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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An upper bound involving the second derivative of the error function

I am trying to bound a function of the form \begin{align} f(x,y) &= \operatorname{erf}(x+y) - 2\operatorname{erf}(x) + \operatorname{erf}(x-y), \end{align} for small values of $y$ and all (...
ors's user avatar
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What's $\int_{-\frac{\pi}{2}}^ {\frac{\pi}{2}}\text{erf}\left(\frac{\sqrt 2 R\cos\theta}{\sigma}\right)\text d\theta$?

The context of $$\int_{-\frac{\pi}{2}}^ {\frac{\pi}{2}}\text{erf}\left(\frac{\sqrt 2 R\cos\theta}{\sigma}\right)\text d\theta$$ is it came up whilst integrating the Rayleigh distribution function over ...
Joan S. Guillamet F.'s user avatar
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Solve or approximate the maxima of a function

Function is given by $$f(\lambda) = \lambda\exp\left(\frac{\lambda}{2}(\lambda a^2 - 2b)\right) \times \left(1 - \text{erf}\left(\frac{\lambda a^2 - b} {\sqrt{2} a}\right)\right)$$ where $\lambda>0$...
Zuba Tupaki's user avatar
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Gaussian integral of squared shifted error function

The integral of a shifted error function with respect to the Gaussian measure admits a nice closed form expression: $$ \int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{1}{2\sigma^{2}}(x-...
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What's wrong in this expansion of this erfc-like function?

Given the function $$H(x)=\int_{-x}^{+\infty}dy \frac{e^{-y^2/2}}{\sqrt{2\pi}}$$, I would like to write it as a series expansion employing integration by parts: $$\int_{-x}^{+\infty}dy \frac{e^{-y^2/2}...
Salvatore Manfredi D's user avatar
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equation related to lagrange error formula.

So im trying to show that $E_2(x)$ = $\int_{a}^{x} \frac{f'''(t)(x-t)^2}{2}dt$. In the problem they want this to be done by using the equation $E_2(x)$ = $E_1(x)$ - $\frac{f''(a)(x-a)^2}{2}$ where we ...
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Seeking Detailed Explanation for Transforming an Integral Equation Using Euler's Formula and Error Function

I am working on understanding the transformation of a specific integral equation into a simpler form using Euler's Formula and the Error Function. The original equation is: $$ u(x, t) = u_0\left\{1 - ...
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Lognormal RVs: How much probable it is finding values near the mode than near the average?

Log-Normal RVs: How much probable it is finding values near the mode than near the average? Intro_______________ I was trying to got an insight into this question, and I think in the following: since ...
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Convergence rate for Chebyshev polynomials to approximate $\text{erf}(x)$ on a subset of $\mathbb{R}$

Let $[-\alpha, \alpha] \subset \mathbb{R}$, and let \begin{equation} \text{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}dt. \end{equation} given projections of $\text{erf}(x)$ onto the first $k$ ...
Cuhrazatee's user avatar
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How to do $\dfrac d{dx}\operatorname{erf}(1/x)$?

So about an hour ago, I asked this question on if my method of finding the antiderivative of $e^{-1/x^2}$ was a valid method. About seven minutes ago (as of writing this) I got this comment by @GEdgar:...
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Pseudo-inverse matrix to minimise $L^1$ norm error

Given the over-determined linear system $A\cdot x = b$, the least-squares solution (minimise $||A\cdot x - b||_2$) can be obtained by matrix multiplication: $$x_\text{ls} = A^\dagger \cdot b $$ where $...
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How to integrate $\frac{1}{\sqrt{1+\ln(x)}}\mathrm{d}x$?

How to evaluate $$\int \frac{1}{\sqrt{1+\ln x}}\mathrm{d}x$$ I am trying to evaluate this integral by substituting $\ln x =u$. So, $dx$ $=$ $e^{u}du$. So, the integral will become $$\int \frac{e^{u}}{\...
Syamaprasad Chakrabarti's user avatar
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Why does the quantile function of bivariate normal variables become non-elementary in one dimension?

