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.

0
votes
3answers
69 views

Can you help me on the Numerical Analysis question

The question: The error function defined by $$erf(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} dt. $$ The error function can also be formed as a series. $$ \frac{2}{\sqrt{\pi}} \sum_{k=0}^{\infty}...
1
vote
1answer
67 views

Closed Form of $\int_{1}^{ \infty } { \frac{1}{x}\operatorname{erfc}\left[\frac{\log x}{2 k} -\frac{k}{2}\right]{\kern 1pt} \,dx}$

I am looking for a closed form of the following $$I = \int_{1}^{ \infty } { \frac{1}{x}\operatorname{erfc}\left[\frac{\log x}{2 k} -\frac{k}{2}\right]{\kern 1pt} \,dx},$$ for $k \ge 0$. Upon ...
0
votes
1answer
341 views

Evaluating The Imaginary Error Function (erfi)

I am trying to understand how I can numerically evaluate the imaginary error function with the incomplete gamma function for all $x \in \mathbb{R}$. I found here that the imaginary error function can ...
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}}{...
1
vote
0answers
77 views

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:...
0
votes
2answers
131 views

How to use maclaruin series of arctanx to calculate $\pi/4$ with 0,01 error?

question is how many should we expand the series to calculate $pi/4$? maclaurin series of arctanx is $\sum _{n=0}^{\infty }\dfrac {\left( -1\right) ^{n}x^{2n+1}} {\left( 2n+1\right) } $ and $...
0
votes
1answer
59 views

Asymptotic Error bound

Asymptotic error bound is the limit on the error when the size of sample goes to infinity. Am I right about this? If not can somebody explain what Asymptotic error bound is? And the situations in ...
1
vote
1answer
106 views

Evaluating $\int_1^{\infty}x\: \text{erfc}(a+b \log (x)) \, dx$

I am trying to evaluate the following integral $$I = \int_1^{\infty } x \mathop{erfc}(a + b \log (x)) \, dx$$ where $a$, $b$ are some positive constants. Using the substitution $t = \log (x)$, ...
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^...
4
votes
1answer
60 views

Why is the domain of the error function scaled by $\sqrt{2}$

The normal distribution function $\Phi(z)$ has the definition $\Phi(z) \equiv \frac{1}{\sqrt{2 \pi}} \int_0^z e^{\frac{-x^2}{2}} \, dx$. However the error function is conventionally defined such that ...
2
votes
5answers
347 views

The integral $\int_0^\infty e^{-t^2}dt$ [duplicate]

Me and my highschool teacher have argued about the limit for quite a long time. We have easily reached the conclusion that integral from $0$ to $x$ of $e^{-t^2}dt$ has a limit somewhere between $0$ ...
3
votes
1answer
181 views

How to evaluate $\int_{0}^{\infty}\arctan (x^2)\sin(x^2)\mathrm dx$

How to evaluate $$I=\int_{0}^{\infty}\arctan (x^2)\sin(x^2)\mathrm dx$$ with the help of Wolfram alpha,I got the answer below $$I=\frac{\pi^{2/3}\text{erfc(1)}(\text{erfi(1)}+1)}{4\sqrt2}$$ But I don'...
0
votes
1answer
46 views

Error function representation

Can someone help about proving the following relationship: \begin{equation} \operatorname{erfc}\left(\sqrt a\right) = \int_{0}^{\infty} e^{-a} \times \frac{\sqrt a}{\sqrt \pi} \times \frac{\...
0
votes
1answer
215 views

Formula for probability of being $\epsilon$ within the mean.

It should be possible to restate that as $P(\mu-\sigma \Phi^{-1}(\frac{p+1}{2})\leq X\leq \mu+\sigma \Phi^{-1}(\frac{p+1}{2}))=p$. In this answer, it says: For a normal distribution, the ...
1
vote
1answer
954 views

How to Show that the Error Function has an Upper Bound?

Here is the error function: $$\mathrm{erf}(x)=\frac{2}{\sqrt\pi}\int^x_0e^{-t^2} dt$$ Here is the question: Show that the odd function erf is bounded, by using the fact that:$$e^{-t^2} \le ...
0
votes
2answers
79 views

Asymptotics of $\sum_{n}e^{-n^{2}}$.

Define the function $S(N)$ as $$S(N)=\sum_{n=0}^{N}e^{-n^{2}}$$ I am interested in the asymptotic behavior of $S(N)$ for large $N$. It is clear by the ratio test that $\lim_{N\rightarrow\infty}S(N)$ ...
1
vote
2answers
898 views

Convolution of Gaussian and error function

I am trying to evaluate the following integral: $$ \int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}\Phi(x-t)dx $$ where $$ \Phi(y) = \frac{1}{2} + \frac{1}{2}erf\left(\frac{y}{\sqrt{2}}\right) $$ I have ...
1
vote
2answers
185 views

$\mathrm{erf}(\eta) = 0.95$, what is a method to get $\eta$ value from the expression?

