}dx$ directly

$1$.$X_n$ is iid Gaussian process and $U_n$ is iid binary random process with $Pr${$U_n=-1$}=$Pr${$U_n=1$}$=0.5$.$X_n$ and $U_n$ are independent $2.Y_n=X_n+U_n$,and $\hat U_n=Q(Y_n),$where $ Q(Y_n)=\...