The function $f(t)$ satisfies $f(t)=0$ for $t<1$ and

$$f(t+1)-\frac{1}{2} f(t)=H(t),$$

where $H(t)$ is the Heaviside step function. By taking Laplace transforms, show that, for $t \geqslant 1$,

$$f(t)=2+2^{1-t} \sum_{n=-\infty}^{\infty} \frac{e^{2 \pi n i t}}{2 \pi n i-\log 2}$$

and verify directly from the inversion integral that your solution satisfies $f(t)=0$ for $t<1$.