# Laplace Transform of a periodic function

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Let f be a piecewise continuous function on [0,T]. Define f on the whole of [0,inf) by f(t+nT) for all t and all integer n. Show that the Laplace transform if f is given by

L[f(t)] = 1/[1-exp(-sT)]*int(exp(-st)*f(t)dt,t=0..T)

By taking the Laplace transform and using the convolution theorem, obtain the solution of the integral equation

f(t) = 1+t+int[(t-u)f(u)du,u=0..t).

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The solution shows how to use the Laplace transform definition and convolution properties to obtain the required results.

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