sb3-hw01-2016

# sb3-hw01-2016 - JHU 580.429 SB3 HW1 In this homework L...

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JHU 580.429 SB3 HW1 In this homework, L stands for the Laplace transform, ˜ f ( s ) = L [ f ( t )] = R 0 dte - st f ( t ) . 1. Simple Laplace transforms. Provide ˜ f ( s ) for the following f ( t ) . (a) f ( t ) = 1 (b) f ( t ) = 0 if t < a , f ( t ) = 1 if t a , and a 0. (c) f ( t ) = e - at (d) f ( t ) = t 2. Laplace transforms of powers through generating functions. (a) Let f ( t ) = lim a 0 ( - d / da ) e - at . What is ˜ f ( s ) ? (b) Let ˜ g ( s , a ) = - ( d / da ) L [ e - at ] . What is ˜ g ( s , a ) ? (c) If functions are well-behaved, then ˜ f ( s ) should equal lim a 0 ˜ g ( s , a ) . Calculate
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