718 Chapter9

1

r sinha(x + i7r) d J sinhaz d 1-R sinha(x + i7r) d

+ R sm. h( x + i7l". ) x + -.-h-sm z z + -r sm. h( x + i7l". ) x

-y-

1

° sinh a(-R + iy). d _

+. h( R. ) z y -^0

11' sm - + zy

(9.11-53)

But

J

R sinh ax dx = 2 {R sinh ax dx

-R sinhx } 0 sinhx

and for R > 1,

0 <, -1 sinha(±R+iy) I < sinhaR+sinay < ----sinhaR+l --7^0

- sinh(±R + iy) - sinhR - sin y - sinhR- 1

as R --+ oo, so that

ll

rr sinha(R+iy) 'd I sinhaR+l

.. zy< 71"--+0

0 smh(R + zy) - sinhR -1

as R --+ oo. Similarly,

Also,

so that

I

{o si~ha(-R+_iy! idyl--+ o'

lrr smh(-R + zy)

as R--+ oo

1

. (. ) sinh az sinh ia7l"

Im z - Z7l" ---= - -zsina7l"

z-+irr sinh z cosh i7l" -

(^1) Im. J -. sinh -h-az d z = z '( -7!" - 0 ) ( -isina7!" ) = -7l"sina7!"
r-+O SIU Z
-y-
by Lemma 9.4.
y

-RI

i'lT R+i'lT

\:: -r

-R - 0 R x

Fig. 9.23

(9.11-54)

(9.11-55)

(9.11-56)