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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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 ...
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0answers
14 views

Sum of a converging series having Error function with a polynomial

I am struggling to find the sum of the following series: $k\sum\limits_{z=1}^{\infty} \frac{(z+1)^2}{4} . erfc(az)$ where $k$ and $a$ are known parameters and $erfc(x)$ is the complementary error ...
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1answer
84 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 $t = \log (x)$, ...
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3answers
68 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} ...
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1answer
42 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 ...
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5answers
153 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 $0$ ...
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1answer
90 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'...
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1answer
27 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{\...
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0answers
23 views

Minimal time to steady-state flow (1st Stokes Problem)

How to obtain a minimal time to steady-state flow? This is related to the Stokes 1st Problem. $$\frac{u}{U}=0.05=1 - \operatorname{erf}(\eta)$$ $$\eta = \frac{y}{2\sqrt{vt}} \approx 1.4$$ $$y=\...
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28 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 ...
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1answer
48 views

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

Here is the error function: $$erf(x)=\frac{2}{\sqrt\pi}\int^x_0e^{-t^2} dt$$ Here is the question: Show that the odd function erf is bounded, by using the fact that:$$e^{-t^2} \le te^{-t^...
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2answers
54 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 $\lim_{N\rightarrow\infty}S(N)$ ...
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1answer
45 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 ...
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30 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'...
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24 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{\...
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1answer
38 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 ...
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1answer
24 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}{\...
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2answers
29 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 ...
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1answer
81 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 $\...
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1answer
35 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 ...
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1answer
46 views

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

The so-called error function is defined as: $$erf(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^{2}}dt$$ show that the function $y(x) = e^{x^2}erf(x)$ satisfies the differential equation: $$\frac{dy}{dx}=...
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55 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}\...
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1answer
40 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)}{...
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2answers
139 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}...
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1answer
18 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 ...
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13 views

Total Error Bound for $D_1(h)$ Approximation

I am asked to find a total error bound for the approximation $D_1(h)$ in terms of $M_2=\max|f''(x)|$ and $\varepsilon=|f(x)-\bar{f}(x)|$. I began research what exactly $D_1(h)$ is and found $D_1(h)=...
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121 views

Computing the inverse error function

I need a formula/series to compute the inverse error function, which is the inverse of $$ \operatorname{erf}(x) = \dfrac{2}{\sqrt{\pi}} \int\limits_{0}^{x} \mathrm{e}^{-t^2} \,\mathrm{d}t. $$ ...
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2answers
62 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'...
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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{...
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63 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 $\...
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1answer
36 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 ...
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3answers
29 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}...
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156 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=\...
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1answer
36 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}...
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0answers
17 views

What are disadvantages of solving difference equations backwards?

Say you have the initial value of 100th and 99th term and you want to see how much error is in 1st term instead of picking values from the table for 1st and 2nd term and propagating to 99th and 100th ...
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29 views

integration of product of error function

Is there a more simple formulae for the following integral $$ \int_{a}^{+\infty} erf(\alpha x).erf(\beta x) \frac{1}{x^2} \:\mathrm{d}x $$ where $a>0$, $\beta>0$ and $\alpha>0$
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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 ...
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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 ...
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21 views

continued fraction expansion of the complementary error function

Could someone please explain to a non-mathematician why the continued fraction expansion of the complementary error function is the following: fraction How does one come to this fraction as ...
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43 views

Integration involving Error function

I am interested in the following integral $$\int_0^\infty x^n\text{Erf}[ax]j_m(bx)dx,$$ where Erf is the error function $j_n$ is the spherical Bessel function of first kind. Does the analytical ...
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1answer
34 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 ...
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73 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 ...
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15 views

Minimizing error of estimation in a differential equation system

I have a system of equations which describe dynamic nature of a system. There is no closed form solution for this system. System of equations is as follow: $$I_c=I_1 + I_2 + I_3$$ $$R_3 = \frac{V_3}{...
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31 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 ...
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301 views

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

I don't manage to solve the following DE $$y''(x)=\int_{-\infty}^{\frac{x^2}{2}} e^{x-\frac{t^2}{2}} \,\mathrm{d}t, \quad x > 0 , \quad y(0) = 0 , \quad y'(0) = 0 $$
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25 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 $erf(\infty) = 1$ ?
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2answers
55 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 $-\infty$ ...
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1answer
57 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 ...
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2answers
75 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 ...
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1answer
45 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 $...