1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

544 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM

formula F (s) might have a domain much larger than this half-plane. Later we
show that F (s) is an analytic function of the complex variables. For most ap-
plications involving Laplace transforms that we present, the Laplace transforms
are rational functions that take the form ~~:~,where P and Qare polynomials;

in other important applications, the functions take the form e~~s~s) •

• EXAMPLE 12. 7 Show that the Laplace transform of the step function given
  by

is

f (t) = { l,

0,

for 0 ::S t < c, and

for c < t,

.l(f (t)) = 1-e- cs.

s