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.

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Is the error function and $\mathcal{N}(0,1)$ the same thing?

Are $\mathcal{N}(0,1)$ and $\Phi(x)$ the same thing? It seems that the derivative $\Phi(x)$ and the density of $\mathcal{N}(0,1)$ are the same, but I'm not sure.
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Prove $ \operatorname{erfi}(x) \in O(e^{x^2})$

How can we prove: $$ \operatorname{erfi}(x) \in O (e^{x^2}) $$ Samely we can prove: $$ \lim_{ x \rightarrow \infty } \frac {\int_0^x e^{t^2} \, dt} {e^{x^2}} = 0 $$ Also it is so awkward. The ...
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Lower bound on complementary error function for $|x|<1$

Let us define the complementary error function $$\mathrm{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2}\, dt$$ I am interested in establishing a lower bound on the region $|x| < 1$ for $x \...
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Maximum error bound for ODE

For $f\in C([0,T]×\mathbb{R})$, let $|f(t,x)-f(t,y)|\leq L|x-y|$ for a $L\geq0$. Also suppose we have a differential equation \begin{align} y'(t) &= f(t,y(t))\\ y(0)&=y_0 \end{align} with $y_1,...
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Deriving the Erf function from an integral

I am reading this mathematical paper, and it states the following on the second page: "This is the classical solution if we note that $$\frac{1}{\pi} \int_0^\infty\ \frac{1}{\alpha} e^{-\alpha t}\...
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How would you derive this expression related to transient temperature rise?

I need help with a math problem for math club tomorrow. Its a fun presentation, but I need help with this one problem. Thanks in advance! Given the transient point source solution valid within an ...
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Minimizing mean absolute percentage error by a constant.

The task is to minimize the mean absolute percentage error (MAPE) by constant C. $$MAPE = \frac{1}{N}\sum_{i=0}^N{|\frac{y_i - C}{y_i}}|$$ Given values $y_0, .., y_N$. Find the constant C at which the ...
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Unscale the mean square error of standardized data

I have a question please about the mean squre error resulting from the analysis of standardized data. How should I unscale it and use the real units of the variable. I have standardized the data (Z) ...
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Rigorous description of the asymptotics for $\operatorname{erfi}(z)$ as $|z|\to\infty$

According to Wolfram functions, the imaginary error function admits an asymptotic expansion for $|z|\to\infty$ of the form $$ \tag{1} \operatorname{erfi}(z)\sim\frac{z}{\sqrt{-z^2}}+\frac{e^{z^2}}{\...
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An complicated integral involving Erfi function

Sir, While, studying the diffraction of the Gaussian beam through apertures, I have faced the following integral (as an expression of diffracted field), $$I= \int_0^R \exp(-\alpha r^2+i\beta r)\bigg[\...
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$\int_{-\infty}^{\infty}\frac{1}{3 t^2+1} {\rm exp}\left(\frac{at^2+ibt}{3 t^2+1}+itx\right){\rm d}t$

How to solve the integral? $$ f(x)=\int_{-\infty}^{\infty}\frac{1}{3 t^2+1} {\rm exp}\left(\frac{a t^2+i b t}{3 t^2+1}+itx\right){\rm d}t\tag{1} \\{\rm with}\,\, x,b\in \mathbb{R},a\in\mathbb{R}_{<...
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Using error function to calculate probability that a random variate falls z standard deviations to the right of the mean

I am imagining a normal distribution with a mean of zero and a standard deviation of S. I know that the following function tells us the probability that a random deviate (under the assumption of ...
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Quadratic-trigonometric integral -- part 2

Problem I need to compute the following integral \begin{equation*}\int_{t_\text{s}}^{t_\text{e}} \cos(a+b\tau+c\tau^2)\text{ d}\tau\end{equation*} where $t_{\text{s}}<t_{\text{e}}$ and $a,b,c>0$...
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Expressing $\int_0^{+ \infty}\frac{\mathrm{e}^{- t^2} \mathrm{d} t}{z^2 - t^2}$ with the error function (complex analysis or Fubini theorem ?)

