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Chapter Contents

• Laplace Transforms
• 1a. The Unit Step Function - Definition
• 1b. The Unit Step Function - Products
• 2. Laplace Transform Definition
• Table of Laplace Transformations
• 3. Properties of Laplace Transform
• 4. Transform of Unit Step Functions
• 5. Transform of Periodic Functions
• 6. Transforms of Integrals
• 7. Inverse of the Laplace Transform
• 8. Using Inverse Laplace to Solve DEs
• 9. Integro-Differential Equations and Systems of DEs
• 10. Applications of Laplace Transform

Following are the original SNB files (.tex or .rap) used in making this chapter. For more information, go to SNB info.

SNB files

1a. The Unit Step Function - Definition (SNB)

1b. The Unit Step Function - Products (SNB)

2. Laplace Transform Definition (SNB)

3. Properties of Laplace Transform (SNB)

4. Transform of Unit Step Functions (SNB)

5. Transform of Periodic Functions (SNB)

6. Transforms of Integrals (SNB)

7. Inverse of the Laplace Transform (SNB)

8. Using Inverse Laplace to Solve DEs (SNB)

9. Integro-Differential Equations and Systems of DEs (SNB)

10. Applications of Laplace Transform (SNB)

• Comments, Questions?

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From the math blog...

5. Laplace Transform of a Periodic Function f(t)

If function f(t) is:

Periodic with period p > 0, so that f(t + p) = f(t), and

f₁(t) is one period (i.e. one cycle) of the function, written using Unit Step functions,

then

 

NOTE: In English, the formula says:

The Laplace Transform of the periodic function f(t) with period p, equals the Laplace Transform of one cycle of the function, divided by (1 − e^-sp).

EXAMPLES

Find the Laplace transforms of the periodic functions shown below:

(a)

Answer

(b) Saw-tooth waveform:

Answer

(c) Full-wave rectification of sin t:

Answer