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The definite integral of $e^{-x^2/2}$ on the interval $[-a/2,a/2]$, Is there any explicit solution?

张Fous
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    Welcome to Stackexchange. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Brian61354270 Jan 21 '20 at 00:54
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    Are you aware of the error function? Of course, this is nothing but giving a new name to the new function and adds no knowledge unless its property is actually studied. For the actual details of this function and how this is related to your question, you may check the article in the link. – Sangchul Lee Jan 21 '20 at 01:06
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    There's no simple expression for this value, unless you allow yourself to use the error function---and if so the only result you get is essentially a tautology. – YiFan Tey Jan 21 '20 at 01:10
  • But it is known that the error function is not an elementary function. So your answer (as a function of $a$) is not an elementary function. – GEdgar Jan 21 '20 at 01:34

1 Answers1

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This is the gaussian integral $$\int e^{-\frac{x^2}{2}}\,dx=\sqrt{\frac{\pi }{2}} \,\text{erf}\left(\frac{x}{\sqrt{2}}\right)$$ So $$I=\int_{-\frac a2}^{\frac a2} e^{-\frac{x^2}{2}}\,dx=\sqrt{2 \pi }\, \text{erf}\left(\frac{a}{2 \sqrt{2}}\right)\tag 1$$ because of the symmetry : $\text{erf}(t)+\text{erf}(-t)=0$.

If you want an approximation, you could use $$I \sim \sqrt{2 \pi }\, \sqrt{1-\exp\Big(-\frac {a^2} {2\pi} \Big)}\tag 2$$ or, for a better accuracy $$I \sim \sqrt{2 \pi }\, \sqrt{1-\exp\Big(-\frac {a^2} {2\pi}\,\frac{8+\alpha\, a^2}{8+\beta \,a^2} \Big)}\tag 3$$ $$\alpha=\frac{10-\pi ^2}{5 (\pi -3) \pi } \qquad \text{and} \qquad \beta=\frac{120-60 \pi +7 \pi ^2}{15 (\pi -3) \pi }$$

Just a few numbers for illustration. $$\left( \begin{array}{cccc} a & (2) & (3) & (1) \\ 0.25 & 0.24938 & 0.24935 & 0.24935 \\ 0.50 & 0.49507 & 0.49484 & 0.49484 \\ 0.75 & 0.73352 & 0.73279 & 0.73279 \\ 1.00 & 0.96150 & 0.95985 & 0.95985 \\ 1.25 & 1.17617 & 1.17317 & 1.17317 \\ 1.50 & 1.37521 & 1.37048 & 1.37049 \\ 1.75 & 1.55691 & 1.55016 & 1.55016 \\ 2.00 & 1.72014 & 1.71123 & 1.71125 \\ 2.25 & 1.86442 & 1.85339 & 1.85343 \\ 2.50 & 1.98985 & 1.97691 & 1.97698 \\ 2.75 & 2.09703 & 2.08257 & 2.08268 \\ 3.00 & 2.18704 & 2.17155 & 2.17171 \\ 3.25 & 2.26129 & 2.24532 & 2.24553 \\ 3.50 & 2.32141 & 2.30553 & 2.30580 \\ 3.75 & 2.36918 & 2.35390 & 2.35424 \\ 4.00 & 2.40642 & 2.39218 & 2.39258 \end{array} \right)$$

Claude Leibovici
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