I would like to show the following (Abrahomitz et Stegun) for any $z\in\mathbb{C}, \Im(z)>0$, $$ \mathrm{e}^{- z^2} \left( 1 + \frac{2 i}{\sqrt{\pi}} \int_0^z \mathrm{e}^{t^2} \mathrm{d} ...
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Question about confidence interval calculation for estimated error (in a paper)

In a classification problem. If there are 250 samples in sum to test (n=250) and number of misclassified samples is 50 (m=50). Therefore according to the paper below (first figure), maximum likelihood ...
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Error Propagation (complicated case)

I have a function: $$G=\frac{(\frac{4\pi^2}{T^2}+\frac{1}{\tau^2})\cdot r^2 \cdot d\cdot S}{4M\cdot L}$$ and I need to do error propagation on this function: $$\delta G=\sqrt{(\frac{\partial G}{\...
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1 vote
1 answer
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Gaussian integral with error function

Given the integral: $$\int_{0}^{+\infty}dz\, e^{-z^2} \text{erf}(z+\beta)=\frac{\sqrt{\pi}}{2}\left(1-\frac{1}{2}\text{erfc}^2\left(\frac{\beta}{\sqrt{2}}\right)\right),$$ where $\text{erf}(z)$ and $\...
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Evaluate $\int_0^t e^{-\lambda s} \,\text{erf}(\ln(t-s))\,\text{d}s$?

I'm trying to efficiently graph the function $I(t)$ for $t>0$ where $$I(t) :=\int_0^t e^{-\lambda s} \,\text{erf}(\ln(t-s))\,\text{d}s,\qquad \lambda>0$$ but its evaluation is beyond my powers ...
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1 answer
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Integral Computation - Error Functions

Consider the following integral: $$ I(t) = \int_{0}^{t} e^{\alpha(\tau+\beta)^2} \ d\tau. $$ I want to compute this integral analytically, which I believe can be done through the use of error ...
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How to Place $\int_{-\infty}^0 e^{-p^2} dp$ in the form of the error function?

The error function is definted as $erf(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-p^2} dp$. I have the function $\frac{2}{\sqrt{\pi}}\int_{-\infty}^0 e^{-p^2}dp$. I can switch the bounds of the integral, ...
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Quantile function of two-term gaussian

I'm trying to find the quantile function of the two-term gaussian. From https://statproofbook.github.io/P/norm-qf.html, I've got that I can take the inverse of the CDF of the two-term gaussian. I've ...
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2 votes
1 answer
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An integral related error function

I am trying to solve the following integral: $$ \int_{-\infty}^{\infty} \big(\text{erf}(x)\big) ^{n} \exp\left(-3x^{2}\right) \ \mathrm dx $$ where $\text{erf}$ is the error function and $n$ is an ...
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Alternative approach to error function for integration

I am integrating $\int_{-\infty}^{\infty}x^2\frac{1}{\sqrt{2\pi}}e^{-x^2/2}dx = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}x^2e^{-x^2/2}dx$ I start with integration by parts: $u=x \rightarrow \frac{...
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What is special about a function being elementary? [duplicate]

Functions such as $ \sin(x) $ are considered to be elementary, however functions like $ \text{erf}(x) $ are considered to be non-elementary. What makes elementary functions different from non-...
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Can I use the asymptotic expansion of the Error function for complex arguments?

I tried to evaluate a limit of the form $$ \lim_{x\rightarrow\infty} \mathrm{erfi}\left(zx\right) $$ for some complex number $z\neq0$. Following this answer, one idea would be to write $iz=\left|z\...
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Though process to calculate error rate for a classification algorithm with 1000 objects?

I am trying to solve this question A classification algorithm classifies 1000 objects in to one of two classes. It incorrectly classifies 13 out of 100 class 1 objects and 53 class 2 objects. (a) What ...
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4 votes
1 answer
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Evaluating $\int e^{-\sqrt{a^2 - b^2} \cosh(x)} \, dx$

I've been trying to solve the definite integral \begin{align*} I = \int_{0}^{u} e^{-\sqrt{a^2 - b^2} \cosh(x)} \, dx \, , \end{align*} with $u = \mathrm{arctanh}(\frac{b}{a})$ and $a > b, \, a &...
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6 votes
2 answers
344 views

How to evaluate $\sum\limits_{x=0}^\infty \text{erfc}(x)= 1.1619990479471263635323…?$

This will be the $5$th in a series of an infinite series of a single function. Here are 2 related sums: A Kelvin-Bessel Sum: $$\mathrm{\sum\limits_{\Bbb N} ker(x)+i\ kei(x)= \sum\limits_1^\infty K_0\...
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Is accuracy a smooth function in binary classification problems?

