Question

Prove this theorem.

The Laplace transorm of a piecewise continuous function
f(t)with period p is
L(f) = [1/(1- e^(-ps)]*integral(e^(-st)*f(t)dt from
0to p.

Answer 1

Proved!

Explanation:

`F(s)=\int\limits^\infty_0 f(t)e^(-st)dt=\\\\=\sum\limits^\infty_(n=0)\int\limits^((n+1)p)_(np)  f(t)e^(-st)dt=\\\\=\sum\limits^\infty_(n=0)\int\limits^p_0 f(u+np)e^(-su-snp)du= \: \:  \: ,(u=t-np)\\\\=\sum\limits^\infty_(n=0) e^(-snp) \int\limits^p_0f(u)e^(-su)du=\\\\=(\int\limits^p_0f(u)e^(-su)du)\sum\limits^\infty_(n=0) e^(-snp)=\\\\=(\int\limits^p_0f(u)e^(-su)du)/(1-e^(-ps))`