Articles of error function

Product of two complementary error functions (erfc)

I believe that (i.e., it would be convenient if, and visually appears that) the product of the two complementary error functions: $$\operatorname{erfc}\left[\frac{a-x}{b}\right]\operatorname{erfc}\left[\frac{a+x}{b}\right]$$ will have a solution, or can be approximated with a solution, of a Gaussian form (i.e., $c\operatorname{exp}\left[-\frac{x^2}{2d^2}\right]$, where $c$ and $d$ are functions of $a$ and $b$) when $a>0$ and $c>0$, however I […]

Prove $e^x=1+\frac{1}{\sqrt{\pi }}{\int_0^x \frac{e^t \text{erf}\left(\sqrt{t}\right)}{\sqrt{x-t}} \, dt}$

It seems to me that $$e^x=1+\frac{1}{\sqrt{\pi }}{\int_0^x \frac{e^t \text{erf}\left(\sqrt{t}\right)}{\sqrt{x-t}} \, dt}$$ This integral seems to converge for all $x\in\mathbb{C}$ I came upon this conjecture by following the instructions here to do a half integral twice. Can anyone prove this conjecture is true?

$\int\text{e}^{-ax^2 } \text{erf}\left(bx + c\right) dx$

I’m hoping to find a closed expression for the following integral. $$ \int\text{e}^{-ax^2 } \text{erf}\left(bx + c\right) dx $$ One can find a solution for a family of products between exponentials and error functions. None of which apparently have the offset term in the error function. I have tried tackling the problem with two approaches. […]

Solving Partial Differential Equation with Self-similar Solution

$$ Greetings, $$ So, I have a heat equation to be solved for in the form of $$ \frac{\partial f(x,t)}{\partial t} = \frac{\partial^2 f(x,t)}{\partial x^2} $$ for t = [0,+inf) and x = (-inf,+inf) and following I.C and B.C: $$ f(x,0)=1,x\geq0 $$ $$ f(x,0)=0,x<0 $$ $$ f(-\inf,t)=0 $$ $$ f(+\inf,t)=1 $$ I work out the […]

Fourier transform of $\operatorname{erfc}^2\left|x\right|$

Could you please help me to find the Fourier transform of $$f(x)=\operatorname{erfc}^2\left|x\right|,$$ where $\operatorname{erfc}z$ denotes the the complementary error function.

How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ \mathrm dx$?

Let $\operatorname{erfc}x$ be the complementary error function. I successfully evaluated these integrals: $$\int_0^\infty\operatorname{erfc}x\ \mathrm dx=\frac1{\sqrt\pi}\tag1$$ $$\int_0^\infty\operatorname{erfc}^2x\ \mathrm dx=\frac{2-\sqrt2}{\sqrt\pi}\tag2$$ (Both $\operatorname{erfc}x$ and $\operatorname{erfc}^{2}x$ have primitive functions in terms of the error function.) But I have problems with $$\int_0^\infty\operatorname{erfc}^3x\ \mathrm dx\tag3$$ and a general case $$\int_0^\infty\operatorname{erfc}^n x\ \mathrm dx.\tag4$$ Could you suggest an approach to evaluate them […]

Integral $\int_0^\infty\left(x+5\,x^5\right)\operatorname{erfc}\left(x+x^5\right)\,dx$

Is it possible to find a closed form (possibly using known special functions) for this integral? $$\int_0^\infty\left(5\,x^5+x\right)\operatorname{erfc}\left(x^5+x\right)\,dx$$ where $\operatorname{erfc}$ is the complementary error function $$\operatorname{erfc} x=\frac{2}{\sqrt{\pi}}\int_x^{\infty}e^{-z^2}dz.$$

Integral of exponential using error function

I’m trying to solve some integrals below $$\int_{-\infty}^{\infty} {x^n e^\frac{-(x – \mu)^2}{\sigma^2}}dx$$ I am interested in the solutions where n = 0, 1, 2, 3, 4. I have learned that integrals of this form can’t be solved the usual way, but can be evaluated in terms of the error function. I’ve checked using wolfram alpha […]

Erf squared approximation

I found a nice and useful approximation of squared error function $$ \mathrm{erf}^{2}\!\left(x\right)=1-\exp\!\left(-\frac{\pi^{2}}{8}x^{2}\right)+\varepsilon\!\left(x\right). $$ I checked numerically that maximum error is bounded by $\left|\varepsilon\!\left(x\right)\right| < 61\cdot10^{-4}$ but I was asked if this could be somehow proved analytically. Or at least if order of the error could be determined in such manner. Error function is defined […]

Prove $\sum_{n=1}^{\infty} \frac{n!}{(2n)!} = \frac{1}{2}e^{1/4} \sqrt{\pi} \text{erf}(\frac{1}{2})$

I would like to prove: $$\sum_{n=1}^{\infty} \frac{n!}{(2n)!} = \frac{1}{2}e^{1/4} \sqrt{\pi} \text{erf}(\frac{1}{2})$$ What I did was consider: $$e^{-t^2}=\sum_{n=0}^{\infty} (-1)^n \frac{t^{2n}}{n!}$$ Then integrate term by term from $0$ to $x$ to get: $$\frac{\sqrt{\pi}}{2}\text{erf} (x)=\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{n!(2n+1)}$$ Then I substituted in $x=\frac{1}{2}$ and tried some manipulations but didn’t get anywhere. May someone help, thanks.