Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

1
vote
3answers
770 views

Do Hermite polynomials exist for negative integers?

I recently asked a question about a differential equation, and received this as an answer. It included a Hermite polynomial of negative degree, namely $H_{-3}$. I searched online and it seems as ...
0
votes
0answers
42 views

Using the inverse error function to show a probability is especially small.

I'm reading through the recent computer science paper [1] and I am stuck on the last line of Corollary 3.2's proof. We are trying to show the probability of an event $A$ is asymptotically small in $n$,...
2
votes
2answers
60 views

Integration by part with substitution

Question: Show that $$\begin{align}u(x,t) &=c\sqrt{\frac{k}{\pi}}\int^t_0s^{-1/2}e^{-x^2/4ks}\,ds\\ &=c\sqrt{\frac{4kt}{\pi}}e^{-x^2/4kt}-cx\,\text{erfc}\frac{x}{\sqrt{4kt}}\end{align}$$ ...
2
votes
2answers
74 views

Integral of Function related to Error Function (e.r.f)

Need help solving this question - I've tried solving it multiple times but to no avail: $$ I=\int_0^\infty e^{-y^3}.\sqrt y~dy $$ My approach has been to substitute: $$ y^3=t\Rightarrow 3.y^2~dy = dt \...
1
vote
0answers
55 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 ....
0
votes
1answer
69 views

Integral with the exponential function

I don't know how to solve the integral $$ \int \exp\left(-\frac{a^2}{2x^2}-\frac{b^2x^2}{2}\right)dx$$ for $a,b \in \mathbb{R}$. My idea was to complete the square to get an expression close to the ...
2
votes
1answer
90 views

Error function in Gams (general algebraic modeling system)

I have a problem, while solving some equations in Gams because there is another expression of so called error function then in my problem.. Error function which I am working with is: $$erf(x)=\frac{2}...
1
vote
0answers
405 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 ...
1
vote
2answers
92 views

How can I show this is equivalent to the error function of $x$?

$$\frac{2}{\pi} \int_0^\infty e^{-t^2}\frac{\sin 2xt}{t}\,dt$$ I know the original $\operatorname{erf}x$ but the infinity in the limit keeps getting in my way. How do I deal with it? Also my hint is ...
2
votes
0answers
197 views

Integral involving error function and exponential [closed]

I have not been able to find a source that gives this indefinite integral: $$ \int e^{-(a+x)^2}\text{erf}(b+x)\,dx.$$ Can someone provide a source or a formula (with our without demonstration)?
1
vote
0answers
111 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 ...
1
vote
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): $$\...
2
votes
2answers
1k views

What is this O function?

I came across such a function written by $O$. Can you please tell me what is this? Actually I see this function during proofs or error finding. I am familiar with big-O notation in algorithmic ...
5
votes
1answer
112 views

Fractional part of normally distributed variable

Let $X$ be a normally distributed variable with mean $0$ and standard deviation $1$. I will consider its fractional part $$\overline{X} = X - \lfloor X \rfloor = X \, \bmod \, 1.$$ I have done some ...
2
votes
1answer
121 views

Definite Integral $x^{n} e^{x^2} $

I want to find an expression for the integral \begin{align*} \int x^{n} e^{x^2}~dx \quad or \int_{a}^{b} x^{n} e^{x^2}~dx. \end{align*} I tried this way: \begin{align*} \int x^{n} e^{x^2}~dx=\frac{1}{...
0
votes
1answer
240 views

Find the solution of the diffusion equation for a given initial condition in terms of the error function

I'm given that the solution to the diffusion equation on $\mathbb{R}$ is $$u(x,t)=\frac{1}{\sqrt{4\pi Dt}} \int_\mathbb{R} e^{-\frac{(x-y)^2}{4Dt}} f(y) \, dy$$ I'm also given that $f(x)=\begin{cases}...
1
vote
1answer
41 views

What is the error of this numerical integration

Consider the following numerical integration rule: $$I = \int_0^1 \sqrt x f(x)\ dx = w_1f(x_1)$$ At part (a) I showed that $w_1=2/3$ and $x_1 = 3/5$ to make the rule exact for linear polynomials. ...
0
votes
1answer
132 views

Which is the correct degree of the taylor polynomial of this question?

