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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30
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789 views

Weber-type integral

In connection with this answer, I came across the following integral: $$\int_{0}^{\infty} \frac{du}{u} \: \,e^{-t u^2} \frac{J_0(u) Y_0(r u)-J_0(r u) Y_0(u)}{J_0^2(u)+Y_0^2(u)}$$ where $r \gt 1$. I ...
25
votes
1answer
535 views

Non-trivial values of error function $\operatorname{erf}(x)$?

The so called error function $\operatorname{erf}(x)$ is defined as $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt,$$ and it is well known that $\operatorname{erf}(\infty)=1$. Are ...
4
votes
1answer
46 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}} \...
1
vote
1answer
26 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 ...
1
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1answer
46 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 ...
1
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1answer
31 views

Alternatives to absolute error?

Let me explain my scenario in which I need to calculate absolute error. Lets say the X is the actual value. And X' is the value of X with some error 'e'. So X' = X + e'. Lets say i = 1 to 10000. I ...
1
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1answer
51 views

Integral of cumulative normal

Let $$\Phi(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} \exp\left({-\dfrac{\omega^2}{2}}\right) d\omega.$$ Question: for what values of $a$, $b$ and for what choices of $f(x)$ would the following ...
0
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1answer
26 views

Error function representation

Can someone help about proving the following relationship: \begin{equation} \operatorname{erfc}\left(\sqrt a\right) = \int_{0}^{\infty} e^{-a} \times \frac{\sqrt a}{\sqrt \pi} \times \frac{\...
5
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0answers
182 views

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 $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
4
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0answers
51 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 $\...
4
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0answers
495 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)\...
3
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0answers
112 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: $$ \...
3
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0answers
230 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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0answers
28 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 ...
2
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0answers
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 \...
2
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0answers
43 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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0answers
42 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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0answers
30 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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0answers
38 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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0answers
18 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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0answers
108 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 ($7.1.26$). $$\operatorname{erf}(x)\...
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135 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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29 views

Why is the Riemann integral for the sum-of-squares error function multiplied by the joint density function of the underlying distribution?

In Bishop's Neural Networks for Pattern recognition , he expresses an error function $$ E= \frac{1}{2}\sum_{n=1}^{N}\sum_{k}\{(y_k(\mathbf{x}^{n};\mathbf{w})-t^{n}_{k}\}^2$$ where $\mathbf{x}^{n}$ ...
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0answers
45 views

How to compute an integral involving the error function

Solving a problem with one dimensional diffusion the following identity naturally arises $$\int _{-\infty }^{\infty }\frac{q}{2}{{\rm e}^{\eta\,t{\alpha}^{2}-\alpha \,x}} \left( {{\rm erf}\left({\...
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48 views

Is the Taylor Expansion a good approximation

Say I use a computer to sum the first 26 terms of $e^{-5}$ (degree 25), will this taylor expansion provide a good approximation? It summed to $.0067$ To me this seems like a good approximation, but I ...
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0answers
61 views

Integral involving complimentary error function.

Essentially, I am trying to work out an integral of the form: $\int_{0}^{\pi }{x\cdot a\cdot \cos \left( x \right)e^{-\left[ a\cdot \sin \left( x \right) \right]^{2}}\mbox{erfc}\left[ -a\cdot \cos \...
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0answers
60 views

Converting this sum to integral (possible?). The goal is to get error function

The solution to the heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ with boundary conditions and ini condition as followed: $$u(0\ or\ 1, t)=0\qquad u(x,0)=1$$ ...
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0answers
65 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)}(...
1
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0answers
261 views

Solving Differential equation, origin Physics

Given constants $u,v,l$ find the solution to the differential equation $$\frac{dx}{dt}+x\left(1+\frac{v}{l}t\right)=u$$ Given that at $(0,l)$ lies on the solution. And hence find the value of $t$ when ...
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0answers
238 views

Grid Function Norm

I am studying about the global truncation error in finite difference methods and I have a question about calculating the error in a Boundary Value Problem (BVP). If we take a simple 1-D problem, the ...
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0answers
221 views

A puzzling inequality involving exp, erf, and log

For $0<y<x<\infty$, I believe the following inequality is true (I've tested it numerically with random values for $x$ and $y$) but have been unable to analytically confirm: \begin{align} s(x,...
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0answers
893 views

What is the significance of error function?

Here's a Wiki article on the subject. Sadly it doesn't do a good job of explaining the significance of the function. Of course it may mean different things to different people (for mathematicians it ...
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0answers
23 views

Minimal time to steady-state flow (1st Stokes Problem)

How to obtain a minimal time to steady-state flow? This is related to the Stokes 1st Problem. $$\frac{u}{U}=0.05=1 - \operatorname{erf}(\eta)$$ $$\eta = \frac{y}{2\sqrt{vt}} \approx 1.4$$ $$y=\...
0
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0answers
23 views

Integral computation with Mathematica and Sympy differ

To compute the integral: $I = \int_{0}^{+oo} ue^{Au^{2}+Bu}du$ where $A<0$ and $B>0$ I have tried both Mathematica and Sympy but they yield different results: Mathematica yields: $ I = \frac{\...
0
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0answers
12 views

Total Error Bound for $D_1(h)$ Approximation

I am asked to find a total error bound for the approximation $D_1(h)$ in terms of $M_2=\max|f''(x)|$ and $\varepsilon=|f(x)-\bar{f}(x)|$. I began research what exactly $D_1(h)$ is and found $D_1(h)=...
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0answers
17 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 ...
0
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0answers
28 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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20 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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0answers
72 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
15 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 = \frac{V_3}{...
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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 $h(...
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62 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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0answers
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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175 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 a2}[{\text{erf}(\frac{...
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46 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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0answers
68 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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0answers
44 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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0answers
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 ...
0
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0answers
59 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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0answers
134 views

Discrete Function Approximation Error - Which type? (Applied math, signals)

I have two functions, one derived via software, and we can call it the exact function, $f_{exact}$. The other is a result I got through hardware, and we can call it the approximation, $f_{approx}$. ...