As far as I understand, for a binary classifier which outputs are either 0 or 1, the accuracy is same as 1 - MSE where MSE stands for Mean Square Error. MSE is a ...
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3 votes
1 answer
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Solving integral using feynman trick

For $x,t > 0 \ $ I want to prove $$\int_0^\infty \frac{\sin(x n)}{x \ n} e^{-t n^2} dn = \text{erf} \left(\frac{x}{2 \sqrt{t}}\right) $$ using the feynman trick. My problem is I don't know which ...
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5 votes
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$\int_0^{\pi/2} e^{-\tan^2(x)}\,dx=\frac{\pi e}{2}\big(1-\operatorname{erf}(1)\big)$ [duplicate]

Prove that $$\int_0^{\pi/2} e^{-\tan^2(x)}\,dx=\frac{\pi e}{2}\big(1-\operatorname{erf}(1)\big)$$ My Attempt Let $u=\tan(x)\implies du=\sec^2(x)\,dx=1+u^2\,dx$ $$I=\int_0^{\pi/2} e^{-\tan^2(x)}\,dx=\...
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3 votes
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ODE: $y''y+ax+by+c=0,y=k\pm\sqrt2\int\sqrt{a\int\ln(y)dx-(ax+c)\,\ln(y)-by+K}dx,\int\frac{dy}{\sqrt{K-(ax+c)\,\ln(y)+a\int\ln(y)dx-by}}=k\pm\sqrt2x$

Imagine we had a differential equation like: $$y’’-\frac xy=0$$ Now let’s standardize the signs. Note we do not need a constant for the first term because of the zero product property. We can ...
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Definite integral of erf, exponential (exp(-(a-x)^2)) and power (x)

I am trying to see if I can have a closed form solution for the following integrals (I give two forms in case one of them is more usefull than the other). $$I_1 = \int_{0}^{t} e^{-(a + b y)^{2}} \...
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1 vote
1 answer
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Positivity of Faddeeva function's imaginary part in the first quadrant of a complex plane

I was playing around with the Faddeeva function $$\omega(z) = e^{-z^2} \left( 1 + \dfrac{2i}{\sqrt{\pi}} \int\limits_{0}^{z} e^{-t^2} dt \right)$$ and noticed numerically that for $z = x+iy$, and $x&...
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Integratation of erfc(a+bz)

Trying to solve $\int$erfc$(a+bz) dz$, but my answer doesn't seem to match what is given here: https://functions.wolfram.com/GammaBetaErf/Erfc/21/ShowAll.html: What I've got so far: $\int\text{erfc}(...
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Statistics of Gaussian Random Walk Passed Through the Heaviside Function

Let $$D=\frac{1}{N}\sum_{n=1}^{N}H\left(\xi_{n}-1\right),$$ such that $\xi$ denotes a Gaussian random walk with mean $\mu$ and $\sigma$, passed through the Heaviside function $$H(x-1)=\begin{cases} 1, ...
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12 votes
2 answers
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Solving the integral $\int_{0}^{1} d{v} \frac{e^{-y^2(1+v^2)}}{1+v^2}e^{\frac{2 t v^2}{v^2+1}}$

For my research, I have to solve many integrals of the Owen's T function. As such, I am having struggles in calculating the integral $$ \int_{0^{-}}^{t}d{s} \ e^{(t+s)} \ \operatorname{T}\left(\frac{...
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2 votes
1 answer
162 views

Is there an analytical solution to $\int_{0^{-}}^{t} d{s} \ T\left(\frac{x}{\sqrt{2t}}, \sqrt{\frac{s}{2t-s}}\right)$?

