1. Translation Theorems and Derivatives of a Transform
Section 7.3
Translation Theorems and
Derivatives of a Transform

2. THE FIRST TRANSLATION THEOREM
Theorem: If a is any real number, then
where F(s) = L { f (t)}.
NOTE: This theorem is sometimes called the First Shifting Theorem.

3. INVERSE FORM OF THE FIRST TRANSLATION THEOREM
where f (t) = L −1{F(s)}.

4. THE UNIT STEP FUNCTION
The unit step function U (t − a) is defined to be

5. THE SECOND TRANSLATION THEOREM
Theorem: If a is a positive constant, then
where F(s) = L { f (t)}.
NOTE: This theorem is also called the Second Shifting Theorem.

6. LAPLACE TRANSFORM OF THE UNIT STEP FUNCTION
Using the Second Translation Theorem with f (t) = 1, f (t − a) = 1, and F(s) = 1/s, we find that the Laplace transform of the unit step function is

7. INVERSE OF THE SECOND TRANSLATION THEOREM
where a > 0 and f (t) = L −1{F(s)}.

8. DERIVATIVES OF TRANSFORMS
Theorem: For n = 1, 2, 3, ,
where F(s) = L { f (t)}.