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.

45
votes
6answers
10k views

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 ...
50
votes
3answers
1k views

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 ...
3
votes
2answers
696 views

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 ...
1
vote
2answers
1k views

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?
8
votes
4answers
194 views

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 ...
5
votes
1answer
104 views

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}}{...
3
votes
1answer
216 views

Closed form of $I(a)=\int_{0}^a {(e^{-x²})}^{\operatorname{erf}(x)}dx $ and is it behave similar with error function?

$\newcommand{\erf}{\operatorname{erf}}$ The computation of $\int_{0}^{a}{(e^{-x²})}^{\erf(x)}dx$ for large $a$ gives $0.972106...$ by wolfram alpha, but according to JJacquelin comments which ...
3
votes
2answers
521 views

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 ...
44
votes
4answers
1k views

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\...
37
votes
2answers
4k views

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{...
18
votes
1answer
460 views

Fourier transform of $\operatorname{erfc}^3\left|x\right|$

(this is a follow-up on my another question) Could you please help me to find the Fourier transform of $$f(x)=\operatorname{erfc}^3\left|x\right|,$$ where $\operatorname{erfc}z$ denotes the the ...
4
votes
2answers
803 views

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.
5
votes
1answer
362 views

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

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 ...
2
votes
1answer
16k views

derivative of error function

How can I calculate the derivatives $$\frac{\partial \mbox{erf}\left(\frac{\ln(t)-\mu}{\sqrt{2}\sigma}\right)}{\partial \mu}$$ and $$\frac{\partial \mbox{erf}\left(\frac{\ln(t)-\mu}{\sqrt{2}\sigma}\...
7
votes
3answers
167 views

Solving the Definite Integral $\int_0^{\infty} \frac{1}{t^{\frac{3}{2}}} e^{-\frac{a}{t}} \, \mathrm{erf}(\sqrt{t})\, \mathrm{d}t$

I would like to solve the following integral $$\int_0^{\infty} \frac{1}{t^{\frac{3}{2}}} e^{-\frac{a}{t}} \, \mathrm{erf}(\sqrt{t})\, \mathrm{d}t$$ with Re$(a)>0$ and erf the error function. Is ...
4
votes
1answer
101 views

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 ...
4
votes
3answers
2k views

Integral of product of exponential function and two complementary error functions (erfc)

I found the following integral evaluation very interesting to me: Integral of product of two error functions (erf) and I hoped that I could use that result to evaluate the following integral: $$ \...
3
votes
3answers
143 views

Show $\frac{2}{\pi} \mathrm{exp}(-z^{2}) \int_{0}^{\infty} \mathrm{exp}(-z^{2}x^{2}) \frac{1}{x^{2}+1} \mathrm{d}x = \mathrm{erfc}(z)$

I used the result $$\frac{2}{\pi} \mathrm{exp}(-z^{2}) \int\limits_{0}^{\infty} \mathrm{exp}(-z^{2}x^{2}) \frac{1}{x^{2}+1} \mathrm{d}x = \mathrm{erfc}(z)$$ to answer this MSE question. As I mentioned ...
3
votes
2answers
2k views

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}\...
2
votes
2answers
116 views

behavior of :$\int_{-1}^1x^{2k} \operatorname{erf}(x)^k \,dx $?

I'm interesting for the evaluation of $\int_{-1}^1 x^{2k} \operatorname{erf}(x)^k \,dx=0$ with $l$ is a real number , I want to get closed form of that integral , i'm coming up to define it as :$\...
1
vote
3answers
776 views

Do Hermite polynomials exist for negative integers?

I recently asked a question about a differential equation, and received this as an answer. It included a Hermite polynomial of negative degree, namely $H_{-3}$. I searched online and it seems as ...
7
votes
3answers
245 views

Closed form for $\int_{0}^{\infty }\!{\rm erf} \left(cx\right) \left( {\rm erf} \left(x \right) \right) ^{2}{{\rm e}^{-{x}^{2}}}\,{\rm d}x$

I encountered this integral in my calculations: $$\int_{0}^{\infty }\!{\rm erf} \left(cx\right) \left( {\rm erf} \left(x \right) \right) ^{2}{{\rm e}^{-{x}^{2}}}\,{\rm d}x$$ where $c>0$ and $c\in ...
3
votes
2answers
150 views

Integral of $x^2 e^{-x^2}$

Like the title says, I'm trying to find $$\int_0^r x^2 e^{-x^2}\,dx$$ Where $r$ is some finite value. I've done one step using integration by parts with $u=x^2$ and $dv=e^{-x^2}dx$, which has left ...
3
votes
2answers
848 views

Value minimizing mean absolute percentage error

What value for $c$ would minimize the formula: $$\frac{1}{n}\;\sum^{n}_{i=1}\left | \frac{y_i-c}{y_i}\right|$$ given the values $y_1, ..., y_n$. For example in the mean squared error we have the ...
2
votes
2answers
2k views

Lagrange polynomial and derivative problem

I need to code this function in matlab $$L'(x) = \sum_{k = 0}^{n} f_k l_k(x) $$ Where's $l_k$ looks like this $$l_k(x) = \sum^{n}_{j=0, j \neq k} \frac{\prod^{n}_{i=0}(x - x_i)}{(x - x_k)(x-x_j)\...
2
votes
1answer
529 views

How to derive integrals with error function?

How to derive this integral $\int_{-\infty}^{\infty}erf(\lambda x)\mathcal{N}(\mu, \sigma ^2)dx$ and this $\int_{-\infty}^{\infty}(erf(\lambda x)-const)^2\mathcal{N}(\mu, \sigma ^2)dx$ where $...
2
votes
2answers
224 views

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}...
2
votes
1answer
73 views

Sum of weighted normal distributions, how to solve $P(X<x) = y$ for $x$?

How do I solve the following equation for $x$ $$\newcommand{\erf}{\operatorname{erf}}\frac{1}{2}\left((f-1)\cdot\erf\left(\dfrac{c-x}{\sqrt2\,b}\right)-f\erf\left(\dfrac{r-x}{\sqrt2\,d}\right)\right)=...
1
vote
0answers
215 views

Leibniz rule for error function

Let $f(x)=erfi(a+x)$ and $g(x)=e^{cx}$ with \begin{align*} f^{(n)}(x)=\frac{2}{\sqrt{\pi}}e^{(a+x)^2}\sum_{m=0}^{n-1}\sum_{j=0}^{m}\frac{\binom{m}{j}(-1)^j(a+x)^{2m-n+1}}{m!}\\ \prod_{p=1}^{n-1}(2m-2j-...
1
vote
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 }{...
1
vote
1answer
75 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 ...
1
vote
1answer
170 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 ...
0
votes
3answers
95 views

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^...
0
votes
1answer
293 views

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 ...
0
votes
0answers
65 views

About L2 error distribution and its STRANGE oscillatory behaviour

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 ...