I have been studying a bivariate random process where $X \sim N(0, \sigma_x), Y \sim N(0, \sigma_y)$. It turns out that finding an ellipse that covers proportion p of samples on this process is given ...
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How to minimize the maximum absolute difference between 2 functions?: example $\min_a\{\|\text{erf}(x)-\tanh(\frac2{\sqrt{\pi}}(x+a x^3))\|_\infty\}$

How to minimize the maximum absolute difference between 2 functions?: example $\min_a\{\|\text{erf}(x)-\tanh(\frac2{\sqrt{\pi}}(x+a x^3))\|_\infty\}$ Intro_______________ In this other question I ...
Joako's user avatar
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Integrable approximation of error function over Gaussian measure

I am interested in a problem that involves computing the expectation of of the CDF $\Phi$ (or equivalently erfc shifted and scaled) for the standard normal distribution, for $x$ normal distributed ...
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Does the rest of this family of continued fractions have closed forms?

The pattern for the continued fractions below is quite straightforward. $F_1$ has numerators with all the integers but, $F_2\; \text{is missing}\; 2m+1 = 3,5,7,\dots\\ F_3\; \text{is missing}\; 3m+1 = ...
Tito Piezas III's user avatar
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Approximating for the Error function $\text{erf}(x)$ through an Hyperbolic tangent function $\text{tanh}\left(\dfrac{4x}{4-x^2}\right)$

Approximating for the Error function $\text{erf}(x)$ through an Hyperbolic tangent function $\text{tanh}\left(\dfrac{4x}{4-x^2}\right)$ I was plotting some functions and I found that the function $$f(...
Joako's user avatar
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9 votes
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On the cubic counterpart of Ramanujan's $\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots$?

We have Ramanujan's well-known, $$\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots\color{blue}+\,\cfrac1{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\...
Tito Piezas III's user avatar
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Expressing Ramanujan's $\sqrt{\frac{\pi\,e}{2}}$ as $two$ continued fractions

Due to a recent comment by Akiva about this post, I decided to revisit Ramanujan's beautiful continued fraction (plus series) relating $\pi$ and $e$, $$\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\...
Tito Piezas III's user avatar
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"$\dfrac{x}{\sqrt{2}}$" is neglected so it proves neither A or B, how to fix my proof?

How to prove A)$$F(\mu+n \sigma)-F(\mu-n \sigma)=\Phi(n)-\Phi(-n)=\operatorname{erf}\left(\frac{n}{\sqrt{2}}\right)$$ B) $$F(x)=\Phi\left(\frac{x-\mu}{\sigma}\right)=\frac{1}{2}\left[1+\operatorname{...
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Error function equality proof

How to prove $$\left(\frac{c}{\pi}\right)^{\frac{1}{2}} \int_p^q e^{-c x^2} \mathrm{~d} x=\frac{1}{2}(\operatorname{erf}(q \sqrt{c})-\operatorname{erf}(p \sqrt{c})) .$$ I know$$ \operatorname{erf}(q ...
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Sum over recursive sequence with non-constant coefficients

I wish to find an expression for $\operatorname{Im}\left(\operatorname{erf}\left(u+i\frac{p \pi}{u}\right)\right)$ that I can integrate over $u$ in some more complicated integral, with $p$ a positive ...
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Evaluate $f(x)=(1−\cos(x))/x$ for arguments $0<x≪1$

How would one usefully evaluate the function $f(x)=\frac{1-\cos (x)}{x}$ for arguments $0<x \ll 1$ evaluate? Calculate $f\left(10^{-4}\right)$ with an error smaller than $10^{-10}$. Unfortunately, ...
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Trapezoidal and Simpsons rule Error Analysis

I was hoping someone could help me understand how to find the upper bound of the second derivative of the function above so I could use it in analyzing the error in the trapezoidal or Simpsons rule.
Brandon Sharp's user avatar
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Solve $y=\operatorname{erf}(x+c)+\operatorname{erf}(x-c)$ for $x$

$\newcommand{\erf}{\operatorname{erf}}\newcommand{\erfc}{\operatorname{erfc}}$ Is there a closed form solution for $y=\erf(x+c)+\erf(x-c)$? More specifically, I want to solve $$\erf(\frac{y}{\sqrt{2}})...
Hyperplane's user avatar
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Simplifying the error function

