18.03 Lecture Notes - Improper Integral, Analytic Continuation
Document Summary
Laplace transform: definition and basic properties: definition of lt; l[1, region of convergence, powers, linearity, s-shift rule, sines and cosines, t-domain and s-domain. We continue to consider functions f(t) which are zero for t < 0 . (i may forget to multiply by u(t) now and then. ) The laplace transform takes a function f(t) (of "time") and uses it to manufacture another function f(s) (where s can be complex). It: [slide] (1) makes explicit long term behavior of f(t) . (2) answers the question: if i know w(t), how can i compute p(s) ? (3) converts differential equations into algebraic equations. But we won"t see these virtues right away. Definition: the laplace transform of f(t) is the improper integral. F(s) = integral_{0}^ infty e^{-st} f(t) dt (formula subject to two refinements). We will often write f(t) ------> f(s) and l[f(t)] = f(s) (this notation isn"t so good, because there"s no room for "s" on the left. )
