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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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maximum error and percentage relative error in surface area.

The radius and altitude of a closed (has a top and bottom) right circular cylinder is measured as 5 inches and 9 inches respectively. There is a possible measurement error in radius of ...
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Solving a Partial Differential Equation using Fourier transform and Convolution

Hello I am having an issue with the solution I have obtained for a problem versus the problem given in a book. The PDE is : $$u_t = \alpha u_{xx}$$ with initial condition: $$\phi = e^{-x^2}$$ where ...
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evaluation of limits of non-elementary functions

I want to try and evaluate the following limit: $$L_1=\lim_{x\to 0}\frac{\text{erf}(x^2)}{\text{erf}(x)}.$$ If I use L'Hopital's rule and then Leibniz' integral rule, I believe I get $$L_1=\lim_{x\to ...
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What is the error of a $\ln(x + R)$? [closed]

I am trying to calculate the error of a $\ln(x)$ function, given my parameter $x$ has an error $R$.
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1answer
56 views

An integral involving a Gaussian, error functions and the Owen's T function.

This question is closely related to An integral involving a Gaussian and an Owen's T function. and An integral involving error functions and a Gaussian . Let $\nu_1 \ge 1$ and $\nu_2 \ge 1$ be ...
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25 views

Contour plot with an error function

I got trouble with the Contourplot function. Probably due to the precision of an error function. The function is wish to plot is $\frac{1}{2}+4.47e^{-1-0.02x^2+25\gamma^2-1+\frac{\gamma^2}{8\sigma^2}}...
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1answer
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Analytical approximate solution to a trascendental equation

I have the following equation to solve $$ z+e^{z^2}\operatorname{erfc}(z)=0 $$ being $$ \operatorname{erfc}(z)=1-\frac{2}{\sqrt{\pi}}\int_0^ze^{-s^2}ds. $$ I solved it numerically and appears to have ...
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closed-form solution to $\int_0^\infty x^a\exp(-bx)\left(\frac{1}{\text{erfc}(c\sqrt{x})}\right)^{2a}$

This integral comes up in a problem in Statistics involving power laws. Here are some notes if anyone is interested. The integral in question would be related to equation (7) therein. I would like ...
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338 views

Evaluating $\int_{-\infty}^0 \log(\frac{1}{2}\operatorname{erfc}(x))\mathrm dx$

I am looking to evaluate $$\int_{-\infty}^0 \log\left(\frac{1}{2}\operatorname{erfc}(x)\right)\mathrm dx = -0.337~668~477...$$ Both Maple and Mathematica have failed to give a closed-form ...
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Derivative of the error function

I got stuck with the derivative of the following function: $$erf(\frac{logit(\theta)-\mu}{\sqrt {2\sigma^2}})$$ with respect to $\theta$. Are there handy approximations with elementary functions in ...
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Integral $\int\limits_0^\infty\frac{e^{-Ak^{2}}}{k}\sin(kr)dk$

I have the following integral $$ f(r)=\int_{0}^{\infty}\frac{\exp(-Ak^{2})}{k}\,\sin(kr)\,\mathrm{d}k $$ with $A>0$ and $r>0$. I know from Wolfram that the result should be $$ f(r)=\frac{\pi}{...
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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
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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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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
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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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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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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
41 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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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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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
51 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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31 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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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
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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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28 views

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
71 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
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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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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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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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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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27 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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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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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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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
52 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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44 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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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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46 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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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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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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137 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
69 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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85 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
84 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
114 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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40 views

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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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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19 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_{...