I have an error function $\mathrm{erf}(\eta) = 0.95$. How can I calculate the value of $\eta$ from this expression? I know that $\eta \approx 1.4$. And I can get the value of $0.95$ by using Excel'...
0
votes
0answers
35 views

Integral computation with Mathematica and Sympy differ

To compute the integral: $I = \int_{0}^{+oo} ue^{Au^{2}+Bu}du$ where $A<0$ and $B>0$ I have tried both Mathematica and Sympy but they yield different results: Mathematica yields: $ I = \frac{\...
0
votes
1answer
195 views

Evaluating a Erfc integral

I am trying to solve the following integral $$\int_0^{\infty } \int_{a-b x}^{\infty } \exp \left(-u^2\right) \, du \, dx.$$ I know it can be represented as an integral of the complementary error ...
2
votes
1answer
38 views

Show that $\displaystyle \int_{\gamma_R}\exp(-w^2)\,\mathrm dw\to 0~(R\to\infty)$ along $\gamma_R(t)=R\exp(\mathrm it)$

Show that $\displaystyle \int_{\gamma_R}\exp(-w^2)\,\mathrm dw\to 0~(R\to\infty)$ along $\gamma_R(t)=R\exp(\mathrm it),t\in\left[0,\frac{\pi}{4}\right]$. Hint. Use that $\cos 2t\geq 1-\frac{4}{\...
2
votes
2answers
249 views

Contour lines of constant absolute value/phase of $z\mapsto\exp(-z^2)$

In the following problem I am going to use terms (denoted with quotation marks) I couldn't properly translate - if you happen to know the particular terms please feel free to edit them. Hopefully you ...
1
vote
1answer
141 views

Seeking help with an error function Integral

I am trying to compute the following Integral $$ I = \int_{0}^\infty x \exp \left(-2 x \right) \operatorname{erf}\left(\frac{x}{t^{H}\sqrt[4]{2}}-\frac{t^H}{2^{3/4}}\right) \, dx $$ where $\...
2
votes
1answer
56 views

Integrating variation of error function: $\int_1^2e^{-nx^2} dx$

Show that $$\lim_{n\to\infty} \int_1^2e^{-nx^2} dx = 0.$$ After much googling, I learned that I am working with a variation of the error function! Yay. I've never heard of it in my life and I ...
1
vote
1answer
204 views

The so-called error function defined as: $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^{2}}dt$

The so-called error function is defined as: $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^{2}}dt$$ show that the function $y(x) = e^{x^2}\operatorname{erf}(x)$ satisfies the ...
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}\...
0
votes
1answer
236 views

Find the 'rough' error bound to the composite simpson rule

Provide a rough error bound for the following composite simpsons rule. I am aware that the upper bound is $f$ to the forth derivative evaluated at some $t$ in the open interval $(a,b)\frac{h^4(b-a)}{...
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}...
0
votes
1answer
572 views

Error function relation to the normal cumulative distribution function

A CDF for a normal standard is the following: $$N(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\phi^2/2} d\phi$$ I have the following relation in my notes which I am not very sure how they ...
5
votes
1answer
549 views

Computing the inverse error function

I need a formula/series to compute the inverse error function, which is the inverse of $$ \operatorname{erf}(x) = \dfrac{2}{\sqrt{\pi}} \int\limits_{0}^{x} \mathrm{e}^{-t^2} \,\mathrm{d}t. $$ ...
2
votes
2answers
76 views

How to compute this integral without the use of the error function?

I was watching this: https://youtu.be/qQ-56b_LvOw?t=4484 And this integral came up. $$\int_{0}^{\infty}{(2x^2+1)e^{-x^2}}dx$$ To which the answer was $\sqrt{\pi}$. They made it clear that you didn'...
1
vote
1answer
80 views

Deriving Separate Forms of the Error Function

I noticed after evaluating a form of the error function $$\text{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt$$ on WolframAlpha that another integral representation for $x\in \mathbb{R}$ is $$\text{...
3
votes
4answers
83 views

How to prove $P(x)=1-\frac{x^2}{2}$ is a good approximation of order $3$ for $f(x)=\cos x$ near $x=0$?

