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Translation Theorems & Derivatives of a Transform

Learn about the first and second translation theorems, the Laplace transform of the unit step function, and the derivatives of transforms.

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Translation Theorems & Derivatives of a Transform

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  1. 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 functionU (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)}.

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