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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20 views

Is the error function in the complex plane bounded? [on hold]

I have to show that the $ erf (\sqrt{(\lambda / 2) }r(t) x)$ is bounded where $r(t)$ is only bounded when $\lambda < 0$
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2answers
69 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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2answers
49 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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60 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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1answer
21 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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1answer
45 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
26 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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0answers
73 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
51 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
20 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
32 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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1answer
45 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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0answers
35 views

Lagrange's form of the remainder vs Cauchy's form

So far (while practicing exercises) I've used Lagrange's form of the remainder. Is there a situation when Cauchy's form comes in handy while Lagrange's form fails for some reason? Is there a rule of ...
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0answers
37 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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21 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
133 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
21 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
23 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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6 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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43 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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13 views

Fitting Error function?

I need to fit a function of the form $$y=A+B\space \text{erf}\left(\frac{\sqrt2(x-a_0)}{w}\right),$$ where $$ \text{erf}(x)=\frac{2}{\sqrt\pi}\int_0^x e^{-t^2} ...
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1answer
34 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
52 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 ...
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2answers
101 views

Integral of exponential using error function

I'm trying to solve some integrals below $$\int_{-\infty}^{\infty} {x^n e^\frac{-(x - \mu)^2}{\sigma^2}}dx$$ I am interested in the solutions where n = 0, 1, 2, 3, 4. I have learned that ...
4
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2answers
76 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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1answer
51 views

Error Term of Chebyshev inequality?

Chebyshev inequality tells us that $$Pr[|X-E[X]|\geq a]\leq \frac{Var[X]^2}{a^2}$$ Do you know an Expression (or a paper where this Expression is mentioned) for the error term?
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1answer
29 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 ...
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1answer
50 views

Integration involving complicated exponential form

I'm trying to simplify the following: $\int_0^ts^{-\frac{3}{2}}e^{-\frac{(a+bs)^2}{2s}}~ds$ Basic substitution always gives a $s^{-\frac{1}{2}}~ds$ counterpart which I don't know how to get rid of. ...
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117 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 = ...
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41 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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1answer
22 views

Determining a value $c$ such that the function $f(c) = \displaystyle\sum_{i=1}^n \left|\frac{y_i-c}{y_i}\right|\times v_i$ is minimized

I'm trying to construct prediction model for a variable of interest, based on a set of input values. I have a set of validation data and their predictions (by my model) and now I need to asses whether ...
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3answers
89 views

Integrate $ x\cdot e^{i\omega x - x^2}$ from $0$ to $\infty$

Can anybody help me in solving the following integral: $$\int_0^\infty x\cdot e^{i \omega x-x^2}\,dx\quad?$$ Any help/hints will be highly appreciated. Thanks
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0answers
26 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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1answer
342 views

Fourier transform of $\operatorname{erfc}^3\left|x\right|$

(this is a follow-up on my another question) Could you please help me to find the Fourier transform of $$f(x)=\operatorname{erfc}^3\left|x\right|,$$ where $\operatorname{erfc}z$ denotes the the ...
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1answer
49 views

Changing the limits of integration from $-\infty$ to $\infty$

Is it fair to change from $$ \int_{-\infty}^{a} \exp \left( -t^2 \right) \mathrm dt $$ replacing $t$ with $-t$ $$ \int_{-a}^{\infty} \exp \left( -t^2 \right) \mathrm dt $$ and thus gaining the ...
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2answers
353 views

Integral $\int_0^\infty F(z)\,F\left(z\,\sqrt2\right)\frac{e^{-z^2}}{z^2}dz$ involving Dawson's integrals

I need you help with evaluating this integral: $$I=\int_0^\infty F(z)\,F\left(z\,\sqrt2\right)\frac{e^{-z^2}}{z^2}dz,\tag1$$ where $F(x)$ represents Dawson's integral: $$F(x)=e^{-x^2}\int_0^x ...
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2answers
173 views

Fourier transform of $\operatorname{erfc}^2\left|x\right|$

Could you please help me to find the Fourier transform of $$f(x)=\operatorname{erfc}^2\left|x\right|,$$ where $\operatorname{erfc}z$ denotes the the complementary error function.
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3answers
602 views

How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ \mathrm dx$?

I successfully evaluated these integrals: $$\int_0^\infty\operatorname{erfc}x\ \mathrm dx=\frac1{\sqrt\pi},\tag1$$ $$\int_0^\infty\operatorname{erfc}^2x\ \mathrm dx=\frac{2-\sqrt2}{\sqrt\pi}\tag2,$$ ...
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0answers
28 views

Calculating error for a division when the number of variables is rather large

Suppose we have $$r = \frac{a_{1}a_{2}...a_{m}}{b_{1}b_{2}...b_{n}}$$ Relative error of each of $a_{i}$ and $b_{i}$ is roughly the same and equals $\delta$. There is a theorem which says: ...
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2answers
140 views

Range of True Value

A calculator is out of order. For all number, before and after any arithmetic operation, the calculator will round up the numerical value to one decimal place if the value at the second decimal digit ...
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0answers
38 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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1answer
253 views

How to derive integrals with error function?

How to derive this integral $\int_{-\infty}^{\infty}erf(\lambda x)\mathcal{N}(\mu, \sigma ^2)dx$ and this $\int_{-\infty}^{\infty}(erf(\lambda x)-const)^2\mathcal{N}(\mu, \sigma ^2)dx$ where ...
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1answer
75 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 ...
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1answer
135 views

Integral of 2D Gaussian over triangle

I am interested in computing (numerically) the integral of a 2D Gaussian (with arbitrary mean and covariance matrix) over a triangle in the plane. I would like the solution to work over a wide range ...
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2answers
171 views

solve equation of erf

I'd like to solve this equation for $\mu$. Is it possible? If not, why? $$ 2 P = \operatorname{erf}\left( \frac{\mu - A}{ \sqrt{2 \sigma^2} } \right) - \operatorname{erf}\left( \frac{\mu - B}{\sqrt{2 ...
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2answers
183 views

Value minimizing mean absolute percentage error

What value for $c$ would minimize the formula: $$\frac{1}{n}\;\sum^{n}_{i=1}\left | \frac{y_i-c}{y_i}\right|$$ given the values $y_1, ..., y_n$. For example in the mean squared error we have the ...
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0answers
107 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 ...
12
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1answer
199 views

Integral $\int_0^\infty\exp\left(-\sqrt2\,x^2\right)\,\operatorname{erfi}(x)\,\log(x)\,x^3\,dx$

Consider the following integral: $$\mathcal{A}=\int_0^\infty\exp\left(-\sqrt2\,x^2\right)\,\operatorname{erfi}(x)\,\log(x)\,x^3\,dx,\tag1$$ where $\operatorname{erfi}(x)$ denotes the imaginary error ...
2
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1answer
62 views

How to show integral is related to complementary error function?

I have to somehow show that: $$ \int_0^\infty{\frac{e^{-a t}}{t^{1/2}(t+x)}} \textrm{d}t=\frac{\pi}{\sqrt{x}} e^{a x} \textrm{erfc}(\sqrt{ax}) $$ I've tried substituting $u=\sqrt{t}$ to get $$ 2 ...