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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How can I calculate percent error with a denominator of 0?

I am using the MAPE formula, ABS(actual-forecast)/actual *100, but this raises problems when the actual values are zero. Is it possible to add 1 to the denominator and calculate this way? Would I get ...
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44 views

Asymptoting to 0: is erfc(z) quicker than exp(-z)?

If I have a function of the form $\mathrm{erfc}\left(z\right)/e^{-z}$, should I expect its limit at large $z$ to be $0$ or $\infty$? My instinct is that it should be $0$, by considering the ...
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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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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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Does anyone know of a closed form solution to the following integral?

Does anyone know of a closed form solution to the following integral? $$ \DeclareMathOperator\erf{erf} \newcommand{d}{\;\mathrm{d}} \int^{+\infty}_{-\infty} \erf^{\;m}\!(x) \frac{\d^n ...
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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?
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inverse complementary error function values near 0

$\newcommand{\erf}{\operatorname{erf}}\newcommand{\erfc}{\operatorname{erfc}}$Let's define for each $x>0$ $$\erf(x)=\frac {2}{\sqrt{\pi}}\int_0^xe^{-t^2} dt$$ and $$\erfc(x)=\frac ...
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1answer
15 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 ...
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1answer
33 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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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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1answer
54 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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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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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}} ...
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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 ...
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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 ...
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1answer
86 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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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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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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118 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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93 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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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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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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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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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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37 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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74 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
180 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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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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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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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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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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51 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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25 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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98 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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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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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) = ...
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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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87 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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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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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 $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
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66 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
26 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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53 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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202 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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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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1answer
224 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 ...