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.

365 questions
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Do Hermite polynomials exist for negative integers?

I recently asked a question about a differential equation, and received this as an answer. It included a Hermite polynomial of negative degree, namely $H_{-3}$. I searched online and it seems as ...
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Using the inverse error function to show a probability is especially small.

I'm reading through the recent computer science paper  and I am stuck on the last line of Corollary 3.2's proof. We are trying to show the probability of an event $A$ is asymptotically small in $n$,...
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Integration by part with substitution

Question: Show that \begin{align}u(x,t) &=c\sqrt{\frac{k}{\pi}}\int^t_0s^{-1/2}e^{-x^2/4ks}\,ds\\ &=c\sqrt{\frac{4kt}{\pi}}e^{-x^2/4kt}-cx\,\text{erfc}\frac{x}{\sqrt{4kt}}\end{align} ...
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Cones, Differentials, and Volume Error Estimates

A cone with a circular base has a height of 40 cm and its radius at the base is 15 cm. Each measurement has 0.3 cm precision. With the help of differentials, estimate the greatest error that is ...
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How can I show this is equivalent to the error function of $x$?

$$\frac{2}{\pi} \int_0^\infty e^{-t^2}\frac{\sin 2xt}{t}\,dt$$ I know the original $\operatorname{erf}x$ but the infinity in the limit keeps getting in my way. How do I deal with it? Also my hint is ...
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Integral involving error function and exponential [closed]

I have not been able to find a source that gives this indefinite integral: $$\int e^{-(a+x)^2}\text{erf}(b+x)\,dx.$$ Can someone provide a source or a formula (with our without demonstration)?
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Solving an integral using the error function

I want to evaluate the following integral $$\int{\frac{1}{\sqrt{2\pi t}} \exp(\frac{-1}{2t}((a-x-wt)^2))dt}$$ where $w>0, 0<x<a$. Using WolframAlpha I obtain an expression for this ...