Laplace Transform A transform is an operation that changes a function into a new function. Examples of this are derivatives and antiderivatives. These.

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Presentation transcript:

Laplace Transform A transform is an operation that changes a function into a new function. Examples of this are derivatives and antiderivatives. These are linear transforms:

Def. Let f (t) be defined for t ≥ 0, then is called the Laplace Transform of f. *** L{ f (t)} = F(s)

Ex. Find L{1}. Find the domain.

Ex. Find L{t}.

Ex. Find L{e-3t}.

Some basic transforms are on p. 277  L is linear Ex. Find L{1 + 5t}.

A function f is of exponential order c if there exist constants c, M > 0, and T > 0 such that | f (t)| < Mect for all t > T. Thm. If f is a piecewise continuous function on [0,∞) and of exponential order c, then L{ f (t)} exists for s > c.  This is not a requirement for the existence of L, others also have an L.

Thm. If f is piecewise continuous on (0,∞) and of exponential order, and if F(s) = L{ f (t)}, then

Ex. Find L{ f (t)}, where