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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95 views

Approximating the erf function

I was trying to find an approximate solution to the following: $\DeclareMathOperator\erf{erf}$ $$\frac12 \sqrt{\pi} \erf\left (\frac{x-2}{\sqrt{10}}\right) + \frac12 \sqrt{\pi} \erf ...
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2answers
141 views

Accurate computation of $\exp(a x^2) Q(x)$ for big values of $x$?

I was wondering how one can accurately compute the value of $\exp(a x^2) Q(b x)$ for large values of $$x \left(Q(x) \triangleq \frac{1}{\sqrt{2\pi}}\int_x^{\infty} e^{-\frac{u^2}{2}} du \right)?.$$ ...
25
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1answer
630 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 ...
19
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1answer
402 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
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1answer
45 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
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1answer
14 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
29 views

Remainder Taylor series two+ variables

Hard to find an article / tutorial specifically about this subject online.... Can anyone explain / send link to explanation or tutorial regarding how to calculate remainder for multi-variable taylor ...
1
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1answer
29 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
45 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
10 views

Equation for standard error of weighted mean

What should I use when to calculate standard errors (and thus a confidence interval) for weighted means? Do I simply substitute the weighted mean for the simple arithmetic mean?
0
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1answer
53 views

Sigmoid function that approaches infinity as x approaches infinity.

The function I'm looking for looks like an error function, but instead of having asymptotes $1$ and $-1$, the function I'm looking for does not have asymptote. It increases to infinity. The ...
0
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1answer
69 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, $f_{approx}$. ...
0
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1answer
223 views

Uncertainty in gradient of data

So I have a set of 9 x,y values, and I need to find the gradient/slope of the data, AND its associated error. Without the error, I would've used Excels LINEST function, but as the errors in my y ...
0
votes
1answer
372 views

analytical filtering of Gaussian function with tophat

I have a Gaussian Function - $$G(a,x) = \sqrt{\frac{6.0}{\pi \cdot a^2}}\cdot \exp\left(\frac{-6.0x^2}{a^2}\right)$$ and I want to filter it with a tophat kernel $$ f(x,\xi) = ...
4
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0answers
117 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 $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
3
votes
0answers
78 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
175 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 = ...
3
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0answers
407 views

Integral of two error functions (erf)

In my research I came across the following integral: \begin{equation} ...
2
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0answers
35 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
67 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 ($7.1.26$). ...
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0answers
40 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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0answers
22 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
41 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 ...
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0answers
46 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
55 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
51 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
44 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: ...
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0answers
184 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
142 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
201 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} ...
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0answers
726 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
12 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 ...
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0answers
45 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
10 views

Difference Between Three Similar Error Reducing Algorithms

I found a Least Square Error Recognition algorithm that finds the least mean square error from a 2-d matrix element by element. Logistic regression from this site, on the other hand, seeks to ...
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0answers
91 views

two dimensional Gaussian integral with complex exponent of an absolute value

I am trying to solve the following two dimensional integral: $$\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}{e^{ia(\left|x\right|-\left|y\right|)} \frac{1}{2\pi\sqrt{1-\rho^2}}e^{-\frac{x^2+y^2-2\rho ...
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0answers
16 views

How to solve this statistics problem? Error function

A store received 1500 glass bottles. The probability that the randomly chosen product is defective is 0.3% . Create the distribution law of the random variable X number of defective products , ...
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0answers
41 views

Proof that $\int_0^\infty t^\gamma e^{t/2} \operatorname{erfc}(\sqrt{t}\,) \,dt$ leads to hypergeometric function

I'm looking for the proof of $$\DeclareMathOperator\erfc{erfc} \int_0^\infty t^\gamma e^{t/2} \erfc(\sqrt{t}\,) \, dt = \frac{2^{-2 \gamma -1} \Gamma (2 \gamma +2) \, _2F_1\left(\gamma +1,\gamma ...
0
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0answers
98 views

error function inverse limits

$$ \lim_{x \to 0+} e^{-(\operatorname{erfc}^{-1}(2x) - 2\operatorname{erfc}^{-1}(x))^2}(e^{\operatorname{erfc}^{-1}(x)^2} - e^{\operatorname{erfc}^{-1}(2x)^2}) $$ Where $\operatorname{erfc}(x) = ...
0
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0answers
91 views

What is the limit of erf in $\infty+i\times\infty$?

What is the following limit: $\lim_{x \to \infty, y \to \infty} \mbox{erf}\left(x+iy\right)$ Wolfram alpha seems to give $1$, but here the unique answer seems to tell that $\mbox{erf}$ diverges. ...