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Gaussian Integral

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The area under e−x2e^{-x^2}
The Gaussian integral, named after the German mathematician Carl Friedrich Gauss, the integral is
∫−∞∞e−x2 dx=π.\begin{equation*}\int_{-\infty}^{\infty} e^{-x^2}\,\mathrm{d}x=\sqrt{\pi}.\end{equation*}
Abraham de Moivre originally discovered this type of integral in 1733, while Gauss [Gauss1809] published the precise integral in 1809, attributing its discovery to Laplace. De Moivre's work also led to the De Moivre–Laplace theorem, an early special case of the central limit theorem.
The standard normal density also appears as the limiting law in the central limit theorem. After a change of scale, it appears as the fundamental solution of the heat equation, where solutions are obtained by convolution with a Gaussian kernel.
Two-dimensional surface e−(x2+y2)e^{-(x^2+y^2)}.
Let
∫−∞∞e−x2 dx.\begin{equation*}\int_{-\infty}^{\infty} e^{-x^2}\,\mathrm{d}x.\end{equation*}
and taking the square gives
∫−∞∞∫−∞∞e−(x2+y2) dx dy.\begin{equation*}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-(x^2+y^2)}\,\mathrm{d}x\,\mathrm{d}y.\end{equation*}
The radius rr satisfies r2=x2+y2r^2=x^2+y^2, so e−(x2+y2)=e−r2e^{-(x^2+y^2)}=e^{-r^2}.
We use polar coordinates,
x=rcos⁡θ,y=rsin⁡θ.\begin{equation*}x=r\cos\theta,\qquad y=r\sin\theta.\end{equation*}
The corresponding Jacobian determinant is
∣∂(x,y)∂(r,θ)∣=∣(cos⁡θ−rsin⁡θsin⁡θrcos⁡θ)∣=r.\begin{equation*}\left|\frac{\partial(x,y)}{\partial(r,\theta)}\right| =\left|\begin{pmatrix}\cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta\end{pmatrix}\right| = r.\end{equation*}
Hence
dx dy=r dr dθ.\begin{equation*}\mathrm{d}x\,\mathrm{d}y= r\,\mathrm{d}r\,\mathrm{d}\theta.\end{equation*}
Substituting these expressions, we obtain
∫02π∫0∞e−r2r dr dθ=2π[−12e−r2]0∞=π.\begin{equation*}\int_0^{2\pi}\int_0^\infty e^{-r^2}r\,\mathrm{d}r\,\mathrm{d}\theta = 2\pi\left[-\frac{1}{2}e^{-r^2}\right]_{0}^{\infty} = \pi.\end{equation*}
Therefore
∫−∞∞e−x2 dx=π\begin{equation*}\int_{-\infty}^{\infty} e^{-x^2}\,\mathrm{d}x = \sqrt{\pi}\end{equation*}
The standard normal distribution distribution is defined by
Φ(t)=P(X≤t)=12π∫−∞te−x2/2 dx.\begin{equation*}\Phi(t) = \mathbb{P}(X\leq t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{t} e^{-x^2/2}\,\mathrm{d}x.\end{equation*}
The rate at which 1−Φ(t)1-\Phi(t) decreases as t→∞t\to\infty can be estimated explicitly.
The distribution function $\Phi(t)$
The distribution function Φ(t)\Phi(t)
LEMMA
For any t>0t>0,
12π(1t−1t3)e−t2/2≤1−Φ(t)≤12π1te−t2/2.\begin{equation*}\frac{1}{\sqrt{2\pi}} \left(\frac{1}{t}-\frac{1}{t^3}\right)e^{-t^2/2} \leq 1-\Phi(t) \leq \frac{1}{\sqrt{2\pi}}\frac{1}{t}e^{-t^2/2}.\end{equation*}
Write
1−Φ(t)=12π∫t∞e−s2/2 ds.\begin{equation*}1-\Phi(t) = \frac{1}{\sqrt{2\pi}}\int_t^\infty e^{-s^2/2}\,\mathrm{d}s.\end{equation*}
For the upper bound, since s/t≥1s/t\geq 1 for s≥ts\geq t,
∫t∞e−s2/2 ds≤1t∫t∞se−s2/2 ds=1te−t2/2.\begin{equation*}\int_t^\infty e^{-s^2/2}\,\mathrm{d}s \leq \frac{1}{t}\int_t^\infty s e^{-s^2/2}\,\mathrm{d}s = \frac{1}{t}e^{-t^2/2}.\end{equation*}
For the lower bound, integration by parts gives
∫t∞e−s2/2 ds=∫t∞se−s2/21s ds=e−t2/2t−∫t∞1s2e−s2/2 ds.\begin{equation*}\int_t^\infty e^{-s^2/2}\,\mathrm{d}s = \int_t^\infty s e^{-s^2/2}\frac{1}{s}\,\mathrm{d}s= \frac{e^{-t^2/2}}{t} - \int_t^\infty \frac{1}{s^2}e^{-s^2/2}\,\mathrm{d}s.\end{equation*}
Using the upper bound just proved,
∫t∞1s2e−s2/2 ds≤1t2∫t∞e−s2/2 ds≤1t3e−t2/2.\begin{equation*}\int_t^\infty \frac{1}{s^2}e^{-s^2/2}\,\mathrm{d}s \leq \frac{1}{t^2}\int_t^\infty e^{-s^2/2}\,\mathrm{d}s \leq \frac{1}{t^3}e^{-t^2/2}.\end{equation*}
Substituting this into the previous identity gives the claimed lower bound.

References

  • [Gauss1809]Gauss, Carl Friedrich. Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium. F. Perthes and I. H. Besser, 1809. PDF.
  • [Bogachev1998]Bogachev, Vladimir I. Gaussian Measures. First edition. American Mathematical Society, 1998.