1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/Applications

On the other hand, from

we get

sinh a( x + i7l") = sinh ax cos a7!" + i cosh ax sin a7!"

sinh(x + i7r) = - sinhx

719

l

r sinha(x. +. i7r) d x = cos a7!" iR -. sinhax --d x + ism.. a7!" iR coshax d x

R smh(x + i7r) r smhx r sinhx

1

-R sinha(x + i7r) d 1R sinhax d .. 1R coshax d

. h(. ) x - cos a7!". h x i sin a7!" .nh x
-r Sill X + Z7!" r Sill X r SI X
and

l

r sinha(x + i7r) d 1-R sinha(x + i7r) d _ JR sinhax d

R sm. h( x + i7!". ) x + -r sm. h( x + i7!". ) x - 2cosa7!" r s1 .nh x x

Hence, (9.11-53) can be written as

1

R sinh ax iR sinh ax

2. h dx + 2 cos a7!" -. -h-dx

o Sill X r Sill X

1

,,. sinha(R+iy) ·a j sinhaz d

+. h(R. ) i y + -.-h- z

0 sm + iy sm z

'Y-

1

0 sinh a( -R + i y). d -0

+ ,,. sm .h(R - + iy .)iy- (9.11-57)

Now, letting r -t 0, then R --+ oo in (9.11-57), and taking (9.11-54),

(9.11-55), and (9.11-56) into account, we obtain

1

(^00) sinhax.
2(1 + cosa7r) -. -h-dx - 7!"Sllla7!" = 0
0 sm x
or
--- x = -7!" == -7!"tan -a7!"
1
(^00) sinhax d 1 sina7r 1 1
0 sinhx^2 l+cosa7!"^2 2
the convergence of the integral at the upper limit being a consequence
of (9.11-57).

2. If f(s) = .C {F(t)}, then the inverse Laplace transform F(t) =

.c-1 {f(s)} is given by

F(t) = 2 ~i 1::: e

8

tf(s)ds

fort > O, and F(t) = 0 fort < 0. This is the so-called complex inversion
integral or Bromwich's integral formula. The integration is supposed to be