4.3 Translation Theorems 4.4 Additional Operational Properies Two functions:

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4.3 Translation Theorems 4.4 Additional Operational Properies Two functions:

Examples

4.3 Translation Theorems 4.4 Additional Operational Properies Two functions:

Examples

Summary

Inverse Laplace

Translation Translation on the s-axis Translation on the t-axis

Heaviside function (Unit step fun) 1)Step function defined on non-negative x-axis (positive x + zero ) 2) f(t)u(t-a) turns off a portion of the graph of f 3)Can be used to write piecewise function in a compact form 1 t

Can be used to write piecewise function in a compact form

1 t 2 Write in compact form

Translation Translation on the s-axis Translation on the t-axis

1 t 2

Inverse Form

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)

Convolution Remarks

Convolution

Convolution (Inverse Form)

Convolution (special case) 7/pp218 8/pp218

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

Volterra Integral Equation

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

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