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.

367 questions
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Is this transformation for the integral $\int_0^\infty \frac{1}{t}e^{-a^2/t^2-b t} \text{d}t$ correct?

Let's consider the integral: $$I(a,b)=\int_0^\infty \frac{1}{t}e^{-a^2/t^2-b t} \text{d}t,~~~~a,b>0$$ We can try to use the integral representation of the part of the function inside the integral:...
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Evaluating a Erfc integral

I am trying to solve the following integral $$\int_0^{\infty } \int_{a-b x}^{\infty } \exp \left(-u^2\right) \, du \, dx.$$ I know it can be represented as an integral of the complementary error ...
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Integrating variation of error function: $\int_1^2e^{-nx^2} dx$

Show that $$\lim_{n\to\infty} \int_1^2e^{-nx^2} dx = 0.$$ After much googling, I learned that I am working with a variation of the error function! Yay. I've never heard of it in my life and I ...
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The so-called error function defined as: $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^{2}}dt$

The so-called error function is defined as: $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^{2}}dt$$ show that the function $y(x) = e^{x^2}\operatorname{erf}(x)$ satisfies the ...