MATH 312 Section 7.1: Definition of a Laplace Transform Prof. Jonathan Duncan Walla Walla University Spring Quarter, 2008
Outline 1 The Laplace Transform 2 The Theory of Laplace Transforms 3 Conclusions
What is a Transform? A common theme in mathematics is the transformation of one sort of object into another. The differential operator D is a transform. It changes a function f into its derivative, f. One reason we to transform one sort of object into another is that we can sometimes more easily work with the second sort of object. Integral Transforms The differential operator above transforms a function into a derivative in a very specific and useful way. In this section we introduce the Laplace Transform which is an integration transformation.
Defining the Laplace Transform The Laplace transform is a result of integrating a function of two variables with respect to only one of the variables. Definition 7.1 Let f be a function defined for t 0. Then the integral L {f (t)} = 0 e st f (t) dt is called the Laplace Transform of f, provided that it converges. Note: What sort of object is L {f (t)}? This integral transform returns a function of s.
Applying the Definition We now apply this definition directly to find the Laplace transform of several basic functions. Use the definition to find L {4}. Use the definition to find L {e kt }. Use the definition to find L {cos 3t}. Question: We have seen that the differential operator D is linear. Is L linear as well?
Transforms of Basic Functions From our work on the last slide, we can now state the Laplace transforms of some basic functions. Theorem 7.1 The following basic functions have the given Laplace Transforms. 1 L {1} = 1 s 2 L {t n } = n! s n+1 3 L {e at } = 1 s a 4 L {sin kt} = k s 2 +k 2 5 L {cos kt} = s s 2 +k 2 6 L {sinh kt} = k s 2 k 2 7 L {cosh kt} = s s 2 k 2
Exponential Order Not all functions will have a Laplace transform as the integral involved may not converge. Definition 7.2 A function f is said to be of exponential order c if there exists constants c, M > 0, and T > 0 such that f (t) Me ct for all t > T. Show that the function e t2 of exponential order. is not The function f pictured below is of exponential order. Note: e t2 > e t for t > 1.
Piecewise Continuous Below is another important definition as we try to determine which functions have Laplace transforms. Piecewise Continuous Functions A function f is piecewise continuous on [0, ) if for any subinterval (a, b) there are at most finitely many points at which f has finite discontinuities and f is continuous elsewhere. A piecewise continuous function is mostly continuous with only a countable number of finite discontinuities.
Existence of a Laplace Transform Putting these two definitions together, we can make the following statement. Theorem 7.2 If f is piecewise continuous on [0, ) and of exponential order c, then L {f (t)} exists for s > c. Find L {f (t)} for the piecewise defined f given below. { 2t + 1 0 t < 3 f (t) = 0 t > 3
Important Concepts Things to Remember from Section 7.1 1 Stating and Applying the definition of a Laplace transform 2 Laplace transforms of basic functions 3 Functions of exponential order 4 Piecewise defined functions 5 Existence theorem for Laplace transforms