# 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.

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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.
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### Prove $\operatorname{erfi}(x) \in O(e^{x^2})$

How can we prove: $$\operatorname{erfi}(x) \in O (e^{x^2})$$ Samely we can prove: $$\lim_{ x \rightarrow \infty } \frac {\int_0^x e^{t^2} \, dt} {e^{x^2}} = 0$$ Also it is so awkward. The ...
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### Using error function to calculate probability that a random variate falls z standard deviations to the right of the mean

I am imagining a normal distribution with a mean of zero and a standard deviation of S. I know that the following function tells us the probability that a random deviate (under the assumption of ...
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### Quadratic-trigonometric integral -- part 2

Problem I need to compute the following integral \begin{equation*}\int_{t_\text{s}}^{t_\text{e}} \cos(a+b\tau+c\tau^2)\text{ d}\tau\end{equation*} where $t_{\text{s}}<t_{\text{e}}$ and $a,b,c>0$...
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### What is special about a function being elementary? [duplicate]

Functions such as $\sin(x)$ are considered to be elementary, however functions like $\text{erf}(x)$ are considered to be non-elementary. What makes elementary functions different from non-...
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### ODE: $y''y+ax+by+c=0,y=k\pm\sqrt2\int\sqrt{a\int\ln(y)dx-(ax+c)\,\ln(y)-by+K}dx,\int\frac{dy}{\sqrt{K-(ax+c)\,\ln(y)+a\int\ln(y)dx-by}}=k\pm\sqrt2x$

Imagine we had a differential equation like: $$y’’-\frac xy=0$$ Now let’s standardize the signs. Note we do not need a constant for the first term because of the zero product property. We can ...
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