Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [error-function]

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.

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.

The error function is an entire function defined as $$\operatorname{erf}(z)\equiv\frac2{\sqrt\pi}\int_0^ze^{-t^2}\,dt.$$ Further details are given here.

The complementary error function is defined as $\operatorname{erfc}(z)=1-\operatorname{erf}(z)$. Further details are given here.

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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 a total of 5005m So Variance of +50 m/hr Variance…
Tim Edwards
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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) = \left\{\begin{aligned} &\frac{1}{\Delta} &&: |x-\xi| < \frac…
user94517
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Integral involving exponentials and error function

For the following integral, $\displaystyle\int_{-\infty}^{\infty}e^{-ax^2}\mathrm{erf}\!\left(\dfrac{x+b}{\sqrt2}\right)\mathrm dx$ we were asked to prove that it yields the following closed form: $\sqrt{\dfrac{\pi}a}\,\mathrm{erf}\!\left(b…
Tasneem Y. Aly
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Explicit form for solutions x in $\operatorname{Erfc}(-x) = a \cdot e^{-x^2}$

I'm looking for closed form of points where $\operatorname{Erfc}(-x)$ intersects with $a \cdot e^{-x^2}$ for $a$ being some fixed non-zero constant. I guess I can use the exponential approximation for $\operatorname{Erfc}$. Then I probably can find…
Piotr Semenov
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Accurate averaging in $Q$ scale

Let $Q(x)$ denote the complementary cumulative distribution function of a standard normal distribution (see here). Given $ 0 \leq a \leq b$, and define the $Q$-scale average of $a$ and $b$ as a $c$ satisfying $$1/2 Q(a) + 1/2 Q(b) = Q(c).$$ A…
MikeL
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Integral of error function times Gaussian

By manipulating equation 4.3.13 from A table of integrals of the error functions, it is possible to derive the following result: $$ \int_{-\infty}^{+\infty} e^{-(ax+b)^2}\text{erf}(cx+d)dx = \frac{\sqrt\pi}{a}\text{erf}\left(\frac{ad-bc}{\sqrt{a^2 +…
NokMok
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Integral involving the Erf function

I'm am trying to solve the following integral $$\int\limits_{-\infty}^{+\infty}dx \; e^{-(ax+b)^2}\mathrm{Erf}(cx+d)\mathrm{Erf}(ex+f)$$ I tried the same reasoning as for these integrals that can be solved analytically but it is not as…
Alexandre Krajenbrink
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How to Show that the Error Function has an Upper Bound?

Here is the error function: $$\mathrm{erf}(x)=\frac{2}{\sqrt\pi}\int^x_0e^{-t^2} dt$$ Here is the question: Show that the odd function erf is bounded, by using the fact that:$$e^{-t^2} \le te^{-t^2} , t \ge1$$ Remark: The normalisation of …
t77
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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?
Expat_Canuck
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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 $$ R=\frac{(A \pm \Delta A)}{(B\pm\Delta B)}.$$
Doon
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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 is 4 and above, or else it rounds down the value…
matin
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Error Calculation With Derivative Respect to Logx

I have a function: $$ LogC= (LogA - 0.80 * LogB - 8.40)/0.50 $$ Here LogA and LogB have errors. So I know that if $$LogC=f$$then the error of f is $$df=\frac{\partial f}{\partial a}da+\frac{\partial f}{\partial b}db$$ and so on. Here i cannot be…
Ege Tunç
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Bounds or approximation on $\frac{1-\mathrm{erf}(x)}{1+\mathrm{erf}(x)}$

Any ideas how I can obtain a tight lower, upper bound, or approximation on $$f(x)=\frac{1-\mathrm{erf}(x)}{1+\mathrm{erf}(x)}$$ for almost any $x$. This function $f(x)$ grows as $e^{x^2}$ on $x\ll 0$ and decays as $e^{-x^2}$ on $x\gg 0$. I need a…
Nik
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Proving $\text{erf}(\sqrt{t})=\frac{1}{\sqrt{\pi}} \int_{0}^{t} \frac{e^{-\tau}}{\sqrt{\tau}} \;d\tau$

I started by using the error function: $$ \begin{align} \text{erf}(y)=\frac{2}{\sqrt{\pi}}\int_{0}^{y} e^{-u^{2}} du, \end{align} $$ where I let $\tau=u^{2}$ then $d\tau=2u\;du$, $du=\frac{d\tau}{2u}=\frac{d\tau}{2\sqrt{\tau}}$. Then substituting…
Dewton
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Is the error function and $\mathcal{N}(0,1)$ the same thing?

Are $\mathcal{N}(0,1)$ and $\Phi(x)$ the same thing? It seems that the derivative $\Phi(x)$ and the density of $\mathcal{N}(0,1)$ are the same, but I'm not sure.
user412026
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