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# 6.3 - Math 334 Lecture#25 6.3 Step Functions(and other...

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Math 334 Lecture #25 § 6.3: Step Functions (and other Piecewise Continuous Functions) A Prototype. The unit step function with an upward step of 1 at c 0 is u c ( t ) = 0 if 0 t < c, 1 if t c. [Sketch graph of the unit step function.] The Laplace transform of the unit step function is L{ u c ( t ) } = lim A →∞ A 0 e - st u c ( t ) dt [ u c ( t ) is zero for t < c ] = lim A →∞ A c e - st dt = lim A →∞ e - st - s A c = lim A →∞ e - sA - s - e - sc - s = e - cs s when s > 0. Example. Can the piecewise defined function h ( t ) = 1 if 0 t < 1 , t if 1 t < 2 , 0 if t 2 , be written as a linear combination of unit step functions? Yes, it can: h ( t ) = 1 u 0 ( t ) - 1 u 1 ( t ) + tu 1 ( t ) - tu 2 ( t ) + 0 u 2 ( t ) = 1 + ( t - 1) u 1 ( t ) - tu 2 ( t ) . [Use appropriate unit step functions to “switch on” or “switch off” pieces of the function.] What is the Laplace transform of a function like h ( t )? The “First” Shifting Theorem. The Laplace transform of the shift or translation u c ( t ) f ( t - c ) is L{ u c ( t ) f ( t - c ) } = e - cs L{ f ( t ) } .

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6.3 - Math 334 Lecture#25 6.3 Step Functions(and other...

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