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Properties and Theorems of the Laplace TransformProof:ft () e – st dt0ft ( + nT)⇔ -----------------------------1–e – sTThe Laplace transform of a periodic function can be expressed as∫T(2.28)∫∞0∫TL { ft ()} = ft () e – st dt = f()t e – st dt + f()t e – st dt + f()t e – st dt + …0In the first integral of the right side, we let t = τ , in the second t = τ + T , in the thirdt = τ + 2T, and so on. The areas under each period of ft () are equal, and thus the upper andlower limits of integration are the same for each integral. Then,∫2TT∫3T2TTL { ft ()} =∫f( τ)e – sτ dτ +∫f ( τ + T ) e – s( τ + T)dτ +∫f ( τ + 2T ) e – s( τ + 2T)dτ + …0T0T0(2.29)Since the function is periodic, i.e., f( τ) = f( τ + T) = f( τ + 2T) = … = f( τ + nT), we can write(2.29) asL { f( τ)} = ( 1 + e – sT + e – 2sT + …) f( τ)e – sτ dτBy application of the binomial theorem, that is,∫T0(2.30)1+ a+ a 2 + a 3 + …=1----------1–a(2.31)we find that expression (2.30) reduces toL{ f( τ)}=∫Tf( τ)e – sτ dτ0---------------------------------1–e – sT2.2.10 Initial Value TheoremThe initial value theorem states that the initial value f( 0 −) of the time function ft () can be foundfrom its Laplace transform multiplied by s and letting s → ∞ .That is,limt→0ft ()= lim sF( s)=s → ∞f ( 0 −)(2.32)Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth EditionCopyright © Orchard Publications2−9