Let $f$ be a function and we want to approximate $f$ using a different function $P$ near $x=0$. The error of approximation is $E(x)=f(x)-P(x)$. If the approximation is going to be any good, we want $\...
0
votes
1answer
159 views

$\int\limits_0^{10}e^{-0.04t}\cdot e^{-0.001t^2}dt$

I need to find the following integral $$\int\limits_0^{10}e^{-0.04t ~-0.001t^2}dt$$ This integral seems to "scream" for the error function, but I have never worked with the error function yet, so I ...
1
vote
3answers
487 views

differentiation of $\operatorname{erfc}(\sqrt{ax})$

I need your help to figure out the derivative of $\operatorname{erfc}(\sqrt{ax})$ with respect to $x$. Based on my knowledge on Wolfram references, they cite that: $$\frac{d \operatorname{erfc}(z)}{dz}...
4
votes
3answers
321 views

Definite integral involving an error function

Let $$\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}}\int\limits_{0}^x e^{-t^2}dt$$ be the error function. Then, I have tow questions. For a positive integer $n$, is there a close-form solution of $f_n=\int_{...
1
vote
1answer
57 views

Stable algorithm for computation of $\Phi(20)$, when $\displaystyle \Phi(x)=\frac{2}{\sqrt{\pi}}\sum_{k=0}^{\infty}(-1)^k\frac{x^{2k+1}}{k!(2k+1)}$

Let $\displaystyle \Phi(x)=\frac{2}{\sqrt{\pi}}\sum_{k=0}^{\infty}(-1)^k\frac{x^{2k+1}}{k!(2k+1)}$, i.e $\Phi$ is the MacLaurin series of the function $\displaystyle \frac{2}{\sqrt{\pi}}\int\limits_{0}...
0
votes
0answers
86 views

Integral of product of two error functions

Is there a simple formula for the integral $$\int_a^{\infty}\!\text{erf}(\alpha x)\,\text{erf}(\beta x)\:\frac{dx}{x^2} $$ where $a, \alpha, \beta > 0$?
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 ...
1
vote
2answers
59 views

Error when approximating $\int_0^{1}(x^{2}+x)dx$ with midpoint rule.

Task is to define the exact error when approximating $\int_0^{1}(x^{2}+x)dx$ with midpoint rule using n subintervals. I know the error term is $E(f)=\frac{1}{24}(b-a)f^{''}(\varepsilon)h^{2}$ but im ...
1
vote
0answers
136 views

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 ...
0
votes
1answer
42 views

Solution for $g(x) - \int_0^y e^{t^2}\,{\rm d}t$.

Given the equation $g(x) - \int_0^y e^{t^2}\,{\rm d}t = 0$, with $g\colon \Bbb R \to \Bbb R$ of class ${\cal C}^\infty$, show that for each $x \in \Bbb R$ there is a unique $y = y(x)$ that solves the ...
1
vote
0answers
355 views

integral involving error function (erf)

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 ...
1
vote
0answers
106 views

How are subtraction operations better conditioned than addition?

I was reading about error analysis in numerical methods, particularly about calculating the sum of arrays. I understand how naively summing all elements can lead to accumulation of error especially ...
2
votes
1answer
312 views

differential equation with $e^{x-t^2/2}$

I don't manage to solve the following DE $$y''(x)=\int_{-\infty}^{\frac{x^2}{2}} e^{x-\frac{t^2}{2}} \,\mathrm{d}t, \quad x > 0 , \quad y(0) = 0 , \quad y'(0) = 0 $$
2
votes
1answer
519 views

Proof that Normal Distribution is Normalized

How do we know that $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{(-x^2/2)}dx$ = 1. Or how do we know that the normal distribution is normalized? Or how do we know $erf(\infty) = 1$ ?
0
votes
2answers
59 views

Hint for integrating exp(x-x^2)

The function $e^{x-x^2}$ is zero if $x \to \infty$ or $x \to -\infty$ it looks like a normal-distribution-curve with the max. value at $x=0.5$. Has somebody a hint for integrating it from $-\infty$ ...
0
votes
1answer
867 views

Inverse Laplace Transform and error function

Express your answer in terms of the error function: $$L^{-1}\left[\frac{1}{\sqrt{s^3+as^2}}\right]$$ Clue: $\qquad L\left[\frac{1}{\sqrt{t}}\right]=\sqrt\frac{π}{s} \qquad , \qquad s>0$ Error ...
1
vote
2answers
581 views

Integral of the error function

So I know that $$\displaystyle \int_{0}^{\infty} \text{erf}(x) dx$$ does not converge so I am assuming that $$\displaystyle \int_{0}^{\infty} \frac{\text{erf}(x)}{x} dx$$ does not converge? Is ...
1
vote
1answer
126 views

Inverse error function, its analytic continuation and Hardy space

Let $\mathrm{erf}(x) := \frac{2}{\sqrt{\pi}} \int_{-\infty}^x \exp(-t^2) \, dt$ be the error function $\mathrm{erf}: \mathbb{R} \to (-1,1)$. It is monotonously increasing and therefore has an inverse $...