4.3 Translation Theorems

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4.3 Translation Theorems 4.4 Additional
Operational Properies
Two functions
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Examples
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4.3 Translation Theorems 4.4 Additional
Operational Properies
Two functions
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Examples
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Summary
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Inverse Laplace
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Translation
Translation on the s-axis
Translation on the t-axis
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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

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Can be used to write piecewise function in a
compact form
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Write in compact form
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Translation
Translation on the s-axis
Translation on the t-axis
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Inverse Form
Translation on the t-axis
Inverse Form
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Inverse Form
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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)
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Convolution
Remarks
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Convolution
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Convolution
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Convolution (Inverse Form)
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Convolution (special case)
7/pp218
8/pp218
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4 types of equations
1) Algebraic 2) Differential 3) Integral 4)
Integrodifferential
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Volterra Integral Equation
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Laplace of Periodic function
Remark The Laplace Transform of a periodic
function can be obtained by integration over one
period
proof
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Example 7
Find the Laplace Transform of the periodic
function shown in the figure.