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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Exact values of error function

The error function is defined as $$\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt.$$ We know that the Gaussian integral is $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
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82 views

closed-form solution to $\int_0^\infty x^a\exp(-bx)\left(\frac{1}{\text{erfc}(c\sqrt{x})}\right)^{2a}$

This integral comes up in a problem in Statistics involving power laws. Here are some notes if anyone is interested. The integral in question would be related to equation (7) therein. I would like ...
7
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1answer
100 views

On $\int_0^\infty \frac{\exp(-x^2)}{1+x^2}dx=\frac{\pi e}2\text{erfc}(1)$

I was attempting to answer this question, but then I came across a question of my own involving my attempt. Task: Prove $$\int_0^\infty\frac{\exp(-x^2)}{1+x^2}\mathrm dx=\frac{\pi e}2\text{erfc}(1)$$ ...
6
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111 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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255 views

Properties and representations of the the rescaled complementary error function $\mathrm{erfcx}{z}$

Consider the rescaled complementary error function: $$ \mathrm{erfcx}(z) = {e^{z^2}} \left( {1-\mathrm{erf}(z)} \right) $$ $z \in \Bbb{C}$ which also has the following integral representation: $$ \...
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556 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 ...
4
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606 views

Integral of two error functions (erf)

In my research I came across the following integral: \begin{equation} \int_{-\infty}^{+\infty}\frac{\partial{p(t)}}{\partial{t}}\frac{1}{4}\Big(1-\operatorname{erf}\Big(\frac{t-a}{\sigma\sqrt{2}}\Big)\...
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116 views

How to construct target objective function for gradient descent?

I have a set of scores from two traits. After applying simple sum rule for fusion i.e. $f_{s}=\sum_{i=1}^{n} w_i s_i$, fused score is achieved. From the fused score, I need to evaluate EER (equal ...
3
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1answer
59 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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287 views

Differentiation under integral sign

There is this integral that I used a lot in my research: $$\int_{-\infty}^{\infty}\exp\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}\left(a\left(x-d\right)\right)\,\mathrm{d}x = \frac{\sqrt{\pi}}{...
2
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1answer
166 views

Asymptotic expansion of Erfi(x)

I know that the imaginary error function, $\mathrm{Erfi}(x)=(2/\sqrt{\pi})\int_0^x \exp{t^2} \mathrm{d}t$, has the asymptotic expansion given in the answer to this question: Asymptotic order of $\frac{...
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30 views

Is $x=-2n$ the solution of :$ \operatorname{erf}(1-x) +\zeta(x)=1$ with $n$ is a positive integer?

let $x$ be a real number , I want to know if $x=-2n$ with $n$ is a positive integer is a solution of this equation:$$ \operatorname{erf}(1-x) +\zeta(x)=1$$ as shown here by wolfram alpha , ...
2
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1answer
132 views

defining inverse error function

The error function is define like this: $$\operatorname{erf}(x):=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^2}dt$$ If i take the derivative i get $$\operatorname{erf'}(x)=\frac{2}{\sqrt\pi}e^{-x^2}$$because ...
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151 views

$\int_c^{\infty} \exp(-u^2) \text{Erf}(au + b) du$

Interested in a closed form solution for $$ \int_0^{\infty} \exp\left(- \left(\frac{x-z_3}{z_4}\right)^2 \right) \text{Erf}(z_1 x + z_2) dx $$ I started by changing the integral to a more nicely ...
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105 views

Gaussian integrals with cumulative normals

Definitions: Let $$\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2} $$ be the standard normal probability density function (pdf) and $$ \Phi(x) = \int_{-\infty}^x \phi(t) dt = \frac{1}{2}\left[ 1 ...
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73 views

How to get minimum and maximum of function with complementary error function

Consider this function with the complementary error function $\mathrm{erfc}(x)$ $$e^{2x} \mathrm{erfc}\left(\sqrt{2} x\right) - \frac{1}{2} e^{\frac{1}{2}x} \mathrm{erfc}\left(\frac {x}{\sqrt{2}}\...
2
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1answer
40 views

fit a line $ax$ to function $\sin(\pi x)$ from $x=-1$ to $1$, that produces the minimal mean square error.

