A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by

where we assume that s is a positive real number. For example, to find the Laplace transform of f(t) = e^-t, the following improper integral is evaluated:

Verify the following Laplace transforms, where a is a real number.

f(t) = 1→ F(s) = 1/s

00 e s" f(t) dt, F(s) = 1 00 F(s) = e s! e dt dt