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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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 ...
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1answer
22 views

Upper and lower bound on different of ${\rm erf}(\frac{x+c}{b})-{\rm erf}(\frac{x-c}{b})$

I am trying to find a good upper bound on \begin{align*} f(x)={\rm erf}\left(\frac{x+d}{b}\right)-{\rm erf} \left(\frac{x-d}{b}\right) \end{align*} here $d>0$ I know that $f(x)$ is symmetric ...
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41 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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1answer
46 views

Closed forms for definite integrals involving error functions

I have been working for a while with these kinds of integrals $$\int_0^\infty dx\,\text{erfc}\left(c +i x\right)\exp \left(-\frac{1}{2}d^2x^2+i cx\right)$$ $$\int_\Lambda^\infty ...
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27 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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1answer
38 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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1answer
34 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 ...
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1answer
17 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 ...
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1answer
78 views

How can I solve the integral in the error function $\mbox{erf}(x)$?

To get from this To this series I can't seem find the step-by-step solution anywhere.
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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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2answers
1k views

Integral involving an error function

For $\sigma>0$, how can we prove that $$\frac{1}{2}\int_{-1}^1 \text{erf}\left(\frac{\sigma}{\sqrt{2}}+\text{erf}^{-1}(x)\right) \, \mathrm{d}x= \text{erf}\left(\frac{\sigma}{2}\right)$$ where erf ...
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2answers
89 views

Challenging integral

I am trying to find a close form representation for the following integral: $$ A(x;a,b,c)= \int_{0}^{x}\frac{\sin\left(a k+b k^{2}\right)+\sin\left(c k-b k^{2}\right)}{k}dk $$ for $0<x \ll ...
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2answers
91 views

Convincing that $\sum_{n=0}^{\infty}\frac{(-1)^n}{2^{n-2}(2n+1)(n!)} \ne \pi$

A friend of mine in high school challenged me to calculate the value of the sum $$\sum_{n=0}^{\infty}\frac{(-1)^n}{2^{n-2}(2n+1)(n!)}$$ And then claimed that the answer was $\pi$ . But when I ...
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18 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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62 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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42 views

Prove that the rounding error can contaminate half the digits of computed root

I am trying to resolve the following problem: If $b^2 \approx 4ac $ the rounding error can contaminate half the digits of the root computed with the formula: $\dfrac {-b \pm \sqrt {b^2 - 4ac}} ...
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13 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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2answers
26 views

Error in ratio of two numbers

How tow find error associated with a ratio $R$, when both the numerator and denominator contains error. For example $$ R=\frac{(A \pm \Delta A)}{(B\pm\Delta B)}.$$
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2answers
71 views

Erf squared approximation

I found a nice and useful approximation of squared error function $$ \mathrm{erf}^{2}\!\left(x\right)=1-\exp\!\left(-\frac{\pi^{2}}{8}x^{2}\right)+\varepsilon\!\left(x\right). $$ I checked ...
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3answers
174 views

What am i doing wrong when solving this differential equation

$$ f(x) = \frac{\frac{d^2}{dx^2}[e^y]}{\frac{d}{dx}[e^y]} $$ Given that $f(x) = cx$ $$ \frac{c}{2}x^2 + k_1 = \ln(e^y y') $$ $$ k_2\int e^{\frac{c}{2}x^2} dx = e^y $$ $$ y = \ln(k_2\int ...
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Integral involving the error function of log(x)

Looking for a closed form for the integral $$\int_0^{\infty } e^{-\left(\frac{a-\log (x)}{b}\right)^2} \left(\frac{1}{2} \text{erf}\left(\frac{a-\log (x)}{b}\right)+\frac{1}{2}\right) \, ...
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1answer
59 views

Inverse Laplace Transform of $e^{c \cdot s^2}$

I am trying to find the Inverse Laplace Transform of the function $$ F(s)=e^{c \cdot s^2} $$ where $c > 0$.
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38 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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44 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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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 ...
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1answer
23 views

What can I imply from $\epsilon^\prime(x) \le -x\epsilon(x)$?

Imagine, that I can prove $$\epsilon^\prime(x) \le -x\epsilon(x)$$ for a function $\epsilon:\mathbb R \rightarrow \mathbb R_0^{+}$. Does this imply $$\epsilon(x) \le c e^{-\frac{x^2}{2}}$$ whereby $c ...
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31 views
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46 views

Integral of a gaussian over a slice of the plane

I need to evaluate the following $n$ real integrals: $$\int_{\frac{\pi}{2}-\frac{\pi}{n}}^{\frac{\pi}{2}+\frac{\pi}{n}}\int_0^\infty\frac{1}{\pi \sigma^2}e^{-\frac{|re^{i\theta}-i|^2}{\sigma^2}} \ dr ...
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24 views

Why is the Wronskian of these two functions equal to $\frac{2}{\sqrt{\pi}}$

I can't seem to get the right answer $ f = e^{\frac{y^2}{2}}$ and $g =e^{\frac{y^2}{2}}erf(y)$ where $erf(y) = \frac{2}{\sqrt\pi}\int_{0}^{y}e^{-\alpha^2}d\alpha$ I get $W = \frac{2}{\sqrt\pi} - ...
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84 views

How to integrate $e^{-x^2 -y^2}$?