I want to fit a line $$ax$$ to function $$\sin(\pi x)$$ from $x=-1$ to $1$, that produces the minimal mean square error. It should be $\int_{-1}^1 (\sin(\pi x)-a x)^2 \, dx$. Then I take derivative ...
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75 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 \sqrt{...
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458 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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58 views

Understanding Variance

I have to analyse the reasons that we did not meet target. Our target is 4750 m Our target speed is 500m/hr Our target hrs are 9.5 We achieved a speed of 550 m/hr We achieved hrs of 9.1 We achieved ...
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233 views

Find the right degree of the Maclaurin polynomial of $e^x$

Here is my question: What degree Maclaurin polynomial of $e^x$ must be taken to guarantee an estimate of $e$ to within $1 \times 10^{-6}?$ I know that the error term is: $$R_n(x)=\frac{f^{(n+1)}(c)...
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71 views

Integral of product of error function and Gaussian

I found the following relation from a 1968 paper: $ \DeclareMathOperator\erf{erf} \int_{-\infty}^{\infty}\erf(x)\exp^{-(ax+b)^2}dx=-\frac{\sqrt\pi}{a}\erf\big(\frac{b}{\sqrt{a^2+1}}\big), Re(a^2)>...
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38 views

Differences between objective function definitions in optimization problems

Can someone explain the differences between the relative error definitions used as an objective function in a constrained optimization problem? I am trying to understand why I get different values of $...
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66 views

An integral leads to complementary error functions

I am reading a paper Albrecher, Constantinescu and Loisel "2011Explicit ruin formulas for models with dependence among risks" and getting stuck at one integral (Example 2.4): $$\int _{\frac{\lambda }{...
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1answer
35 views

Calculating the error function

If I want to calculate the following integral in terms of the Error function, is this correct? $$\frac{1}{\sqrt{2\pi}}\int_{f(x)}^{-\infty}e^{-p^2}\mathrm{d}p = \mathrm{Erf}(-\infty) - \mathrm{Erf}(f(...
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52 views

Solution for the Fourier transform of multiplication of two error functions.

I'm hoping someone can help me with what a Fourier transform problem. I seek the Fourier transform of $f(t)$, where: $$f(t) = a\left(1+\mathrm{erf}\left(\frac{\ln(t)-u_1}{\sigma_1\sqrt{2}}\right)\...
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75 views

Correctly Differentiating (Cross Entropy and Other) Error Functions

The following is a lecture slide from a machine learning class: Cross Entropy For classification tasks, target $t$ is either $0$ or $1$, so better to use $$E=-t\log(z)-(1-t)\log(1-z)$$ ...
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1answer
36 views

Is :$I(t)= \int_{0}^{t} ( \sin x+\cos x)^{\operatorname{erf}(x)}dx$ complex or real for $t \to \infty $?

let us to check the behavior of this integral :$I(t)= \int_{0}^{t} ( \sin x+\cos x)^{\operatorname{erf}(x)}dx$, Really this integral gives to me for small $t$ values close to $t$ as shown here , ...
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2answers
67 views

Laplace transform ambiguity

According to Laplace transform tables $\mathcal{L}\{f(t,k)\}=e^{-k\sqrt s}/\sqrt s $ has the solution ${f(t,k)=e^{-c^2}/\sqrt{t\pi}}$ where k must be a positive real value and ${c=k/\sqrt4t}$ ...
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0answers
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Find the number of significant figure in the approximate number $0.39865$ for the given relative error of it as $0.2\times 10^{-2}$.

Find the number of significant figure in the approximate number $0.39865$ for the given relative error of it as $0.2\times 10^{-2}$. Adding $0.2\times 10^{-2}$ to $0.39865$ gives $0.40065$ and it ...
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0answers
56 views

What is the asymptotic series of :$I(a)=\int_{0}^a {(e^{-x²})}^{\operatorname{erf}(x)}dx $ and does it have a complementary function?

The same topic of the OP of this question is posted here seeking for closed form of $I(a)=\int_{0}^a {(e^{-x²})}^{\operatorname{erf}(x)}dx $ and it's behavior compared to error function, the ...
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99 views

Remainder in Asymptotic Expansion of Erfc

According to Abramowitz and Stegun: Handbook of Mathematical Functions (7.1.23 and 7.1.24, http://people.math.sfu.ca/~cbm/aands/page_298.htm), we have Asymptotic expansion of Erfc is given by \begin{...
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177 views

What is a good approximation for the inverse of the cumulative distribution function?