I have two functions. $$ y=a\operatorname{erf}\left(\frac{t-\mu}{2^{1/2}\sigma}\right), \quad\mu=t_1+c\sigma $$ I want to simplify $y$ as much as possible. Is there any way to do it?
nchom's user avatar
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Explicit form for solutions x in $\operatorname{Erfc}(-x) = a \cdot e^{-x^2}$

I'm looking for closed form of points where $\operatorname{Erfc}(-x)$ intersects with $a \cdot e^{-x^2}$ for $a$ being some fixed non-zero constant. I guess I can use the exponential approximation for ...
Piotr Semenov's user avatar
2 votes
5 answers
179 views

Closed form for the similar integrals $\int_0^\infty1-\text{erf}(x)dx$ and $\int_0^\infty x(1-\text{erf}(x))dx$

How do I find a closed form of $$\int_0^\infty1-\text{erf}(x)dx\tag{1}$$? A follow up is $$\int_0^\infty x(1-\text{erf}(x))dx\tag{2}$$Where $$\text{erf}(x)=\frac2{\sqrt{\pi}}\int_0^xe^{-t^2}dt$$The ...
Kamal Saleh's user avatar
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Integrating x,y gaussian distribution over a square area, expressed in terms of error (erf)

I am struggling to wrap my head around the math of a paper I need to understand for my work. In essence, it has to do with super-resolution microscopy, and determining the expected value of a pixel in ...
JHS's user avatar
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Type of singularity at infinity for Faddeeva function, error function?

I am a bit confused with the type of singularity at infinity for the following function. $f(z) = z^2e^{(z-1/2)^2}\text{erfc}(z-1/2)$. Alternatively, we can also use the Faddeeva function to re-write ...
Ranger's user avatar
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Solving a first order non-linear ordinary differential equation

I would like to know how to solve the following differential equation $$ \frac{dy}{dx} = 4y \biggl(\!-\frac{x}{2} - y \biggr) - \frac12 $$ but I'm not sure how to go about doing so given it is non-...
Chris's user avatar
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Which error calculation is the correct one?

I have a function $$ LogC= (LogA - 0.80 * LogB - 8.40)/0.50 $$ Here i have error values for LogA and LogB. So i want to calculate error of LogC. I'm using the formula: $$\Delta LogC=\sqrt{(\frac{\...
Ege Tunç's user avatar
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Error Calculation With Derivative Respect to Logx

I have a function: $$ LogC= (LogA - 0.80 * LogB - 8.40)/0.50 $$ Here LogA and LogB have errors. So I know that if $$LogC=f$$then the error of f is $$df=\frac{\partial f}{\partial a}da+\frac{\partial f}...
Ege Tunç's user avatar
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1 answer
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Define 'accuracy' for numerical data?

Normally, people use 'accuracy' to describe the output quality (from a model or methodology https://en.wikipedia.org/wiki/Precision_and_recall) for categorical data. However, I am wondering could the ...
Edamame's user avatar
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What is the best metric to compute the error for complex time series data?

I have a complex number time series of data of ground truth and estimated values. I am currently using Mean square error metric to check the quality of my method: $$mse = \frac{1}{N} \sum |true-...
Sagar's user avatar
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About the Errors in Finite Approximation

I am reading up on finite approximation recentlyenter link description here, can someone help me explain why the author expressed the interval h as a sinusoidal function in this paper? And how did ...
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Computing $\int_{0}^{\infty }x^a\,e^{-bx}\,Q(x)\,dx$, where $Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty }e^{-t^2/2}\,dt$.

I want to know if this integral can be solved: $$\int_{0}^{\infty }x^a\ e^{-bx}\ Q(x)\ dx\ .$$ where $a,b >0$ are real numbers and $$ Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty }e^{-t^2/2}\,dt = -\...
why_me's user avatar
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Integrating a product of exponential and complementary error function in a communication scenario.

