14.5: Shifting Theorem
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This is a very useful theorem, and one that is almost trivial to prove. (Try it!) It is
L(e−aty(t))=ˉy(s+a).
For example, from the table, we have L(t)=1/s2. The shifting theorem tells us that L(te−at)=1/(s+a)2. I'm sure you will now want to expand your table even more. Or you may want to go the other way, and cut down the table a bit! After all, you know that L(1)=1/s. The shifting theorem, then, tells you that L(eat)=1/(s−a), so that entry in the table is superfluous! Note that you can use the theorem to deduce either direct or inverse transforms.