MATH 255 Lecture Notes - Lecture 8: Improper Integral, Multiple Integral, Umber
Document Summary
Laplace transform (i) not every function f (t) has a laplace transform. For example, f (t) = 1 t does not have a laplace transform. The reason is that the improper integral does not converge for any s 2 r: = 1: e e (cid:0)st 1 t (cid:0)s 1 t. Note that f (t) = 1 t is not continuous on [0;1), although it tends to zero as t ! Theorem 6. 1. 2: if f (t) is continuous for t 2 [0;1), and of exponential order, then the laplace transform lff (t)g = f (s) is de(cid:12)ned for s > c. Note: every polynomial is of exponential order, as one can check that lim t!1 tn et = 0; for any positive integer n. on the other hand, function f (t) = et2 is not of expo- nential order. 1 (ii) the laplace transform is linear so that the inverse laplace transform is also linear:
