1550251515-Classical_Complex_Analysis__Gonzalez_

720 Chapter9

made along the line Res = c where the arbitrary constant c is chosen so

that all singularities of f ( s) lie to the left of the line Res = c, and the

integral is understood in the sense of a principal value.

To evaluate the Bromwich integral, we consider the complex integral

2 ~i j e

(^8) tf(s)ds
c+
where c+ is the contour shown in Fig. 9.24, i.e., c+ = L + r+, where
L: s = c + ir, -R' ::; T ::; R', R' = JR^2 - c^2 and r+: s = Rei^8 , 81 ::;

8 ::; 27r - 81 , 81 = Arc cos( cf R). However, if f( s) has branch points, this

contour must be conveniently modified so as to have f ( s) single-valued on

D = Int c+ and continuous on D

If f ( s) has a finite number of isolated singularities bk ( k = 1, ... , m ),

by choosing R > max lbkl we have

1 J 1 1c-l-iR

1

-^1 J

2

: e^8 tf(s) ds = -

2

. e^8 tf(s) ds + -
2
. e^81 f(s) ds
7l"Z 7l"Z c-iR' ?rt
c+ r+

(9.11-58)

and letting R -too (in which case R' -too also), we find that

l lc-1-ioo m

P(t) = 2"7 e^8 tf(s)ds = L Res e^81 f(s)

7rZ c-ioo k=l s=bk

(9.11-59)

'T

~ c+iR'

f+

b; R

01

0 c CT

•b2

L

b" 3

c-iR'

Fig. 9.24