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 $\mathrm{Re}\alpha >0$, 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 $b=-1$ giveIn particular, for positive real $\alpha$ the transform is real and even, as expected from the symmetry of $h$.