Ok, so for context purposes the problem is: to determine value of x for which the function can be replaced by the taylor polynomial if the error cannot exceed 0.001 My only confusion with the answer ...
1
vote
1answer
48 views

Show: $\lim\limits_{y \to +\infty}\left [\frac{1}{\sqrt{\pi}e^{y^2}(1-\text{erf}(y))}-y\right]=0.$

I would like to show the following: $$\lim_{y \to +\infty} \left[\frac{1}{\sqrt{\pi}e^{y^2}(1-\text{erf}(y))}-y\right]=0.$$ Here $$\text{erf}(y)= \frac{2}{\sqrt{\pi}} \int_{0}^{y}e^{-x^2}dx$$ ...
0
votes
1answer
181 views

Good upper bound for $1-\operatorname{Erf}(x)$

For error function $\text{Erf}(x)$ I mean $$\operatorname{Erf}(x) = \int_{-\infty}^x\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}u^2\right)\mathrm{d} u.$$ My statistics professor said that $$1-\...
7
votes
3answers
244 views

Closed form for $\int_{0}^{\infty }\!{\rm erf} \left(cx\right) \left( {\rm erf} \left(x \right) \right) ^{2}{{\rm e}^{-{x}^{2}}}\,{\rm d}x$

I encountered this integral in my calculations: $$\int_{0}^{\infty }\!{\rm erf} \left(cx\right) \left( {\rm erf} \left(x \right) \right) ^{2}{{\rm e}^{-{x}^{2}}}\,{\rm d}x$$ where $c>0$ and $c\in ...
0
votes
1answer
65 views

Aproximation to function of error function

The function: $p = \dfrac{\mathrm{erf}(z)-\mathrm{erf}(f(x))}{\mathrm{erf}(z)-\mathrm{erf}(y)}$ All the parameters of the error function ($z, y, f(x)$) are very large. This leads to catastrophic ...
2
votes
2answers
232 views

Integral of product of two error complementary functions (erfc)

Could you please help me to show that the integral $$ \int_0^{\infty} \mathrm{erfc}(ax) \, \mathrm{erfc}(bx)\, \mathrm{d}x $$ is equal to $$ \frac{1}{ab\sqrt{\pi}} (a+b-\sqrt{a^2+b^2}), $$ where $$ ...
1
vote
0answers
215 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-...
2
votes
0answers
95 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 ...
6
votes
1answer
171 views

Prove $1+\frac13 \left(1+\frac15 \left(1+\frac17 (1+\dots ) \right) \right)=\sqrt{\frac{\pi e}{2}} \text{erf} \left( \frac{1}{\sqrt{2}} \right)$

How to prove: $$1+\frac13 \left(1+\frac{1}{5}\left(1+\frac{1}{7}\left(1+\frac{1}{9}\left(1+\dots \right) \right) \right) \right)=\sqrt{\frac{\pi e}{2}} \text{erf} \left( \frac{1}{\sqrt{2}} \right)$$ ...
0
votes
0answers
29 views

Find $\inf\limits_{p_{n}\in\bar{\prod}_{n}}\max\limits_{x\in[a,b]}|p_{n}(x)|$ where $p_{n}$: monic polynomial.

I want to find $M_{n}$ such that $$ M_{n}=\inf_{p_{n}\in\bar{\prod}_{n}}\max\limits_{x\in[a,b]}|p_{n}(x)|, $$ where $\overline{\prod}\limits_{n}$ is the set of all monic polynomial degree $n$. ...
2
votes
0answers
70 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
votes
1answer
131 views

Taylor Series Error Bound

The question: Let $f(x) = \sin(x)$. The taylor polynomial is the following: $P_{n}(x) = \sum_{n =1}^{\infty} (-1)^{n} \frac{x^{2n+1}}{(2n+1)!}$. Find the smallest value of n such that $|P_{n}(x) - f(...
1
vote
0answers
53 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 ...
1
vote
0answers
50 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^...
1
vote
1answer
46 views

Using implicit method to solve system analytically and finding error

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

Linear regression model and the error vector

In the linear regression model, since the true error vector $U=Y-X\beta$ is based upon the true value of the unknown coefficient vector $\beta$ and the LS residual vector $U^* =Y-X\beta^*$ uses the LS ...
-3
votes
1answer
122 views

What 's equal this :$\lim_{x\to \infty} \operatorname{erf}(\operatorname{erf}(\operatorname{erf}(\cdots \operatorname{erf}((x)))))) $? [closed]

Error function is a special function (non-elementary ) ,it is defined as follow :$$\operatorname{erf}(x)= \frac{1}{\sqrt{\pi}}\int_{-x}^x e^{-t^2}\ dt,$$ I would like to know what's equal this limit :...
0
votes
2answers
60 views

Can one show this is equivalent to erf(x) by elementary means?

