1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

560 CHAPTER 12 • FOURJER SERIES AND THE LAPLACE TRANSFORM

82 - b2

•EXAMPLE12. 21 Showthat.C(tcosbt)= 2.

(s2 + b2)

S olution If we let f (t) = cosbt, then F (s) = .C (cosbt) = 2

8

L'>. Hence we

s +v-

can differentiate F ( s) to obtain the desired result:

.C (tcosbt) = -F' (s) =

s^2 + b^2 - 2s^2 s^2 - b^2

=

(s2 + />2)2 (s2 + b2) ·

• EXAMPLE 12.22 Show that .C ( si: t) = Arctan~.

Solution We let f (t) =sin t and F (s) = +-· Because lim sin t = 1, we
8 + l t-+O+ t
can integrate F ( s) to obtain the desired result:

C ( ~. t) = J. ~ = -Arctan- = Arctan-.

00

d 11"=

00

1

t • Cl + 1 Cl <1=• s

Some types of differential equations involve the terms ty' (t) or ty" (t). We can

use Laplace transforms to find the solution if we use the additional substitutions

C (ty' (t)) = -sY' (s) - Y (s), and (12-32)

.C (ty" (t)) = - s^2 Y' (s) - 2sY (s) + y (0). (12-33)