 Laplace transform of a periodic function

Let $T>0$, and let $g$ be a locally integrable function with period $T$ on nonnegative time. For $\mathrm{Re}s>0$, its unilateral Laplace transform isSplit the defining integral into successive periods and put $t=u+jT$. Periodicity makes each piece $e^{-sjT}{\int }_0^T{e}^{-su}g(u)du$; summing the absolutely convergent geometric series proves the formula. Local integrability on one period also controls the corresponding absolute integrals.

For a square pulse equal to one on $[0,T/2)$ and zero on $[T/2,T)$, repeated periodically, the result is $(1-e^{-sT/2})/[s(1-e^{-sT})]=1/[s(1+e^{-sT/2})]$ in that half-plane. At $s=0$ the defining improper integral generally fails; continuing the quotient algebraically does not enlarge the defining region of absolute convergence of a Laplace transform.