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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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Proof that $\text{erfc}(x)\leqslant e^{-x^2}$

I'm looking for a proof that $\text{erfc}(x)\leqslant e^{-x^2}$ without invoking any probability theory. Some sources say this inequality is a result of the "Chernoff bound," which seems to come ...
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2answers
62 views

Integral $ \int_\sigma^\infty r^2 {e^{-A/r^6}} dr $ [closed]

Below integral can be calculated by using taylor expansion for the $ e^{-A/r^6} $ term. I want to know how to solve this integral analytically? $$ \int_\sigma^\infty r^2 {e^{-A/r^6}} dr $$ Hint: I ...
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0answers
18 views

Integration of a Product of a Complementary Error Function and Exponential function

I have derived a solution that describes heat diffusion in a linear model that is subjected to an initial temperature distribution. The resulting solution is obtained using the method of separation of ...
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0answers
41 views

Inverse Laplace transform of $F(s) = \exp(-a\sqrt{s})/s$, $a > 0$

Show that the inverse Laplace transform of F(s) = $e^{-as^{1/2}}/s$, $a > 0$, is given by $$f(x) = 1 - \frac{1}{\pi}\int^{\infty}_{0} \frac{\sin(a\sqrt{r})}{r}e^{-rx}dr$$ Note that the integral ...
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38 views

Differences between objective function definitions in optimization problems

Can someone explain the differences between the relative error definitions used as an objective function in a constrained optimization problem? I am trying to understand why I get different values of $...
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0answers
5 views

Estimating the discrete random walk probability by error function

I am trying to work out the asymptotic large $t$ behavior of following function \begin{equation} f(t ) = \sum_{x = 0}^{2t} { 2t \choose t + x} p^{ t+x } (1 - p)^{ t - x} = \sum_{x = 0}^{2t} { 2t \...
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42 views

What is the best way to compare the numerical solution to the exact solution?

I have 2000 approximate solutions (2000 arrays of points (x,y)). In the following figure, I have randomly shown three approximate solutions versus the exact solution. By increasing the case number, ...
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24 views

A definite integral involving a Gaussian and shifted error functions.

Let $d\ge 1$ be an integer and let $\vec{a} \in {\mathbb R}^d$ and $\vec{b} \in {\mathbb R}^d$. We consider a following integral: \begin{eqnarray} {\mathfrak J}^{(d)}(\vec{a},\vec{b}):=\int\limits_{\...
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14 views

finding the overhead and distance of an unknown code based on message making algorithm

For an information word M with m bits that is coded as following: M Is coded into a word A using an unknown code that allows detection of not more than one error. The code word is the word obtained ...
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32 views

Can any Gaussian integral with complex limits be written as a (complex) error function?

Can the integral, $$I = \int^{-x}_{-\infty} e^{-at^2}\ \mathrm dt,\ a \in \mathbb{C},\ Re(a) > 0,$$ be written as an error function? I tried, by substitution, $$\int^{-x}_{-\infty} e^{-at^...
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18 views

Modified Error Function Integral

I was given an identity recently without any stated derivation, so I am attempting a derivation on my own. The identity in question is $$\int_0^\infty\frac{e^{-s^2t}\sin(sy)}{s}\,ds=erf(\frac{y}{2\...
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20 views

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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2answers
34 views

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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2answers
35 views

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
90 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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27 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
47 views

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 ...
7
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76 views

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 ...
7
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2answers
355 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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1answer
81 views

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 ...
4
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1answer
80 views

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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80 views

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
87 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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0answers
60 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 }{...
7
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1answer
88 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
139 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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0answers
97 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
43 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,...
3
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1answer
106 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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40 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
82 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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0answers
48 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
83 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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17 views

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
17 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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32 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
74 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
38 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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22 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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26 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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0answers
62 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
31 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
79 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
29 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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0answers
10 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
60 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
48 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 ...
3
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2answers
69 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
57 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 $ ...
2
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1answer
90 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 ...