Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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**Inverse Laplace Transform as Convolution Integrals** **Objective:** Evaluate the given inverse Laplace transforms as convolution integrals, and then evaluate the integrals without directly referencing the table of Laplace transforms. --- **Exercises:** (a) Evaluate the inverse Laplace transform: \[ \mathcal{L}^{-1} \left\{ \frac{1}{s(s + 3)^2} \right\} \] (b) Evaluate the inverse Laplace transform: \[ \mathcal{L}^{-1} \left\{ \frac{s}{(s^2 + 4)^2} \right\} \] **Hint:** For part (b), you may want to use the trigonometric identity: \[ \sin(a) \cos(b) = \frac{1}{2} \sin(a+b) + \frac{1}{2} \sin(a-b) \]
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Transcribed Image Text:**Inverse Laplace Transform as Convolution Integrals** **Objective:** Evaluate the given inverse Laplace transforms as convolution integrals, and then evaluate the integrals without directly referencing the table of Laplace transforms. --- **Exercises:** (a) Evaluate the inverse Laplace transform: \[ \mathcal{L}^{-1} \left\{ \frac{1}{s(s + 3)^2} \right\} \] (b) Evaluate the inverse Laplace transform: \[ \mathcal{L}^{-1} \left\{ \frac{s}{(s^2 + 4)^2} \right\} \] **Hint:** For part (b), you may want to use the trigonometric identity: \[ \sin(a) \cos(b) = \frac{1}{2} \sin(a+b) + \frac{1}{2} \sin(a-b) \]
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