One dimension:
G(x)=1√2πσ2e−x22σ2
Two dimensions:
G(x,y)=12πσ2e−x2+y22σ2
This is because:
I(t)=∫t−te−x2dxI2(t)=(∫t−te−x2dx)⋅(∫t−te−y2dy)=∫t−t∫t−te−(x2+y2)dxdy∫t0∫2π0e−r2rdrdθ<I2(t)<∫t√20∫2π0e−r2rdrdθ∫t02πe−r2rdrdθ<I2(t)<∫t√202πe−r2rdrdθπ(1−e−t2)<I2(t)<π(1−e−2t2)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
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