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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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 ...
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10 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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22 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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77 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 ...
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39 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 ...
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11 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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28 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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32 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 ...
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46 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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13 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 = ...
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25 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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1answer
294 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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1answer
17 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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49 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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48 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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70 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
24 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 ...
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36 views

Existence of solution for equation with erfc

I'm looking for the self-consistent (e.g. input needs to be the same as output) solution of $r$ in $$ r = \frac{1}{2}\operatorname{erfc}(z(r)) $$ where $\operatorname{erfc}$ is complimentary error ...
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22 views

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 ...
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12 views

Show that this conversion between $\Phi$ and erf(z) holds for all z

I am trying to wrap my head around the connections between the standard Normal distribution and the error function. I could use some help working through the following problem. Show that the ...
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3answers
98 views

Integral of $e^{-x^{2}}$ and the error function

How to integrate $e^{-x^{2}}$? When I used geogebra I got the answer as $\frac{1}{2}\sqrt{\pi}\operatorname{erf}(x)$. What is $\operatorname{erf}(x)$ ? How to arrive at this answer?
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19 views

Integrating the product of two imaginary error functions

I want to evaluate the integral of a product of two imaginary error functions as in the following: $$\int^{L_2}_{-L_2} \operatorname{erfi}(g(y)) \operatorname{erfi}(h(y)) \, dy$$ where $g(y)$ and ...
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57 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 ...
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20 views

Modeling Gaussian components with standard vs exact functions

I'm studying a paper on modeling DNA histograms. It presents two alternative formulas for modeling Gaussian components: Standard form: $G(x) = ...
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1answer
59 views

Difference between the Error function and Normal distribution function?

I have just started reading about the Error function but Distribution theory is not my strong point. So I apologize in advance for asking really simple questions about it. So the Error function is ...
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46 views

How do I symbolically compute $\int_{a}^{b} e^{x^2}(\textrm{erf}(x) - \textrm{erf(a)})\;\textrm{d}x$?

I want to symbolically write (in the form of a series), the integral of: $$ \int_{a}^{b} e^{x^2}(\textrm{erf}(x) - \textrm{erf(a)})\;\textrm{d}x, \text{where }\{x, a, b\} \subset \mathbb{R} $$ The ...
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41 views

Convert an integral with exponential of cosine to the error function

I have an expression that looks like $$\frac{1}{T}\int_0^T\left(e^{\alpha_\theta(\cos\frac{\pi t}{T})^2}-1\right)\,dt,$$ where $\alpha_\theta,T$ are constants. I'd like to rewrite it into a form ...
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12 views

Taylor series expansion - error term

Consider the equation $$ f'(x) \approx \frac{1}{12h} \left( -f(x+2h) + 8f(x+h) - 8f(x-h) + f(x-2h) \right)$$ I wish to determine the $\textbf{error}$ term of this approximation (and I don't know how ...
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14 views

Where does mean and standard deviation go in the error function?

The error function is defined as $$ \textrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt~.$$ However, the normal distribution can take a more general form than the definition of the error ...
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96 views

Integration with Log of error function (erf)

Can anybody help me evaluating the closed-form or an approximate form of $H(x) = \int P(x) \ln(P(x)) \Bbb dx$ where $P(x) = \frac{C(x)}{v\int C(x) \Bbb dx}$ and $C(x) = {\frac ...
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40 views

Solving an integral that includes an exponential function and the error function

This question contains all the values needed to compute an equation. My question is, do you get the same result I get? Or do you get the result in the paper I've linked to? I'm trying to decipher ...
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58 views

Solve:$\int_{0}^{t}{{\left(\cos({…})+\sin({…})\right)} \frac{\lambda^2 e^{(…)}}{\sqrt{\pi (t-r)}} \text{Erfc}{\left(… \right)} }~\mathrm{d}r$

I have another nasty integral to solve as follow: $$ I(t)=\int_{0}^{t}{{\left(\cos({\frac{\gamma}{4(t-r)}})+\sin({\frac{\gamma}{4(t-r)}})\right)} \frac{{\lambda^2} e^{2 \lambda^2 r+ \lambda ...
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44 views