What is a formula to approximate the (left-tail) inverse of the cumulative distribution function? I've tried the following which produced incorrect results. I can't use an exact (as it will be done ...
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159 views

How to isolate x in equation with ln(erfc(x))?

I was wondering whether it is possible to isolate $x$ in this equation: $$ f = 1- \exp\left(\frac{xt}{C^2}\right)\operatorname{erfc}\left(\frac{\sqrt(xt)}{C}\right) $$ I have not come further than $$...
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56 views

How to solve this differential equation using Laplace transform?

$$\dfrac{dy(t)}{dt}+\dfrac{1}{2}y(t).t=\dfrac{1}{2} $$ My attempt : $$sY(s)-\dfrac{1}{2}Y'(s)=\dfrac{1}{2s} \implies Y(s)=\dfrac{1}{2s^{2}}+\dfrac{1}{2s}Y'(s) $$ Now taking inverse laplace transform ....
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452 views

Cones, Differentials, and Volume Error Estimates

A cone with a circular base has a height of 40 cm and its radius at the base is 15 cm. Each measurement has 0.3 cm precision. With the help of differentials, estimate the greatest error that is ...
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0answers
119 views

Solving an integral using the error function

I want to evaluate the following integral $$ \int{\frac{1}{\sqrt{2\pi t}} \exp(\frac{-1}{2t}((a-x-wt)^2))dt}$$ where $w>0, 0<x<a$. Using WolframAlpha I obtain an expression for this ...
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2answers
1k views

Laplace Transform of erfc( \frac{k}{2\sqrt t}).

I am trying to show that: $$\mathcal{L}\{erfc( \frac{k}{2\sqrt t})\} = \frac{1}{s}e^{-k\sqrt s}$$ The hint given for this question is the Laplace Transform of an integral (from convolution): $$\...
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0answers
232 views

Leibniz rule for error function

Let $f(x)=erfi(a+x)$ and $g(x)=e^{cx}$ with \begin{align*} f^{(n)}(x)=\frac{2}{\sqrt{\pi}}e^{(a+x)^2}\sum_{m=0}^{n-1}\sum_{j=0}^{m}\frac{\binom{m}{j}(-1)^j(a+x)^{2m-n+1}}{m!}\\ \prod_{p=1}^{n-1}(2m-2j-...
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54 views

Integral involving error function

Could someone help me in evaluating this integral please? : $$\int_{0}^x t^2\exp(-t^2) dt$$ By error function method please I spliced the integrand into $t\cdot t\cdot \exp(-t^2)$ then doing ...
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51 views

Doing the actual integration in the error-function.

I got a question. The error-function is defined as: $erfc(z)=\frac{2}{\sqrt{\pi}}\int_0^zdw\exp(-w^2)$. How do I do the actual integration? I can't simply say that $\frac{2}{\sqrt{\pi}}\int_0^...
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1answer
47 views

Using implicit method to solve system analytically and finding error

How do I solve this? Please help! Given the following problem; $$u_t = u_{xx} + u_x; \quad\text{for} \quad 0 < x < 1, \quad t > 0$$ $$u(0,t) = 0 = u(1,t); \quad\text{for} \quad t > 0$$ $$...
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0answers
131 views

Calculate the difference between two error functions with argument having different real parts and same imaginary part

Does anyone know how to calculate the difference between two error functions with arguments having different real parts and the same imaginary part? Basically I'm looking to simplify: $$\mathrm{erf}(...
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0answers
248 views

Integral involving the Erf function

I'm am trying to solve the following integral $$\int\limits_{-\infty}^{+\infty}dx \; e^{-(ax+b)^2}\mathrm{Erf}(cx+d)\mathrm{Erf}(ex+f)$$ I tried the same reasoning as for these integrals that can ...
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0answers
662 views

Partial Differential Equations - Error Functions

So I am doing a course this semester in PDEs and we are currently doing the heat/diffusion equation $(u_t +ku_{xx}=0)$ on the whole line and the half line. In solving these equations we have ...
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94 views

MATLAB implementation of erf(x)

I have implemented erf(x) using its Taylor expansion in Matlab. But even after repeated attempts to correct it, it shows wrong answer for x>1. I am not able to understand why it is so. Any help will ...
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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:...
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0answers
145 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 ...
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0answers
368 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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0answers
112 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 ...