While studying the communication signal processing, it was proved that the following integrals play an important role in the block error rate approximation of the MPSK signals: \begin{equation} \int_{...
LZ981ko's user avatar
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3 answers
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Calculate the integral $\int_{0}^{\frac{\pi}{2}} \cos(x) \left( \int_{1}^{\sin(x)} e^{t^2} dt \right) dx$

Calculate the following integral:$\int_{0}^{\frac{\pi}{2}} \cos(x) \left( \int_{1}^{\sin(x)} e^{t^2} dt \right) dx$. I tried integrating by parts but ended up getting Gauss error function (which hasn'...
Arthur De Arola Brito's user avatar
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The error function for the numerical Dedekind eta function?

The Dedekind eta function \begin{equation} \eta(\tau)=e^{\frac{\pi i \tau}{12}} \prod_{n=1}^\infty (1-e^{2n\pi i \tau}) =q^{\frac{1}{24}} \prod_{n=1}^\infty (1-q^n) \end{equation} where the Eulear ...
ShoutOutAndCalculate's user avatar
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1 answer
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How does this expression reduce to the erfc?

I'm working on characteristic functions, and in this process I have came across the following step in a proof $$ \frac{i}{2\pi}e^{-\frac{1}{2}\tau_k\mu^2/\sigma^2}\int_{-\infty}^{\infty}e^{-\frac{1}{2}...
Jord van Eldik's user avatar
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Finding polynomial degree $1$ of $f(x)=\text{erf}(x-1)$ at $x=1$

Question: Find the polynomial of degree $1$ that has the highest possible order of contact with $f(x)=\text{erf}(x-1)$ at $x=1$. Plot the spline knotted at $(1,0)$ with $f(x)$ on the right and your ...
l0calgod's user avatar
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Does representing the fresnal integral in terms of the error and imaginary error function have a use?

The Fresnal S integral can be represented with the following: $$F(a,b)=\int_a^b{\sin(x^2)}\,dx = \frac{-\sqrt{i\pi}}{4}[\operatorname{erfi}(\sqrt{i}\,b)-\operatorname{erf}(\sqrt{i}\,b)]+\frac{\sqrt{i\...
Npola The Maths Guy's user avatar
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Gaussian integral over a circle

Problem definition I'm finding some difficulties in the computation of the following integral \begin{equation*} I_C(y)\triangleq \int_{C(a)} \exp\left[-\frac{1}{2}\left(\frac{\nu_1-y_1}{\sigma_v}\...
matteogost's user avatar
1 vote
2 answers
53 views

Asymptotic expression for complex root of $\text{erf}(z)$

I tried to figure out if the complex roots of $\text{erf}$ function follows some kind of rule: First of all I wrote it with integral representation: $$\text{erf}(z)=\text{erf}(x+iy)=0 \quad \...
Math Attack's user avatar
2 votes
0 answers
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Definite integral involving the error function

This is my very first question on stack exchange, nice to meet you all! Let me apologise in advance in case I did not respect some conventions for a first question. I came across this integral: $$I:=\...
Matthieu's user avatar
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double integral of two Gaussians and complex poles

Recently encountered an integral: $$ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \dfrac{ (x_1+x_2) e^{-i(x_1+x_2)k} \exp\left(-\frac{(x_1-x_0)^2}{2\sigma^2} -\frac{(x_2-x_0)^2}{2\sigma^2}\right) }{...
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Is the error function essential to $\int_0^{\infty} e^{-a x^2} \sinh (b x) d x$?

In the post, there are 4 methods to find the integral $$ \int_0^{\infty} e^{-a x^2} \cosh (b x) d x= \frac{e^{\frac{b^2}{4 a}}}{2} \sqrt{\frac{\pi}{a}}, $$ I believe that we can evaluate its partner ...
Lai's user avatar
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maximize a product of error function

I want to maximize the value of product of two error functions. Given a function \begin{align} f(a) = erf\left(\frac{c}{a}\right) erf\left(\frac{c}{\sqrt{1-a^2}}\right) , \end{align} where $c\gg 1$, ...
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Soundness of changing the error function of a squared loss to force a good Hessian condition number

Suppose we have a homogeneous linear model $y = w^T x$ and define the following error function to minimize: \begin{align} E(w) = \alpha||w||^2 + \frac{1}{N}||X^Tw - t||^2 \end{align} This is just the ...
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