4.3 Translation Theorems Two functions: First Translation Theorem

Examples First Translation Theorem

PROOF First Translation Theorem PROOF : The proof is by Definition

Inverse Form of the First Translation Theorem First Translation Theorem Inverse

Inverse Laplace Inverse

Initial-Value Problem

Examples

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 Second Translation Theorem

1 t 2

Inverse Form Translation on the t-axis

Inverse Form

Alternative Form 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) Translation on the t-axis Alternative Form Proof: Definition and u = t - a

An Initial-Value Problem Example: