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One dimension:

G(x)=12πσ2ex22σ2

Two dimensions:

G(x,y)=12πσ2ex2+y22σ2

This is because:

I(t)=ttex2dxI2(t)=(ttex2dx)(ttey2dy)=tttte(x2+y2)dxdyt02π0er2rdrdθ<I2(t)<t202π0er2rdrdθt02πer2rdrdθ<I2(t)<t202πer2rdrdθπ(1et2)<I2(t)<π(1e2t2)lim

so we obtain:

\lim_{t \to +\infty} I^2(t) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2+y^2)}dx dy = \pi \\ \lim_{t \to +\infty} I(t) = \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt \pi \\ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{2\pi \sigma^2} e^{-\frac{x^2+y^2}{2\sigma^2 }} dx dy = 1

2016-10-10 Comments

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