How do you compute the following integral:$e^{-x^2 -y^2}$? Wolfram already gives the answer as: $\frac{1}{2} \sqrt{\pi} e^{-y^2} \text{erf}(x) + C$, but I have no idea how to get there. I tried ...
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52 views

probability: When $X\sim\mathcal{N}(\mu, \sigma^2)$, the med = $\mu$

The median of a PDF is defined as the point $x = $med for which $P[X\leq\text{med}] = 1/2$. Prove that if $X\sim\mathcal{N}(\mu, \sigma^2)$, then med $= \mu$. Since $P[X\leq\text{med}] = 1/2$, ...
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95 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) = ...
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23 views

If $y$ is the solution of $\left\{y'=-y+\sqrt{t},y(0)=y_0>0\right\}$, then $\lim_{t\to\infty}\frac{y(t)}{\sqrt{t}}=1$

The homogeneous equation $$y'=-y$$ has the solution $$y_h(t)=ce^{-t}\;\;\;\;\;(c,t\in\mathbb{R})$$ In order to find a particular solution we can take the approach $$y_p(t)\stackrel{!}{=}c(t)e^{-t}$$ ...
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73 views

Arctan(f(x)) is almost the same as Erf(f(x)) for many f(x). Is the just coincidence or is there a reason?

For example: Arctan(x) is almost Erf(x) (subtle differences in absolute value and curve) Arctan(x^50) is almost Erf(x^50) (difference in absolute value) and many others, so we can conclude: ...
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1answer
29 views

Error function - Not seeming to come out right

I have reached two integrals: $$\int_{(x+L)/(2c\sqrt{t})}^\infty e^{-z^2} dz + \int_{-\infty}^{(x-L)/(2c\sqrt{t})} e^{-z^2} dz$$ Now the first evaluates just fine to ...
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102 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 $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
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1answer
62 views

Integral involving the error function

Is there a closed form solution to the integrals \begin{align} I_{c} &= \int_{0}^{\infty} \cos(a x) \, \operatorname{erf}(b x) \, dx \\ I_{s} &= \int_{0}^{\infty} \sin(a x) \, ...
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2answers
24 views

Behaviour of the error function as $z \rightarrow -\infty$?

I'm trying to find the behaviour of the error function, $erf(z)$ as $z \rightarrow -\infty$ $$erf(z) = \frac{2}{\sqrt{\pi}}\int_0^{z} e^{-s^2}\mathrm{ds}$$ I know that we can find the limit of ...
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1answer
50 views

approximation of x=sin(x) error

The larger $x$ becomes, the worst the approximation of $x=\sin(x)$ becomes. How large can $x$ get before the error is no longer less than $1/(2\times10^{14})$? What I have tried: I know that the ...
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140 views

Minimizing an error function by deriving a system of linear equations

Consider the following formula: $$E(\mathbf{w}) = \frac{1}{2}\sum_{n=1}^{N}\{y(x_n,\mathbf{w})-t_n\}^2$$ where $\mathbf{w}$ is a vector of weights; $x_n$ and $t_n$ come from two vectors of length ...
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51 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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52 views

what are some good ways to define the errors between two functions?

I have two functions. One is the original function (that contains 4 variables), and the second one is the approximation to the first one (also contains 4 variables). The question is, if I want to ...
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1answer
188 views

Is there an analytical solution to Gaussian integral $\int_{-\infty}^{\infty} \frac{e^{-x^2}}{(x+a)^2+b} dx$?

I wonder if there is an analytical solution to $$\int_{-\infty}^{\infty} \frac{e^{-x^2}}{(x+a)^2+b} dx,$$ where $a, b>0$. I know, of course, that the antiderivative of the fraction is a version ...
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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 ...
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1answer
25 views

Where does the “2” come from in deriving Normal PDF from its kernel?

I'm trying to train myself to recognize probability densities by deriving PDFs from bare kernel functions. In other words, I find a constant expression by integrating a kernel function over its ...
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1answer
38 views

Integrating the error function in a calculation related to Brownian motion

I wish to calculate the probability that a standard linear Brownian motion $B(t)$, $t\ge 0$, will be at time $t_0$ inside the interval $[a,b]$, and at time $t_1$ in the interval $[c,\infty)$. To do ...
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8 views

To find error in time histogram

I have a data which is recorded from a detector. Whenever the detector produce signal it records the time. I have recorded the data for several cycles, one cycle is 0 to 1 second. Finally I made the ...
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72 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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1answer
68 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, $f_{approx}$. ...
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1answer
56 views

Inverse Laplace transform of a given function

1) The Laplace transform of f(t) is $\overline{f}(p)=\frac{1}{p}$ when $f(t)=1$ 2) The Laplace transform of $f(at)$ is $\frac{1}{a}\overline{f}(\frac{p}{a})$ 3) The Laplace transform of the ...