I am trying to solve the integral \begin{align} \int_{0^{-}}^{t} d{s} \ \ T\left(\frac{x}{\sqrt{2t}}, \sqrt{\frac{s}{2t-s}}\right) \end{align} where $T$ is Owen's T function. I have been trying to ...
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4 votes
3 answers
179 views

Solving the integral $\int_{0}^{t} ds \sqrt{\frac{s}{t-s}} \operatorname{erfc}\left(\frac{a}{\sqrt{s}}\right)$

I want to solve the following integral $$ \int_{0}^{t} ds \sqrt{\frac{s}{t-s}} \operatorname{erfc}\left(\frac{a}{\sqrt{s}}\right) $$ with $a\in\mathbb{R}$ and $t\in\mathbb{R}^+$. I have tried some ...
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1 vote
1 answer
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Define integral containing error function and exponential function

I am trying to solve the following improper integral $$I(G) = \int_{-\infty}^{\infty} \! \exp(i G r) \, \textrm{erf}(r) \, \dfrac{1}{r^2} \, dr $$ for $\{G \in \mathbb{R}: G \neq 0 \}$. For the case ...
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Simple approximations for $\operatorname{erf}(x)$

I am looking for some potentially unexpected approximations of $\operatorname{erf}(x)$, by which I mean not a really long series expansion, and something that resembles the correct shape for an ...
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Prove that $E_θ(θ-\frac{X}{100})^2 = \frac {θ(1-θ)}{100}$

$X \text {~} Binomial(100,θ)$, $δ(X)=\frac {X}{100}$, $g(θ)=θ$ and the loss function is given by $L(θ,d)=(θ-d)^2$. The risk function for δ is $R(θ,δ) = E_θ(θ-\frac{X}{100})^2 = \frac {θ(1-θ)}{100}$ I ...
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Simplifying an expression with erf and exponential integral functions

For determining the spread of particles in a force field due to the initial speeds having a normal distribution, I was able to derive the following simple approximation: $f(x) = erf(x) - \frac{x}{\...
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Gaussian Integrals of Powers of Complementary Error Function

Can anyone help me with the following integral: $$ \int_{-\infty}^\infty dt \frac{\exp\left\{-t^2/2\right\}}{\sqrt{2\pi}} \left(\frac{1}{2}\text{erfc}\left\{\frac{a-bt}{\sqrt{2}}\right\}\right)^n, $$ ...
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How to compute $\int \sqrt x e^{-x/2} \operatorname{erfc}(\sqrt{x/2})^{m}dx $ [closed]

Hello I want to compute $\int_0^\infty \sqrt xe^{-x/2} (\operatorname{erfc}(\sqrt{x/2}))^{m}dx $ where $\operatorname{erfc}(x)=1-\operatorname{erf}(x)$ with $\operatorname{erf}(x)$ the standard error ...
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How to compute $\int_0^\infty \sqrt{x}e^{-x/2}(erf({\sqrt{\frac{x}{2}}}))^{r-1} (1-erf(\sqrt{\frac{x}{2}}))^{m-r}dx$

I want to compute $\frac{m!}{(r-1)!(m-r)!} \frac{1}{\sqrt{2\pi}} \int_0^\infty \sqrt{x}e^{-x/2}(erf({\sqrt{\frac{x}{2}}}))^{r-1} (1-erf(\sqrt{\frac{x}{2}}))^{m-r}dx$ where $m, r \in \mathbb{N}$ ...
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Get the Q function value for a Probability Density Function with an absolute value

Let $X$ be a random variable with mean $ \mu $ and variance $ \sigma ^ 2 $. Then how do we express the probability $P(|X| \ge t)$ in terms of the Q function (error function) where $t\geq 0$ is a ...
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2 votes
1 answer
83 views

How to calculate relative error when values are close to zero or negative?

I'm trying to calculate the relative error of a machine learning model prediction. Normally I'd calculate the relative error this way $RelError=\left | \frac{y-\hat{y}}{y} \right |$ However, since <...
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1 vote
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Accurate averaging in $Q$ scale

Let $Q(x)$ denote the complementary cumulative distribution function of a standard normal distribution (see here). Given $ 0 \leq a \leq b$, and define the $Q$-scale average of $a$ and $b$ as a $c$ ...
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1 vote
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Taking derivative of an equation

I'm trying to apply a boundary condition to the following equation: $$c_A = {K_o}\int_{0}^{u} e^{-u^2} du + K_1$$ where $$u = \frac{x}{\sqrt{4Dt}}$$ and ${K_o}$ and $K_1$ are constants. The boundary ...
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