47
edits
Changes
Jump to navigation
Jump to search
Line 1:
Line 1: − + − + − The '''Gaussian integral''' is the integral:+ − + − It has a value of <math>\sqrt\pi</math>. The value is needed to normalize the [[Normal distribution]].+ − + − First look at the [[Double integral|double integral]]+ − + − Separating it,+ − + − So, the double integral is merely the square of the Gaussian integral. + − Now, do the double integral in polar co-ordinates. <math>-x^2-y^2= -(x^2+y^2) = -r^2</math> and <math>dxdy=rdrd\theta</math>, so:+ − + − + − + − Substituting <math>z=r^2</math> into the integral,+ − + − + − Therefore,+ − + − + − +
no edit summary
{{move|Gaussian integral}} (The current version uses quotation marks in the title!)
{{| }} ( !)
{{Template:Math-h}}
{{:-}}
''' ''' :
:<math> \int_{-\infty}^{\infty} e^{-x^2}dx. </math>
:<> \{-\}^{\} ^{-^}. </>
<>\\</>. [[ ]].
==Derivation==
====
[[ | ]]
<math>\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2-y^2}dxdy.</math>
<>\{-\}^{\} \{-\}^{\} ^{-^-^}.</>
,
<math>\int_{-\infty}^{\infty} e^{-x^2}dx \int_{-\infty}^{\infty} e^{-y^2}dy = (\int_{-\infty}^{\infty} e^{-x^2}dx)^2 </math>
<>\{-\}^{\} ^{-^} \{-\}^{\} ^{-^} = (\{-\}^{\} ^{-^})^ </>
, .
, -. <>-^-^= -(^+^) = -^</> <>=\</>, :
<math>\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2-y^2}dxdy = \int_{0}^{2\pi}\int_{0}^{\infty} e^{-r^2}rdrd\theta</math>
<>\{-\}^{\} \{-\}^{\} ^{-^-^} = \{}^{\}\{}^{\} ^{-^}\</>
<math> = (\int_{0}^{2\pi}d\theta)(\int_{0}^{\infty} e^{-r^2}rdr)</math>
<> = (\{}^{\}\)(\{}^{\} ^{-^})</>
<math> = 2\pi \int_{0}^{\infty} e^{-r^2}rdr.</math>
<> = \ \{}^{\} ^{-^}.</>
<>=^</> ,
<math> = 2\pi \int_{0}^{\infty} e^{-z}(z^{1/2})(\frac{dz}{2z^{1/2}})</math>
<> = \ \{}^{\} ^{-}(^{/})(\{}{^{/}})</>
<math> = \pi \int_{0}^{\infty} e^{-z}dz = \pi \left[-e^{-z} \right]_0^{\infty} = \pi.</math>
<> = \ \{}^{\} ^{-} = \ \[-^{-} \]^{\} = \.</>
,
<math>(\int_{-\infty}^{\infty} e^{-x^2}dx)^2 = \pi </math>
<>(\{-\}^{\} ^{-^})^ = \ </>
<math>\int_{-\infty}^{\infty} e^{-x^2}dx = \sqrt\pi.</math>
<>\{-\}^{\} ^{-^} = \\.</>
[[category:mathematics]]
[[:]]