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Physics LibreTexts

14.5: Shifting Theorem

This is a very useful theorem, and one that is almost trivial to prove.  (Try it!)   It is

                                                            File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif                                                14.5.1

For example, from the table, we have File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.gif   The shifting theorem tells us that

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif  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.