Page 70 - Calculus with Complex Numbers
P. 70
Ffgure 8.5
On p We have z = --t (R k: t k: r). Therefore
d i . R - i t
d
z
dz = - dt .
/3 z r t
Combining these two integrals we get
i k;
e
i k;
e t? R i t - e R jyj j
--i t
S
- - dz + -- dz = dt = li dt.
1
p' Z py Z r t r t
On yz We have (integrating by parts)
d iz d iz d iz
dz = + dz .
p' z iz izl
z
z pv p'
The lirst term on the right-hand side
- i R t?
i k;
t? e i R g COS p
-s- = - = - --> O
l .! / - i R i R i R
:
2
as R --> co. W hilst the second term
eiz a.w a.
dz :% = - --> 0
iz1 42 R
as R --> co. Therefore
d iz
- dz --> 0
Z
as R --> co.
On y4 We have z = reit (zr k: t k: 0). Therefore,
eiz Jc 1 co (jcln
dz = - + - )-) Jc.
v 2 :. 2 :. 2 n !
4
4
4
1
