Definition of Laplace Transform

Definition of Laplace Transform 1. Definition of Laplace Transform The Laplace transform of a function f (t) of a real variable t is another function depending on a new variable s, which is in general complex. We will denote the Laplace transform of f by L f . It is defined by the integral (L f )(s) = � ∞ 0− f (t)e−st dt, (1) for all values of s for which the integral converges. There are a few things to note. • L f is only defined for those values of s for which the improper inte­ gral on the right-hand side of (1) converges. • We will allow s to be complex. • As with convolution the use of 0− , in the definition (1) is necessary to accomodate generalized functions containing δ(t). Many textbooks do not do this carefully, and hence their definition of the Laplace transform is not consistent with the properties they assert. In those cases where 0− isn’t needed we will use the less precise form (L f )(s) = � ∞ f (t)e−st dt. 0 (1’) • Also, as with convolution, the limits of integration mean that the Laplace transform is only concerned with functions on (0− , ∞). 2. Notation, F (s) We will adopt the following conventions: 1. Writing (L f )(s) can be cumbersome so we will often use an uppercase letter to indicate the Laplace transform of the corresponding lowercase function: (L f )(s) = F (s), (L g)(s) = G (s), etc. For example, in the formula L( f � ) = sF (s) − f (0− ) it is understood that we mean F (s) = L( f ). 18.03SC 2. If our function doesn’t have a name we will use the formula instead. For example, the Laplace transform of the function t2 is written L(t2 )(s) or more simply L(t2 ). 3. If in some context we need to modify f (t), e.g. by applying a translation by a number a, we can write L( f (t − a)) for the Laplace transform of this translation of f . 4. You’ve already seen several different ways to use parentheses. Some­ times we will even drop them altogether. So, if f (t) = t2 then the fol­ lowing all mean the same thing (L f )(s) = F (s) = L f (s) = L( f (t))(s) = L(t2 )(s); L f = F = L(t2 ). 3. First Examples For the first few examples we will explicitly use a limit for the improper integral. Soon we will do this implicitly without comment. Example 1. Let f (t) = 1, find F (s) = L f (s). Solution. Using the definition (1’) we have L(1) = F (s) = � ∞ e −st 0 e−st dt = lim T →∞ − s �T 0 e−sT − 1 = lim T →∞ −s �T . 0 The limit depends on whether s is positive or negative. � 0 if s > 0 −sT lim e = T →∞ ∞ if s < 0. Therefore, � L(1) = F (s) = 1 s diverges if s > 0 if s ≤ 0. (We didn’t actually compute the case s = 0, but it is easy to see it diverges.) Example 2. Compute L(e at ). Solution. Using the definition (1’) we have at L(e ) = � ∞ 0 at −st e e e( a−s)t dt = lim T →∞ a − s �T 0 e( a−s) T − 1 = lim T →∞ a−s �T . 0 18.03SC The limit depends on whether s > a or s < a. � 0 if s > a ( a−s) T lim e = T →∞ ∞ if s < a. Therefore, � L(e at ) = 1 s− a diverges if s > a if s ≤ a. (We didn’t actually compute the case s = a, but it is easy to see it diverges.) We have the first two entries in our table of Laplace transforms: f (t) = 1 ⇒ F (s) = 1/s, f (t) = e at ⇒ F (s) = 1/(s − a), s > a. s>0 4. Linearity You will not be surprised to learn that the Laplace transform is linear. For functions f , g and constants c1 , c2 L(c1 f + c2 g) = c1 L( f ) + c2 L( g) This is clear from the definition (1) of L and the linearity of integration.