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Laplace Transform Translation Theorems: Examples & Properties

Explore translation theorems in Laplace transform, including turning off portions of graphs and writing piecewise functions compactly, with detailed examples and operational properties. Learn about convolution, alternative forms, and solving different types of equations. Discover the Laplace transform of periodic functions and how to integrate over one period.

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Laplace Transform Translation Theorems: Examples & Properties

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  1. 4.3 Translation Theorems 4.4 Additional Operational Properies Two functions:

  2. Examples

  3. 4.3 Translation Theorems 4.4 Additional Operational Properies Two functions:

  4. Examples

  5. Summary

  6. Inverse Laplace

  7. Translation Translation on the s-axis Translation on the t-axis

  8. 1 t Heaviside function (Unit step fun) • Step function defined on non-negative x-axis (positive x + zero ) • f(t)u(t-a) turns off a portion of the graph of f • Can be used to write piecewise function in a compact form

  9. Can be used to write piecewise function in a compact form

  10. 2 1 t Write in compact form

  11. Translation Translation on the s-axis Translation on the t-axis

  12. 2 1 t

  13. Inverse Form Translation on the t-axis Inverse Form

  14. Inverse Form

  15. Alternative Form Translation on the t-axis We are frequently confronted with the problem of finding the Laplace Transform of a product of a function g and a unit step function u(t-a)

  16. Convolution Remarks

  17. Convolution

  18. Convolution

  19. Convolution (Inverse Form)

  20. Convolution (special case) 7/pp218 8/pp218

  21. 4 types of equations 1) Algebraic 2) Differential 3) Integral 4) Integrodifferential

  22. Volterra Integral Equation

  23. Laplace of Periodic function Remark: The Laplace Transform of a periodic function can be obtained by integration over one period proof

  24. 1 t Example 7 Find the Laplace Transform of the periodic function shown in the figure.

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