# 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.

35 questions
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### What is the antiderivative of $e^{-x^2}$

I was wondering what the antiderivative of $e^{-x^2}$ was, and when I wolfram alpha'd it I got $$\displaystyle \int e^{-x^2} \textrm{d}x = \dfrac{1}{2} \sqrt{\pi} \space \text{erf} (x) + C$$ So, I ...
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### Non-trivial values of error function $\operatorname{erf}(x)$?

The so called error function $\operatorname{erf}(x)$ is defined as $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt,$$ and it is well known that $\operatorname{erf}(\infty)=1$. Are ...
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### 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 ...
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### How to prove error function $\mbox{erf}$ is entire (i.e., analytic everywhere)?

How do I prove the error function $$\mbox{erf}(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^{2}} dt.$$ is entire? Could you give me some scratch proof?
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### Sum: $\sum\limits_{n=0}^\infty \frac{n!}{(2n)!}$

I'm struggling with the following sum: $$\sum_{n=0}^\infty \frac{n!}{(2n)!}$$ I know that the final result will use the error function, but will not use any other non-elementary functions. I'm ...
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### Series expansion for integral including error function

$\DeclareMathOperator{\erfc}{erfc} \DeclareMathOperator{\Ei}{Ei}$ What is the series expansion of $f$ for small $q$? \begin{align} U(q) &= q e^{q^2}\erfc q\\ I(q,q') &= \int_0^{2\pi}...
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### 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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### 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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### Compute $\mathbb{P}(1<X^2+Y^2<2)$ when $(X,Y)$ is i.i.d. standard normal

Assume that $(X,Y)$ is i.i.d. standard normal. Compute $\mathbb{P}(1<X^2+Y^2<2)$. So I've decided to use polar coordinates to solve and I've gotten to this point: \iint_{1\lt X^2+Y^2\lt2} e^...
### Upper and lower bound on different of ${\rm erf}(\frac{x+c}{b})-{\rm erf}(\frac{x-c}{b})$
I am trying to find a good upper bound on \begin{align*} f(x)={\rm erf}\left(\frac{x+d}{b}\right)-{\rm erf} \left(\frac{x-d}{b}\right) \end{align*} here $d>0$ I know that $f(x)$ is symmetric ...
Lets assume we are performing an physical experiment and in that experiment we get $10^{6}$ number of data values. All those data values follows some distribution function. Now, in next step what we ...