Maximum Error Bound on a Cubic Spline, Chebyshev Polynomials

Most notations for the error resulting from interpolation of cubic splines (e.g. http://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture12.pdf) requires knowledge of the original function to ...
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34 views

how to derive error function

I saw many places where erf(x) is used. As per wiki it is defined as gauss error function shown here . I recently used this to derive the cumulative probability density function for normal ...
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16 views

How can I calculate percent error with a denominator of 0? [duplicate]

I am using the MAPE formula, ABS(actual-forecast)/actual *100, but this raises problems when the actual values are zero. Is it possible to add 1 to the denominator and calculate this way? Would I get ...
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1answer
46 views

Asymptoting to 0: is erfc(z) quicker than exp(-z)?

If I have a function of the form $\mathrm{erfc}\left(z\right)/e^{-z}$, should I expect its limit at large $z$ to be $0$ or $\infty$? My instinct is that it should be $0$, by considering the ...
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16 views

Proving equally probability functions - general to specific case

I want to verify that my general probability expression can be reduced to a familiar case that was published. I can see that if I assign the parameters right in my case, I can get the same results ...
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101 views

Derivation of approximation of Error function

In Abramowitz and Stegun there are some formulas for approximation of Error funtion. I am intrested in the formulas $7.1.25$ to $7.1.28$, here is one of them ($7.1.26$). ...
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91 views

Does anyone know of a closed form solution to the following integral?

Does anyone know of a closed form solution to the following integral? $$ \DeclareMathOperator\erf{erf} \newcommand{d}{\;\mathrm{d}} \int^{+\infty}_{-\infty} \erf^{\;m}\!(x) \frac{\d^n ...
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14 views

Equation for standard error of weighted mean

What should I use when to calculate standard errors (and thus a confidence interval) for weighted means? Do I simply substitute the weighted mean for the simple arithmetic mean?
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99 views

inverse complementary error function values near 0

$\newcommand{\erf}{\operatorname{erf}}\newcommand{\erfc}{\operatorname{erfc}}$Let's define for each $x>0$ $$\erf(x)=\frac {2}{\sqrt{\pi}}\int_0^xe^{-t^2} dt$$ and $$\erfc(x)=\frac ...
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29 views

Standard Error for Weighted Values

I want to calculate the standard error for an experimental measurement. The data is stored as a 2D image which is circularly symmetric about a center point. To reduce the data we radially integrate ...
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53 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 ...
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57 views

Closed form for integral of an error function

My question is similar to that posted here. I have the following integral that I want to determine in a closed form. My uncertainty arises due to the addition term within the Error function: ...
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72 views

Closed forms for definite integrals involving error functions

I have been working for a while with these kinds of integrals $$\int_0^\infty dx\,\text{erfc}\left(c +i x\right)\exp \left(-\frac{1}{2}d^2x^2+i cx\right)$$ $$\int_\Lambda^\infty ...
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85 views

Integrating a product of to error functions and an exponential

I have the following integral that I need to solve. $\int_{-\infty}^\infty \exp(-\frac{x^2}{2})*\text{erf}(x-\delta)*\text{erf}(x-\gamma)dx$ I was hoping I could use this: Integral of product of ...
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45 views

Solving $\int dx {\sqrt{x^2+a}} e^{-A x^2} erf \left( c(x-b) \right)$

I got as far as: $$\int dx {\sqrt{x^2+a}} e^{-A x^2} erf \left( c(x-b) \right) $$ $$=\frac{2}{\sqrt{\pi}} \int dx \int^{c(x-b)}_0 dy {\sqrt{x^2+a}} e^{-A x^2 - y^2}$$ $$=\frac{-2 c}{\sqrt{\pi}} ...
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1answer
92 views

Sigmoid function that approaches infinity as x approaches infinity.

The function I'm looking for looks like an error function, but instead of having asymptotes $1$ and $-1$, the function I'm looking for does not have asymptote. It increases to infinity. The ...
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1answer
373 views

Remainder Taylor series two+ variables

Hard to find an article / tutorial specifically about this subject online.... Can anyone explain / send link to explanation or tutorial regarding how to calculate remainder for multi-variable taylor ...
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106 views

How can I solve the integral in the error function $\mbox{erf}(x)$?

To get from this To this series I can't seem find the step-by-step solution anywhere.