 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-1/4/d/solution

For

Re α > 0

, the even exponential is

h (t) = g (t) + g (- t)

except for an irrelevant single value at zero. Linearity and the scaling property of the Fourier transform at

b = - 1

give

h (ω) = \frac{1}{α + iω} + \frac{1}{α - iω} = \frac{2 α}{α ^{2} + ω ^{2}} .

In particular, for positive real

α

the transform is real and even, as expected from the symmetry of

h

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