According to the PDF at this link (eqn 12), the following is true: $$\text{erf}(x)=\frac{1}{\pi}\int_0^\infty e^{-t} \sin(2x\sqrt{t})\frac{dt}{t} $$ I would like to use this identity as part of a ...
1
vote
0answers
124 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}(...
1
vote
0answers
244 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 ...
2
votes
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 ...
3
votes
3answers
143 views

Show $\frac{2}{\pi} \mathrm{exp}(-z^{2}) \int_{0}^{\infty} \mathrm{exp}(-z^{2}x^{2}) \frac{1}{x^{2}+1} \mathrm{d}x = \mathrm{erfc}(z)$

I used the result $$\frac{2}{\pi} \mathrm{exp}(-z^{2}) \int\limits_{0}^{\infty} \mathrm{exp}(-z^{2}x^{2}) \frac{1}{x^{2}+1} \mathrm{d}x = \mathrm{erfc}(z)$$ to answer this MSE question. As I mentioned ...
0
votes
2answers
135 views

Determine Gaussian integral without using erf(z)

How can I compute the well-known integral, $$ I(t,\mu,\sigma) = \frac{1}{\sqrt{2\pi} \sigma} \int_0^t e^{-\frac{(\tau - \mu)^2}{2\sigma^2}} d\tau$$ without using the $erf(z)$ definition. I am ...
0
votes
1answer
58 views

Solving or approximating an equation dealing with the error function

In my studies of probability theory I have come across the need to solve this equation, on which Mathematica got stuck: $ \sqrt{\pi} t_0 \left( \text{erf}(\frac{t_0}{\sqrt{2}}) + 1\right) - \sqrt{2}...
1
vote
0answers
608 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 ...
1
vote
0answers
90 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 ...
1
vote
2answers
55 views

Solve $ \int_0^\infty \exp(-(\rho t)^k)dt$

Let $T$ a random variable that represents a failure time with $S(t)=\exp(-(\rho t)^k)$ where $\rho,k\in\mathbb{R}$. Find the mean survival time of $T$. I will not give much statistical details ...
1
vote
1answer
48 views

Cost Function: Does it matter what order the y_predicted and y_actual are in?

I'm learning neural networks, specifically backpropagation and am reviewing the cost function. When looking at different educational sources for backprop, I'm seeing the cost function written in ...
1
vote
3answers
141 views

How to compute this limit involving complementary error functions

I am trying to take the following limit $$\lim_{x\to \infty } \, \frac{2 x \operatorname{erfc}\left[\frac{x}{\sqrt{2} t}\right]}{t \operatorname{erfc}\left[-\frac{x}{\sqrt{2}}\right]}$$ my first ...
0
votes
3answers
276 views

Solid Ellipse Fitting on 2D Image Using Gradient Descent

I am attempting to fit an ellipse for a specific color, $\mu$ at grayscale, on an image that will cover as much of the region as possible, with the targeted color inside the ellipse. This is not ...
7
votes
3answers
167 views

Solving the Definite Integral $\int_0^{\infty} \frac{1}{t^{\frac{3}{2}}} e^{-\frac{a}{t}} \, \mathrm{erf}(\sqrt{t})\, \mathrm{d}t$

I would like to solve the following integral $$\int_0^{\infty} \frac{1}{t^{\frac{3}{2}}} e^{-\frac{a}{t}} \, \mathrm{erf}(\sqrt{t})\, \mathrm{d}t$$ with Re$(a)>0$ and erf the error function. Is ...
0
votes
3answers
69 views

Can you help me on the Numerical Analysis question

The question: The error function defined by $$erf(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} dt. $$ The error function can also be formed as a series. $$ \frac{2}{\sqrt{\pi}} \sum_{k=0}^{\infty}...
1
vote
1answer
67 views

Closed Form of $\int_{1}^{ \infty } { \frac{1}{x}\operatorname{erfc}\left[\frac{\log x}{2 k} -\frac{k}{2}\right]{\kern 1pt} \,dx}$

I am looking for a closed form of the following $$I = \int_{1}^{ \infty } { \frac{1}{x}\operatorname{erfc}\left[\frac{\log x}{2 k} -\frac{k}{2}\right]{\kern 1pt} \,dx},$$ for $k \ge 0$. Upon ...