Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2014 / 6 / c

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- Solution c

Solution

Complete the square inside the Fourier transform of the Gaussian heat kernel:

- \frac{x ^{2}}{4 t} - iω x = - (\frac{x}{2 t} + i t ω)^{2} - t ω^{2} .

After

z = x / (2 t)

,

q_{t} (ω) = \frac{e ^{- t ω^{2}}}{π} \int_{- \infty}^{\infty} e^{- (z + i t ω)^{2}} d z .

The supplied shifted Gaussian integral equals

π

for real

t ω

. Hence

q_{t} (ω) = e^{- t ω^{2}} .

At zero this says

\int q_{t} = 1

, so convolution with the heat kernel averages rather than rescales the initial mass. The formula is an instance of the Fourier transform of a Gaussian.

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