Gaussian Integral: Difference between revisions
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{{refactor|Definition and proof separate}} | {{refactor|Definition and proof separate}} | ||
{{rename|Gaussian Integral over Reals, or something similar<br />In light of GFPs intended work, I propose [[Value of | {{rename|Gaussian Integral over Reals, or something similar<br />In light of GFPs intended work, I propose [[Value of Gaussian Integral]]}} | ||
== Theorem == | == Theorem == | ||
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It has been suggested that this page be renamed. In particular: Gaussian Integral over Reals, or something similar In light of GFPs intended work, I propose Value of Gaussian Integral To discuss this page in more detail, feel free to use the talk page. |
Theorem
The Gaussian Integral is the integral over $\R$ of the function $f(x) = e^{-x^2}$.
Its value is $\sqrt{\pi}$.
That is:
- $\displaystyle \int_{-\infty}^{+\infty} e^{-x^2} \ \mathrm d x = \sqrt \pi$
Proof
One of the most famous proofs of this result uses the trick of calculating instead a two-dimensional integral in polar coordinates, as follows.
Notice that, as $e^{-x^2 -y^2} = e^{-x^2} e^{-y^2}$:
- $\displaystyle (1) \qquad \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} e^{-x^2 - y^2} \ \mathrm d x \ \mathrm d y = \left({\int_{-\infty}^{+\infty} e^{-x^2} \ \mathrm d x}\right)^2$
so if we calculate this two-dimensional integral, we will also have the value we want to find.
Changing to polar coordinates:
\(\ds \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} e^{-x^2 -y^2} \ \mathrm d x \ \mathrm d y\) | \(=\) | \(\ds \int_0^{2\pi} \int_0^{+\infty} r e^{-r^2} \ \mathrm d r \ \mathrm d \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \pi \left[-\frac 1 2 e^{-r^2} \right]_0^{+\infty}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \pi\) |
The result follows from $(1)$.
$\blacksquare$
Source of Name
This entry was named for Carl Friedrich Gauss.