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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Exact values of error function

The error function is defined as $$\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt.$$ We know that the Gaussian integral is $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
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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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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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Integral involving the Erf function

I'm am trying to solve the following integral $$\int\limits_{-\infty}^{+\infty}dx \; e^{-(ax+b)^2}\mathrm{Erf}(cx+d)\mathrm{Erf}(ex+f)$$ I tried the same reasoning as for these integrals that can ...
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Partial Differential Equations - Error Functions

So I am doing a course this semester in PDEs and we are currently doing the heat/diffusion equation $(u_t +ku_{xx}=0)$ on the whole line and the half line. In solving these equations we have ...
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MATLAB implementation of erf(x)

I have implemented erf(x) using its Taylor expansion in Matlab. But even after repeated attempts to correct it, it shows wrong answer for x>1. I am not able to understand why it is so. Any help will ...
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Is this transformation for the integral $\int_0^\infty \frac{1}{t}e^{-a^2/t^2-b t} \text{d}t$ correct?

Let's consider the integral: $$I(a,b)=\int_0^\infty \frac{1}{t}e^{-a^2/t^2-b t} \text{d}t,~~~~a,b>0$$ We can try to use the integral representation of the part of the function inside the integral:...
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Integration involving Error function

I am interested in the following integral $$\int_0^\infty x^n \,\textbf{Erf}[ax]\,j_m(bx) \,\mathrm{d}x,$$ where Erf is the error function $j_n$ is the spherical Bessel function of first kind. Does ...
Does anybody know if a closed form of this integral exist? $\int \mbox{erf}(x) \ln(\mbox{erf}(x)) \Bbb dx$ where erf is so called error function. In case there is no closed form solution. Is it ...