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Lecture Notes for Laplace Transform

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Lecture Notes for Laplace TransformWen ShenApril 2009NB! These Notes are used by myself. They are provided to students as a supplement to thetextbook. They can not substitute the textbook. Laplace Transform is used to handle piecewise continuous or impulsive : Definition of the Laplace Transform (1)Topics: Definition of Laplace Transform , Compute Laplace Transform by definition, including piecewise continuous :Given a functionf(t),t 0, its Laplace transformF(s) =L{f(t)}is defined asF(s) =L{f(t)}.= 0e stf(t) limA A0e stf(t)dtWe say the Transform converges if the limit exists, and diverges if we will give examples on computing the Laplace Transform of given functions by (t) = 1 fort (s) =L{f(t)}= limA A0e st 1dt= limA 1se st A0= limA 1s[e sA 1]=1s,(s >0) (t) = (s) =L{f(t)}= limA A0e steatdt= limA A0e (s a)tdt= limA 1s ae (s a)t A0= limA 1s a(e (s a)A 1)=1s a,(s > a) (t)

† Deflnition of Laplace transform, † Compute Laplace transform by deflnition, including piecewise continuous functions. Deflnition: Given a function f(t), t ‚ 0, its Laplace transform F(s) = Lff(t)g is deflned as F(s) = Lff(t)g: = Z 1 0 e¡stf(t)dt = lim: A!1 Z A 0 e¡stf(t)dt We say the transform converges if the limit exists, and ...

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